TPTP Problem File: ITP216_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP216_2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Time_Reasoning 00074_002979
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0029_Time_Reasoning_00074_002979 [Des22]
% Status : Theorem
% Rating : 0.50 v8.1.0
% Syntax : Number of formulae : 8903 (2602 unt;1558 typ; 0 def)
% Number of atoms : 18435 (6857 equ)
% Maximal formula atoms : 38 ( 2 avg)
% Number of connectives : 12938 (1848 ~; 191 |;1199 &)
% (1222 <=>;8478 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 6 avg)
% Maximal term depth : 40 ( 2 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 1509 (1073 >; 436 *; 0 +; 0 <<)
% Number of predicates : 259 ( 256 usr; 2 prp; 0-7 aty)
% Number of functors : 1289 (1289 usr; 78 con; 0-8 aty)
% Number of variables : 27626 (24979 !; 382 ?;27626 :)
% (2265 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 16:14:01.156
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
tff(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
tff(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
tff(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
tff(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
tff(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
tff(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
tff(ty_t_Product__Type_Oprod,type,
product_prod: ( $tType * $tType ) > $tType ).
tff(ty_t_Old__Datatype_Onode,type,
old_node: ( $tType * $tType ) > $tType ).
tff(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
tff(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
tff(ty_t_Assertions_Oassn,type,
assn: $tType ).
tff(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
tff(ty_t_Sum__Type_Osum,type,
sum_sum: ( $tType * $tType ) > $tType ).
tff(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
tff(ty_t_Option_Ooption,type,
option: $tType > $tType ).
tff(ty_t_String_Ochar,type,
char: $tType ).
tff(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
tff(ty_t_List_Olist,type,
list: $tType > $tType ).
tff(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
tff(ty_t_HOL_Obool,type,
bool: $tType ).
tff(ty_t_Set_Oset,type,
set: $tType > $tType ).
tff(ty_t_Rat_Orat,type,
rat: $tType ).
tff(ty_t_Num_Onum,type,
num: $tType ).
tff(ty_t_Nat_Onat,type,
nat: $tType ).
tff(ty_t_Int_Oint,type,
int: $tType ).
tff(ty_t_itself,type,
itself: $tType > $tType ).
tff(ty_t_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tf_a,type,
a: $tType ).
% Explicit typings (1530)
tff(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
tff(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[A: $tType] : $o ).
tff(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
tff(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : $o ).
tff(sy_cl_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
tff(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
tff(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
quickc658316121487927005ustive:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
quickc3360725361186068524ustive:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
tff(sy_c_ATP_058Lamp__a____,type,
aTP_Lamp_a:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aa____,type,
aTP_Lamp_aa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ab____,type,
aTP_Lamp_ab:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ac____,type,
aTP_Lamp_ac:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ad____,type,
aTP_Lamp_ad:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ae____,type,
aTP_Lamp_ae:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__af____,type,
aTP_Lamp_af:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ag____,type,
aTP_Lamp_ag:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ah____,type,
aTP_Lamp_ah:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ai____,type,
aTP_Lamp_ai:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aj____,type,
aTP_Lamp_aj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ak____,type,
aTP_Lamp_ak:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__al____,type,
aTP_Lamp_al:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__am____,type,
aTP_Lamp_am:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__an____,type,
aTP_Lamp_an:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ao____,type,
aTP_Lamp_ao:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ap____,type,
aTP_Lamp_ap:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aq____,type,
aTP_Lamp_aq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ar____,type,
aTP_Lamp_ar:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__as____,type,
aTP_Lamp_as:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__at____,type,
aTP_Lamp_at:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__au____,type,
aTP_Lamp_au:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__av____,type,
aTP_Lamp_av:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aw____,type,
aTP_Lamp_aw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ax____,type,
aTP_Lamp_ax:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ay____,type,
aTP_Lamp_ay:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__az____,type,
aTP_Lamp_az:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ba____,type,
aTP_Lamp_ba:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bb____,type,
aTP_Lamp_bb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bc____,type,
aTP_Lamp_bc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bd____,type,
aTP_Lamp_bd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__be____,type,
aTP_Lamp_be:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bf____,type,
aTP_Lamp_bf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bg____,type,
aTP_Lamp_bg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bh____,type,
aTP_Lamp_bh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bi____,type,
aTP_Lamp_bi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bj____,type,
aTP_Lamp_bj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bk____,type,
aTP_Lamp_bk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bl____,type,
aTP_Lamp_bl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bm____,type,
aTP_Lamp_bm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bn____,type,
aTP_Lamp_bn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bo____,type,
aTP_Lamp_bo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bp____,type,
aTP_Lamp_bp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bq____,type,
aTP_Lamp_bq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__br____,type,
aTP_Lamp_br:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bs____,type,
aTP_Lamp_bs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bt____,type,
aTP_Lamp_bt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bu____,type,
aTP_Lamp_bu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bv____,type,
aTP_Lamp_bv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bw____,type,
aTP_Lamp_bw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bx____,type,
aTP_Lamp_bx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__by____,type,
aTP_Lamp_by:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bz____,type,
aTP_Lamp_bz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ca____,type,
aTP_Lamp_ca:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cb____,type,
aTP_Lamp_cb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cc____,type,
aTP_Lamp_cc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cd____,type,
aTP_Lamp_cd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ce____,type,
aTP_Lamp_ce:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cf____,type,
aTP_Lamp_cf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cg____,type,
aTP_Lamp_cg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ch____,type,
aTP_Lamp_ch:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ci____,type,
aTP_Lamp_ci:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cj____,type,
aTP_Lamp_cj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ck____,type,
aTP_Lamp_ck:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cl____,type,
aTP_Lamp_cl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cm____,type,
aTP_Lamp_cm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cn____,type,
aTP_Lamp_cn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__co____,type,
aTP_Lamp_co:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cp____,type,
aTP_Lamp_cp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cs____,type,
aTP_Lamp_cs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ct____,type,
aTP_Lamp_ct:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cu____,type,
aTP_Lamp_cu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cv____,type,
aTP_Lamp_cv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cw____,type,
aTP_Lamp_cw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cx____,type,
aTP_Lamp_cx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cy____,type,
aTP_Lamp_cy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cz____,type,
aTP_Lamp_cz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__da____,type,
aTP_Lamp_da:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__db____,type,
aTP_Lamp_db:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dc____,type,
aTP_Lamp_dc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dd____,type,
aTP_Lamp_dd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__de____,type,
aTP_Lamp_de:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__df____,type,
aTP_Lamp_df:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dg____,type,
aTP_Lamp_dg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dj____,type,
aTP_Lamp_dj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dk____,type,
aTP_Lamp_dk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dl____,type,
aTP_Lamp_dl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dm____,type,
aTP_Lamp_dm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dn____,type,
aTP_Lamp_dn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__do____,type,
aTP_Lamp_do:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dp____,type,
aTP_Lamp_dp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dr____,type,
aTP_Lamp_dr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ds____,type,
aTP_Lamp_ds:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dt____,type,
aTP_Lamp_dt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__du____,type,
aTP_Lamp_du:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dv____,type,
aTP_Lamp_dv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dw____,type,
aTP_Lamp_dw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dx____,type,
aTP_Lamp_dx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dy____,type,
aTP_Lamp_dy:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dz____,type,
aTP_Lamp_dz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ea____,type,
aTP_Lamp_ea:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eb____,type,
aTP_Lamp_eb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ec____,type,
aTP_Lamp_ec:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ed____,type,
aTP_Lamp_ed:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ee____,type,
aTP_Lamp_ee:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ef____,type,
aTP_Lamp_ef:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eg____,type,
aTP_Lamp_eg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eh____,type,
aTP_Lamp_eh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ei____,type,
aTP_Lamp_ei:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ej____,type,
aTP_Lamp_ej:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ek____,type,
aTP_Lamp_ek:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__el____,type,
aTP_Lamp_el:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__em____,type,
aTP_Lamp_em:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__en____,type,
aTP_Lamp_en:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eo____,type,
aTP_Lamp_eo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ep____,type,
aTP_Lamp_ep:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eq____,type,
aTP_Lamp_eq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__er____,type,
aTP_Lamp_er:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__es____,type,
aTP_Lamp_es:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__et____,type,
aTP_Lamp_et:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eu____,type,
aTP_Lamp_eu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ev____,type,
aTP_Lamp_ev:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ew____,type,
aTP_Lamp_ew:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ex____,type,
aTP_Lamp_ex:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ey____,type,
aTP_Lamp_ey:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ez____,type,
aTP_Lamp_ez:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fa____,type,
aTP_Lamp_fa:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fb____,type,
aTP_Lamp_fb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fc____,type,
aTP_Lamp_fc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fd____,type,
aTP_Lamp_fd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fe____,type,
aTP_Lamp_fe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ff____,type,
aTP_Lamp_ff:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fg____,type,
aTP_Lamp_fg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fh____,type,
aTP_Lamp_fh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fi____,type,
aTP_Lamp_fi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fj____,type,
aTP_Lamp_fj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fk____,type,
aTP_Lamp_fk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fl____,type,
aTP_Lamp_fl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__fm____,type,
aTP_Lamp_fm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fn____,type,
aTP_Lamp_fn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fo____,type,
aTP_Lamp_fo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fp____,type,
aTP_Lamp_fp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fq____,type,
aTP_Lamp_fq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fr____,type,
aTP_Lamp_fr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fs____,type,
aTP_Lamp_fs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ft____,type,
aTP_Lamp_ft:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fu____,type,
aTP_Lamp_fu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fv____,type,
aTP_Lamp_fv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fw____,type,
aTP_Lamp_fw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fx____,type,
aTP_Lamp_fx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fy____,type,
aTP_Lamp_fy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fz____,type,
aTP_Lamp_fz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ga____,type,
aTP_Lamp_ga:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gb____,type,
aTP_Lamp_gb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gc____,type,
aTP_Lamp_gc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gd____,type,
aTP_Lamp_gd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ge____,type,
aTP_Lamp_ge:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gf____,type,
aTP_Lamp_gf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gg____,type,
aTP_Lamp_gg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gh____,type,
aTP_Lamp_gh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gi____,type,
aTP_Lamp_gi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gj____,type,
aTP_Lamp_gj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gk____,type,
aTP_Lamp_gk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gl____,type,
aTP_Lamp_gl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gm____,type,
aTP_Lamp_gm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gn____,type,
aTP_Lamp_gn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__go____,type,
aTP_Lamp_go:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gp____,type,
aTP_Lamp_gp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gq____,type,
aTP_Lamp_gq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gr____,type,
aTP_Lamp_gr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gs____,type,
aTP_Lamp_gs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gt____,type,
aTP_Lamp_gt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gu____,type,
aTP_Lamp_gu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gv____,type,
aTP_Lamp_gv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gw____,type,
aTP_Lamp_gw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__gx____,type,
aTP_Lamp_gx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gy____,type,
aTP_Lamp_gy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gz____,type,
aTP_Lamp_gz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ha____,type,
aTP_Lamp_ha:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hb____,type,
aTP_Lamp_hb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hc____,type,
aTP_Lamp_hc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hd____,type,
aTP_Lamp_hd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hg____,type,
aTP_Lamp_hg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hh____,type,
aTP_Lamp_hh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hi____,type,
aTP_Lamp_hi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hk____,type,
aTP_Lamp_hk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hl____,type,
aTP_Lamp_hl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hm____,type,
aTP_Lamp_hm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hn____,type,
aTP_Lamp_hn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ho____,type,
aTP_Lamp_ho:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hp____,type,
aTP_Lamp_hp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hq____,type,
aTP_Lamp_hq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hr____,type,
aTP_Lamp_hr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hs____,type,
aTP_Lamp_hs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ht____,type,
aTP_Lamp_ht:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hu____,type,
aTP_Lamp_hu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hv____,type,
aTP_Lamp_hv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hw____,type,
aTP_Lamp_hw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hx____,type,
aTP_Lamp_hx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hy____,type,
aTP_Lamp_hy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hz____,type,
aTP_Lamp_hz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ia____,type,
aTP_Lamp_ia:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ib____,type,
aTP_Lamp_ib:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ic____,type,
aTP_Lamp_ic:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__id____,type,
aTP_Lamp_id:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ie____,type,
aTP_Lamp_ie:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__if____,type,
aTP_Lamp_if:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ig____,type,
aTP_Lamp_ig:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ih____,type,
aTP_Lamp_ih:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ii____,type,
aTP_Lamp_ii:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ij____,type,
aTP_Lamp_ij:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ik____,type,
aTP_Lamp_ik:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__il____,type,
aTP_Lamp_il:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__im____,type,
aTP_Lamp_im:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__in____,type,
aTP_Lamp_in:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__io____,type,
aTP_Lamp_io:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ip____,type,
aTP_Lamp_ip:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iq____,type,
aTP_Lamp_iq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ir____,type,
aTP_Lamp_ir:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__is____,type,
aTP_Lamp_is:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__it____,type,
aTP_Lamp_it:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__iu____,type,
aTP_Lamp_iu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iv____,type,
aTP_Lamp_iv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iw____,type,
aTP_Lamp_iw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ix____,type,
aTP_Lamp_ix:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iy____,type,
aTP_Lamp_iy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iz____,type,
aTP_Lamp_iz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ja____,type,
aTP_Lamp_ja:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jb____,type,
aTP_Lamp_jb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jc____,type,
aTP_Lamp_jc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jd____,type,
aTP_Lamp_jd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__je____,type,
aTP_Lamp_je:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jf____,type,
aTP_Lamp_jf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jg____,type,
aTP_Lamp_jg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jh____,type,
aTP_Lamp_jh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ji____,type,
aTP_Lamp_ji:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jj____,type,
aTP_Lamp_jj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jk____,type,
aTP_Lamp_jk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jl____,type,
aTP_Lamp_jl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jm____,type,
aTP_Lamp_jm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jn____,type,
aTP_Lamp_jn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jo____,type,
aTP_Lamp_jo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jp____,type,
aTP_Lamp_jp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jq____,type,
aTP_Lamp_jq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jr____,type,
aTP_Lamp_jr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__js____,type,
aTP_Lamp_js:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jt____,type,
aTP_Lamp_jt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ju____,type,
aTP_Lamp_ju:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jv____,type,
aTP_Lamp_jv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jw____,type,
aTP_Lamp_jw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jx____,type,
aTP_Lamp_jx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jy____,type,
aTP_Lamp_jy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jz____,type,
aTP_Lamp_jz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ka____,type,
aTP_Lamp_ka:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kb____,type,
aTP_Lamp_kb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kc____,type,
aTP_Lamp_kc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kd____,type,
aTP_Lamp_kd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ke____,type,
aTP_Lamp_ke:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kf____,type,
aTP_Lamp_kf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kg____,type,
aTP_Lamp_kg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kh____,type,
aTP_Lamp_kh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ki____,type,
aTP_Lamp_ki:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kj____,type,
aTP_Lamp_kj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kk____,type,
aTP_Lamp_kk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kl____,type,
aTP_Lamp_kl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__km____,type,
aTP_Lamp_km:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__kn____,type,
aTP_Lamp_kn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ko____,type,
aTP_Lamp_ko:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kp____,type,
aTP_Lamp_kp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kq____,type,
aTP_Lamp_kq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kr____,type,
aTP_Lamp_kr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ks____,type,
aTP_Lamp_ks:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kt____,type,
aTP_Lamp_kt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ku____,type,
aTP_Lamp_ku:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kv____,type,
aTP_Lamp_kv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kw____,type,
aTP_Lamp_kw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kx____,type,
aTP_Lamp_kx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ky____,type,
aTP_Lamp_ky:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kz____,type,
aTP_Lamp_kz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__la____,type,
aTP_Lamp_la:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lb____,type,
aTP_Lamp_lb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lc____,type,
aTP_Lamp_lc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ld____,type,
aTP_Lamp_ld:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__le____,type,
aTP_Lamp_le:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lf____,type,
aTP_Lamp_lf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lg____,type,
aTP_Lamp_lg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lh____,type,
aTP_Lamp_lh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__li____,type,
aTP_Lamp_li:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lj____,type,
aTP_Lamp_lj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lk____,type,
aTP_Lamp_lk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ll____,type,
aTP_Lamp_ll:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lm____,type,
aTP_Lamp_lm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ln____,type,
aTP_Lamp_ln:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lo____,type,
aTP_Lamp_lo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lp____,type,
aTP_Lamp_lp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lq____,type,
aTP_Lamp_lq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lr____,type,
aTP_Lamp_lr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ls____,type,
aTP_Lamp_ls:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lt____,type,
aTP_Lamp_lt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lu____,type,
aTP_Lamp_lu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lv____,type,
aTP_Lamp_lv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lw____,type,
aTP_Lamp_lw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lx____,type,
aTP_Lamp_lx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ly____,type,
aTP_Lamp_ly:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lz____,type,
aTP_Lamp_lz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ma____,type,
aTP_Lamp_ma:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mb____,type,
aTP_Lamp_mb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mc____,type,
aTP_Lamp_mc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__md____,type,
aTP_Lamp_md:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__me____,type,
aTP_Lamp_me:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mf____,type,
aTP_Lamp_mf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mg____,type,
aTP_Lamp_mg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mh____,type,
aTP_Lamp_mh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mi____,type,
aTP_Lamp_mi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mj____,type,
aTP_Lamp_mj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mk____,type,
aTP_Lamp_mk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ml____,type,
aTP_Lamp_ml:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mm____,type,
aTP_Lamp_mm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mn____,type,
aTP_Lamp_mn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mo____,type,
aTP_Lamp_mo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mp____,type,
aTP_Lamp_mp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mq____,type,
aTP_Lamp_mq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mr____,type,
aTP_Lamp_mr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ms____,type,
aTP_Lamp_ms:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mt____,type,
aTP_Lamp_mt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mu____,type,
aTP_Lamp_mu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mv____,type,
aTP_Lamp_mv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mw____,type,
aTP_Lamp_mw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mx____,type,
aTP_Lamp_mx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__my____,type,
aTP_Lamp_my:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mz____,type,
aTP_Lamp_mz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__na____,type,
aTP_Lamp_na:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nb____,type,
aTP_Lamp_nb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nc____,type,
aTP_Lamp_nc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nd____,type,
aTP_Lamp_nd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ne____,type,
aTP_Lamp_ne:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nf____,type,
aTP_Lamp_nf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ng____,type,
aTP_Lamp_ng:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nh____,type,
aTP_Lamp_nh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ni____,type,
aTP_Lamp_ni:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nj____,type,
aTP_Lamp_nj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nk____,type,
aTP_Lamp_nk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nl____,type,
aTP_Lamp_nl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nm____,type,
aTP_Lamp_nm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nn____,type,
aTP_Lamp_nn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__no____,type,
aTP_Lamp_no:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__np____,type,
aTP_Lamp_np:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nq____,type,
aTP_Lamp_nq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nr____,type,
aTP_Lamp_nr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ns____,type,
aTP_Lamp_ns:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nt____,type,
aTP_Lamp_nt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nu____,type,
aTP_Lamp_nu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nv____,type,
aTP_Lamp_nv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nw____,type,
aTP_Lamp_nw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nx____,type,
aTP_Lamp_nx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ny____,type,
aTP_Lamp_ny:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nz____,type,
aTP_Lamp_nz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oa____,type,
aTP_Lamp_oa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ob____,type,
aTP_Lamp_ob:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oc____,type,
aTP_Lamp_oc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__od____,type,
aTP_Lamp_od:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oe____,type,
aTP_Lamp_oe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__of____,type,
aTP_Lamp_of:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__og____,type,
aTP_Lamp_og:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oh____,type,
aTP_Lamp_oh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oi____,type,
aTP_Lamp_oi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oj____,type,
aTP_Lamp_oj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ok____,type,
aTP_Lamp_ok:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ol____,type,
aTP_Lamp_ol:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__om____,type,
aTP_Lamp_om:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__on____,type,
aTP_Lamp_on:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oo____,type,
aTP_Lamp_oo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__op____,type,
aTP_Lamp_op:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oq____,type,
aTP_Lamp_oq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__or____,type,
aTP_Lamp_or:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__os____,type,
aTP_Lamp_os:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ot____,type,
aTP_Lamp_ot:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ou____,type,
aTP_Lamp_ou:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ov____,type,
aTP_Lamp_ov:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ow____,type,
aTP_Lamp_ow:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ox____,type,
aTP_Lamp_ox:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oy____,type,
aTP_Lamp_oy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oz____,type,
aTP_Lamp_oz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pa____,type,
aTP_Lamp_pa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pb____,type,
aTP_Lamp_pb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pc____,type,
aTP_Lamp_pc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pd____,type,
aTP_Lamp_pd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pe____,type,
aTP_Lamp_pe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pf____,type,
aTP_Lamp_pf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pg____,type,
aTP_Lamp_pg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ph____,type,
aTP_Lamp_ph:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pi____,type,
aTP_Lamp_pi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pk____,type,
aTP_Lamp_pk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pl____,type,
aTP_Lamp_pl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pm____,type,
aTP_Lamp_pm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pn____,type,
aTP_Lamp_pn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__po____,type,
aTP_Lamp_po:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pp____,type,
aTP_Lamp_pp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pq____,type,
aTP_Lamp_pq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pr____,type,
aTP_Lamp_pr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ps____,type,
aTP_Lamp_ps:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pt____,type,
aTP_Lamp_pt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pu____,type,
aTP_Lamp_pu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pv____,type,
aTP_Lamp_pv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pw____,type,
aTP_Lamp_pw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__px____,type,
aTP_Lamp_px:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__py____,type,
aTP_Lamp_py:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pz____,type,
aTP_Lamp_pz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qa____,type,
aTP_Lamp_qa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qb____,type,
aTP_Lamp_qb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qc____,type,
aTP_Lamp_qc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qd____,type,
aTP_Lamp_qd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qe____,type,
aTP_Lamp_qe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qf____,type,
aTP_Lamp_qf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qg____,type,
aTP_Lamp_qg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qh____,type,
aTP_Lamp_qh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qi____,type,
aTP_Lamp_qi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qj____,type,
aTP_Lamp_qj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qk____,type,
aTP_Lamp_qk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ql____,type,
aTP_Lamp_ql:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qm____,type,
aTP_Lamp_qm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qn____,type,
aTP_Lamp_qn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qo____,type,
aTP_Lamp_qo:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qp____,type,
aTP_Lamp_qp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qq____,type,
aTP_Lamp_qq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qr____,type,
aTP_Lamp_qr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qs____,type,
aTP_Lamp_qs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qt____,type,
aTP_Lamp_qt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qu____,type,
aTP_Lamp_qu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qv____,type,
aTP_Lamp_qv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qw____,type,
aTP_Lamp_qw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qx____,type,
aTP_Lamp_qx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qy____,type,
aTP_Lamp_qy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qz____,type,
aTP_Lamp_qz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ra____,type,
aTP_Lamp_ra:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rb____,type,
aTP_Lamp_rb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rc____,type,
aTP_Lamp_rc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rd____,type,
aTP_Lamp_rd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__re____,type,
aTP_Lamp_re:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rf____,type,
aTP_Lamp_rf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rg____,type,
aTP_Lamp_rg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rh____,type,
aTP_Lamp_rh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ri____,type,
aTP_Lamp_ri:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rj____,type,
aTP_Lamp_rj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rk____,type,
aTP_Lamp_rk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rl____,type,
aTP_Lamp_rl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rm____,type,
aTP_Lamp_rm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rn____,type,
aTP_Lamp_rn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ro____,type,
aTP_Lamp_ro:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rp____,type,
aTP_Lamp_rp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rq____,type,
aTP_Lamp_rq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rr____,type,
aTP_Lamp_rr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rs____,type,
aTP_Lamp_rs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rt____,type,
aTP_Lamp_rt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ru____,type,
aTP_Lamp_ru:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rv____,type,
aTP_Lamp_rv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rw____,type,
aTP_Lamp_rw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rx____,type,
aTP_Lamp_rx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ry____,type,
aTP_Lamp_ry:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rz____,type,
aTP_Lamp_rz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sa____,type,
aTP_Lamp_sa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sb____,type,
aTP_Lamp_sb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sc____,type,
aTP_Lamp_sc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sd____,type,
aTP_Lamp_sd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__se____,type,
aTP_Lamp_se:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sf____,type,
aTP_Lamp_sf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sg____,type,
aTP_Lamp_sg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sh____,type,
aTP_Lamp_sh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__si____,type,
aTP_Lamp_si:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sj____,type,
aTP_Lamp_sj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sk____,type,
aTP_Lamp_sk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sl____,type,
aTP_Lamp_sl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sm____,type,
aTP_Lamp_sm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sn____,type,
aTP_Lamp_sn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__so____,type,
aTP_Lamp_so:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sp____,type,
aTP_Lamp_sp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sq____,type,
aTP_Lamp_sq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sr____,type,
aTP_Lamp_sr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ss____,type,
aTP_Lamp_ss:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__st____,type,
aTP_Lamp_st:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__su____,type,
aTP_Lamp_su:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sv____,type,
aTP_Lamp_sv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sw____,type,
aTP_Lamp_sw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sx____,type,
aTP_Lamp_sx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sy____,type,
aTP_Lamp_sy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sz____,type,
aTP_Lamp_sz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ta____,type,
aTP_Lamp_ta:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tb____,type,
aTP_Lamp_tb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tc____,type,
aTP_Lamp_tc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__td____,type,
aTP_Lamp_td:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__te____,type,
aTP_Lamp_te:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tf____,type,
aTP_Lamp_tf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tg____,type,
aTP_Lamp_tg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__th____,type,
aTP_Lamp_th:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ti____,type,
aTP_Lamp_ti:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tj____,type,
aTP_Lamp_tj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tk____,type,
aTP_Lamp_tk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tl____,type,
aTP_Lamp_tl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tm____,type,
aTP_Lamp_tm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tn____,type,
aTP_Lamp_tn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__to____,type,
aTP_Lamp_to:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tp____,type,
aTP_Lamp_tp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tq____,type,
aTP_Lamp_tq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tr____,type,
aTP_Lamp_tr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ts____,type,
aTP_Lamp_ts:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tt____,type,
aTP_Lamp_tt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tu____,type,
aTP_Lamp_tu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tv____,type,
aTP_Lamp_tv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tw____,type,
aTP_Lamp_tw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tx____,type,
aTP_Lamp_tx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ty____,type,
aTP_Lamp_ty:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tz____,type,
aTP_Lamp_tz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ua____,type,
aTP_Lamp_ua:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ub____,type,
aTP_Lamp_ub:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__uc____,type,
aTP_Lamp_uc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ud____,type,
aTP_Lamp_ud:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ue____,type,
aTP_Lamp_ue:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uf____,type,
aTP_Lamp_uf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ug____,type,
aTP_Lamp_ug:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uh____,type,
aTP_Lamp_uh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ui____,type,
aTP_Lamp_ui:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uj____,type,
aTP_Lamp_uj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uk____,type,
aTP_Lamp_uk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ul____,type,
aTP_Lamp_ul:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__um____,type,
aTP_Lamp_um:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__un____,type,
aTP_Lamp_un:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uo____,type,
aTP_Lamp_uo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__up____,type,
aTP_Lamp_up:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uq____,type,
aTP_Lamp_uq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ur____,type,
aTP_Lamp_ur:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__us____,type,
aTP_Lamp_us:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ut____,type,
aTP_Lamp_ut:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__uu____,type,
aTP_Lamp_uu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uv____,type,
aTP_Lamp_uv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uw____,type,
aTP_Lamp_uw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ux____,type,
aTP_Lamp_ux:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uy____,type,
aTP_Lamp_uy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uz____,type,
aTP_Lamp_uz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__va____,type,
aTP_Lamp_va:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vb____,type,
aTP_Lamp_vb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vc____,type,
aTP_Lamp_vc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vd____,type,
aTP_Lamp_vd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ve____,type,
aTP_Lamp_ve:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vf____,type,
aTP_Lamp_vf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vg____,type,
aTP_Lamp_vg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vh____,type,
aTP_Lamp_vh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vi____,type,
aTP_Lamp_vi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vj____,type,
aTP_Lamp_vj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vk____,type,
aTP_Lamp_vk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vl____,type,
aTP_Lamp_vl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vm____,type,
aTP_Lamp_vm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vn____,type,
aTP_Lamp_vn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vo____,type,
aTP_Lamp_vo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vp____,type,
aTP_Lamp_vp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vq____,type,
aTP_Lamp_vq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vr____,type,
aTP_Lamp_vr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vs____,type,
aTP_Lamp_vs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vt____,type,
aTP_Lamp_vt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vu____,type,
aTP_Lamp_vu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vv____,type,
aTP_Lamp_vv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vw____,type,
aTP_Lamp_vw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vx____,type,
aTP_Lamp_vx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vy____,type,
aTP_Lamp_vy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vz____,type,
aTP_Lamp_vz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wa____,type,
aTP_Lamp_wa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wb____,type,
aTP_Lamp_wb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wc____,type,
aTP_Lamp_wc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wd____,type,
aTP_Lamp_wd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__we____,type,
aTP_Lamp_we:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wf____,type,
aTP_Lamp_wf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wg____,type,
aTP_Lamp_wg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wh____,type,
aTP_Lamp_wh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wi____,type,
aTP_Lamp_wi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wj____,type,
aTP_Lamp_wj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wk____,type,
aTP_Lamp_wk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wl____,type,
aTP_Lamp_wl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wm____,type,
aTP_Lamp_wm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wn____,type,
aTP_Lamp_wn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wo____,type,
aTP_Lamp_wo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wp____,type,
aTP_Lamp_wp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wq____,type,
aTP_Lamp_wq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wr____,type,
aTP_Lamp_wr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ws____,type,
aTP_Lamp_ws:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wt____,type,
aTP_Lamp_wt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wu____,type,
aTP_Lamp_wu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wv____,type,
aTP_Lamp_wv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ww____,type,
aTP_Lamp_ww:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wx____,type,
aTP_Lamp_wx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wy____,type,
aTP_Lamp_wy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wz____,type,
aTP_Lamp_wz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xa____,type,
aTP_Lamp_xa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xb____,type,
aTP_Lamp_xb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xc____,type,
aTP_Lamp_xc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xd____,type,
aTP_Lamp_xd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xe____,type,
aTP_Lamp_xe:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xf____,type,
aTP_Lamp_xf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xg____,type,
aTP_Lamp_xg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xh____,type,
aTP_Lamp_xh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xi____,type,
aTP_Lamp_xi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xj____,type,
aTP_Lamp_xj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xk____,type,
aTP_Lamp_xk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xl____,type,
aTP_Lamp_xl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xm____,type,
aTP_Lamp_xm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xn____,type,
aTP_Lamp_xn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xo____,type,
aTP_Lamp_xo:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xp____,type,
aTP_Lamp_xp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xq____,type,
aTP_Lamp_xq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xr____,type,
aTP_Lamp_xr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xs____,type,
aTP_Lamp_xs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xt____,type,
aTP_Lamp_xt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xu____,type,
aTP_Lamp_xu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xv____,type,
aTP_Lamp_xv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xw____,type,
aTP_Lamp_xw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xx____,type,
aTP_Lamp_xx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xy____,type,
aTP_Lamp_xy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xz____,type,
aTP_Lamp_xz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ya____,type,
aTP_Lamp_ya:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__yb____,type,
aTP_Lamp_yb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__yc____,type,
aTP_Lamp_yc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__yd____,type,
aTP_Lamp_yd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ye____,type,
aTP_Lamp_ye:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__yf____,type,
aTP_Lamp_yf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__yg____,type,
aTP_Lamp_yg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__yh____,type,
aTP_Lamp_yh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__yi____,type,
aTP_Lamp_yi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list(A) * heap_ext(product_unit) ) > product_prod(array(A),heap_ext(product_unit)) ) ).
tff(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).
tff(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * array(A) ) > list(A) ) ).
tff(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).
tff(sy_c_Array__Time_Olen_H,type,
array_len2:
!>[B: $tType,A: $tType] : ( array(B) > heap_Time_Heap(A) ) ).
tff(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( heap_ext(product_unit) > fun(array(A),nat) ) ).
tff(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Omake_H,type,
array_make2:
!>[B: $tType,A: $tType] : ( ( code_integer * fun(B,A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : fun(nat,fun(A,heap_Time_Heap(array(A)))) ).
tff(sy_c_Array__Time_Onew_H,type,
array_new2:
!>[A: $tType] : fun(code_integer,fun(A,heap_Time_Heap(array(A)))) ).
tff(sy_c_Array__Time_Onoteq,type,
array_noteq:
!>[A: $tType,B: $tType] : ( ( array(A) * array(B) ) > $o ) ).
tff(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( array(A) > fun(nat,heap_Time_Heap(A)) ) ).
tff(sy_c_Array__Time_Onth_H,type,
array_nth2:
!>[A: $tType] : ( array(A) > fun(code_integer,heap_Time_Heap(A)) ) ).
tff(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Opresent,type,
array_present:
!>[A: $tType] : ( heap_ext(product_unit) > fun(array(A),bool) ) ).
tff(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array(A) * list(A) ) > fun(heap_ext(product_unit),heap_ext(product_unit)) ) ).
tff(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).
tff(sy_c_Array__Time_Oupd_H,type,
array_upd2:
!>[A: $tType] : ( ( array(A) * code_integer * A ) > heap_Time_Heap(product_unit) ) ).
tff(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: ( assn * product_prod(heap_ext(product_unit),set(nat)) ) > $o ).
tff(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( fun(A,assn) > assn ) ).
tff(sy_c_Assertions_Oin__range,type,
in_range: product_prod(heap_ext(product_unit),set(nat)) > $o ).
tff(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
tff(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: product_prod(heap_ext(product_unit),set(nat)) > $o ).
tff(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: fun(product_prod(heap_ext(product_unit),set(nat)),fun(product_prod(heap_ext(product_unit),set(nat)),bool)) ).
tff(sy_c_Assertions_Oproper,type,
proper: fun(product_prod(heap_ext(product_unit),set(nat)),bool) > $o ).
tff(sy_c_Assertions_Opure__assn,type,
pure_assn: bool > assn ).
tff(sy_c_Assertions_Opure__assn__raw,type,
pure_assn_raw:
!>[A: $tType,B: $tType] : ( ( bool * product_prod(A,set(B)) ) > $o ) ).
tff(sy_c_Assertions_Opure__assn__raw__rel,type,
pure_assn_raw_rel:
!>[A: $tType,B: $tType] : fun(product_prod(bool,product_prod(A,set(B))),fun(product_prod(bool,product_prod(A,set(B))),bool)) ).
tff(sy_c_Assertions_OrelH,type,
relH: ( set(nat) * heap_ext(product_unit) * heap_ext(product_unit) ) > $o ).
tff(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array(A) * list(A) ) > assn ) ).
tff(sy_c_Assertions_Osnga__assn__raw,type,
snga_assn_raw:
!>[A: $tType] : ( ( array(A) * list(A) * product_prod(heap_ext(product_unit),set(nat)) ) > $o ) ).
tff(sy_c_Assertions_Osnga__assn__raw__rel,type,
snga_assn_raw_rel:
!>[A: $tType] : fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref(A) * A ) > assn ) ).
tff(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref(A) * A * product_prod(heap_ext(product_unit),set(nat)) ) > $o ) ).
tff(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) * product_prod(heap_ext(product_unit),set(nat)) ) > $o ).
tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_Assertions_Owand__assn,type,
wand_assn: ( assn * assn ) > assn ).
tff(sy_c_Assertions_Owand__raw,type,
wand_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),bool) * fun(product_prod(heap_ext(product_unit),set(nat)),bool) * product_prod(heap_ext(product_unit),set(nat)) ) > $o ).
tff(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * set(product_prod(A,A)) ) > set(product_prod(fun(A,B),fun(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set(product_prod(product_unit,product_unit)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set(product_prod(bool,bool)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
bNF_Ca6246979054910435723ardSuc:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(set(A),set(A))) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,bool)) ) ).
tff(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( set(fun(B,set(A))) > fun(B,set(A)) ) ).
tff(sy_c_BNF__Def_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,B) * fun(A,C) ) > fun(A,product_prod(B,C)) ) ).
tff(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( fun(A,bool) > fun(A,fun(A,bool)) ) ).
tff(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) * product_prod(A,C) ) > product_prod(A,B) ) ).
tff(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,bool)) * fun(A,fun(C,bool)) * B * C ) > A ) ).
tff(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).
tff(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,bool)) * fun(A,fun(B,bool)) * product_prod(C,B) ) > product_prod(A,B) ) ).
tff(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( fun(A,D) * fun(B,E) ) > fun(fun(D,fun(E,C)),fun(A,fun(B,C))) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(A,B) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(A) * fun(product_prod(A,B),C) * A ) > fun(B,C) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * A ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).
tff(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * sum_sum(A,B) ) > nat ) ).
tff(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),set(A)) ).
tff(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) * fun(A,B) ) > $o ) ).
tff(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(product_prod(A,B),bool) ) ).
tff(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).
tff(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).
tff(sy_c_Binomial_Obinomial,type,
binomial: nat > fun(nat,nat) ).
tff(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > fun(nat,A) ) ).
tff(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).
tff(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: ( num * num ) > option(num) ).
tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).
tff(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: ( nat * int * int ) > int ).
tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: ( num * num ) > num ).
tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : fun(A,fun(nat,bool)) ).
tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).
tff(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: ( nat * num ) > option(num) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
bit_un6697907153464112080or_num: ( num * num ) > num ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
bit_un4773296044027857193um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: ( num * num ) > option(num) ).
tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).
tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).
tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).
tff(sy_c_Cancellation_Oiterate__add,type,
iterate_add:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Code__Numeral_ONat,type,
code_Nat: fun(code_integer,code_natural) ).
tff(sy_c_Code__Numeral_ONeg,type,
code_Neg: num > code_integer ).
tff(sy_c_Code__Numeral_OPos,type,
code_Pos: fun(num,code_integer) ).
tff(sy_c_Code__Numeral_OSuc,type,
code_Suc: fun(code_natural,code_natural) ).
tff(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).
tff(sy_c_Code__Numeral_Ocr__integer,type,
code_cr_integer: fun(int,fun(code_integer,bool)) ).
tff(sy_c_Code__Numeral_Ocr__natural,type,
code_cr_natural: fun(nat,fun(code_natural,bool)) ).
tff(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odup,type,
code_dup: fun(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: fun(code_integer,int) ).
tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: fun(int,code_integer) ).
tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: fun(nat,code_integer) ).
tff(sy_c_Code__Numeral_Ointeger__of__natural,type,
code_i5400310926305786745atural: fun(code_natural,code_integer) ).
tff(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: fun(num,code_integer) ).
tff(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: fun(code_integer,nat) ).
tff(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( ( T * fun(code_natural,T) ) > fun(code_natural,T) ) ).
tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: fun(code_natural,nat) ).
tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: fun(nat,code_natural) ).
tff(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) ) > fun(code_natural,T) ) ).
tff(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
code_rec_set_natural:
!>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) * code_natural ) > fun(T,bool) ) ).
tff(sy_c_Code__Numeral_Onatural__of__integer,type,
code_n4118661773612635043nteger: fun(code_integer,code_natural) ).
tff(sy_c_Code__Numeral_Onegative,type,
code_negative: fun(num,code_integer) ).
tff(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: fun(code_integer,num) ).
tff(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: fun(int,fun(code_integer,bool)) ).
tff(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: fun(nat,fun(code_natural,bool)) ).
tff(sy_c_Code__Numeral_Opositive,type,
code_positive: fun(num,code_integer) ).
tff(sy_c_Code__Numeral_Osize__nat,type,
code_size_nat: fun(code_natural,nat) ).
tff(sy_c_Code__Numeral_Osub,type,
code_sub: fun(num,fun(num,code_integer)) ).
tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,bool) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,A) ) > A ) ).
tff(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,fun(B,bool)) * fun(A,B) ) > $o ) ).
tff(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Countable_Onth__item,type,
nth_item:
!>[A: $tType] : ( nat > set(old_node(A,product_unit)) ) ).
tff(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: fun(nat,fun(nat,bool)) ).
tff(sy_c_Countable_Oto__nat,type,
to_nat:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Divides_Oadjust__div,type,
adjust_div: product_prod(int,int) > int ).
tff(sy_c_Divides_Odivmod__nat,type,
divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).
tff(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( product_prod(A,A) > $o ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).
tff(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > set(A) ) ).
tff(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).
tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( set(B) > nat ) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
finite1981172056906289455axioms:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).
tff(sy_c_Finite__Set_Ofolding,type,
finite_folding:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem,type,
finite_folding_idem:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__axioms,type,
finite7837460588564673216axioms:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
finite6916993218817215295axioms:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,B) * fun(B,C) ) > fun(A,C) ) ).
tff(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).
tff(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).
tff(sy_c_Fun_Omono__on,type,
mono_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).
tff(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).
tff(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).
tff(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).
tff(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).
tff(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_Obezw,type,
bezw: ( nat * nat ) > product_prod(int,int) ).
tff(sy_c_GCD_Obezw__rel,type,
bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).
tff(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).
tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).
tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).
tff(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).
tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
groups3894954378712506084id_sum:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A * fun(B,A) * set(B) ) > A ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).
tff(sy_c_Groups__List_Omonoid__add_Osum__list,type,
groups4543113879258116180m_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * list(A) ) > A ) ).
tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : fun(list(A),A) ).
tff(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(list(A),A) ) ).
tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : fun(list(A),A) ).
tff(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
tff(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
tff(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( array(A) > nat ) ).
tff(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ref(A) > nat ) ).
tff(sy_c_Heap_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > array(A) ) ).
tff(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( ( heap_ext(Z) * typerep * nat ) > list(nat) ) ).
tff(sy_c_Heap_Oheap_Oarrays__update,type,
arrays_update:
!>[Z: $tType] : ( fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat)))) > fun(heap_ext(Z),heap_ext(Z)) ) ).
tff(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( heap_ext(Z) > nat ) ).
tff(sy_c_Heap_Oheap_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( fun(nat,nat) * heap_ext(Z) ) > heap_ext(Z) ) ).
tff(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext(Z) * typerep * nat ) > nat ) ).
tff(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ref(A) ) ).
tff(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A)) ).
tff(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap,type,
heap_Time_case_Heap:
!>[A: $tType,B: $tType] : fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(heap_Time_Heap(A),B)) ).
tff(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
heap_Time_map_Heap:
!>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(heap_Time_Heap(A),heap_Time_Heap(Aa))) ).
tff(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
heap_Time_pred_Heap:
!>[A: $tType] : fun(fun(A,bool),fun(heap_Time_Heap(A),bool)) ).
tff(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
heap_Time_rec_Heap:
!>[A: $tType,C: $tType] : fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C)) ).
tff(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
heap_Time_rel_Heap:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))) ).
tff(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
heap_Time_set_Heap:
!>[A: $tType] : ( heap_Time_Heap(A) > set(A) ) ).
tff(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
heap_Time_size_Heap:
!>[A: $tType] : ( fun(A,nat) > fun(heap_Time_Heap(A),nat) ) ).
tff(sy_c_Heap__Time__Monad_OHeap__lub,type,
heap_Time_Heap_lub:
!>[A: $tType] : fun(set(heap_Time_Heap(A)),heap_Time_Heap(A)) ).
tff(sy_c_Heap__Time__Monad_OHeap__ord,type,
heap_Time_Heap_ord:
!>[A: $tType] : fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)) ).
tff(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( fun(A,bool) * A ) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).
tff(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).
tff(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : fun(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
tff(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( fun(heap_ext(product_unit),bool) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Olift,type,
heap_Time_lift:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Time_Heap(B)) ) ).
tff(sy_c_Heap__Time__Monad_Oraise,type,
heap_Time_raise:
!>[A: $tType] : ( list(char) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : fun(A,heap_Time_Heap(A)) ).
tff(sy_c_Heap__Time__Monad_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).
tff(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Time_Heap(A) ) ).
tff(sy_c_Heap__Time__Monad_OtimeFrame,type,
heap_Time_timeFrame:
!>[A: $tType] : ( ( nat * option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ).
tff(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
heap_T5500966940807335491me_rel:
!>[A: $tType] : fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool)) ).
tff(sy_c_Heap__Time__Monad_Oureturn,type,
heap_Time_ureturn:
!>[A: $tType] : fun(A,heap_Time_Heap(A)) ).
tff(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > heap_Time_Heap(product_unit) ).
tff(sy_c_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( fun(A,A) > $o ) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).
tff(sy_c_Hoare__Triple_Onew__addrs,type,
hoare_new_addrs: ( heap_ext(product_unit) * set(nat) * heap_ext(product_unit) ) > set(nat) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Int_OAbs__Integ,type,
abs_Integ: fun(product_prod(nat,nat),int) ).
tff(sy_c_Int_ONeg,type,
neg: num > int ).
tff(sy_c_Int_OPos,type,
pos: num > int ).
tff(sy_c_Int_ORep__Integ,type,
rep_Integ: fun(int,product_prod(nat,nat)) ).
tff(sy_c_Int_Odup,type,
dup: int > int ).
tff(sy_c_Int_Ointrel,type,
intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).
tff(sy_c_Int_Onat,type,
nat2: fun(int,nat) ).
tff(sy_c_Int_Opcr__int,type,
pcr_int: fun(product_prod(nat,nat),fun(int,bool)) ).
tff(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( ( A * int ) > A ) ).
tff(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : set(A) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : fun(int,A) ).
tff(sy_c_Int_Osub,type,
sub: ( num * num ) > int ).
tff(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > fun(A,A) ) ).
tff(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) * fun(A,fun(B,bool)) ) > $o ) ).
tff(sy_c_List_Oappend,type,
append:
!>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).
tff(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).
tff(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),bool)) ).
tff(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list(A) * A ) > nat ) ).
tff(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).
tff(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).
tff(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).
tff(sy_c_List_Ofilter,type,
filter:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(A) ) ).
tff(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( fun(B,fun(A,B)) > fun(B,fun(list(A),B)) ) ).
tff(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).
tff(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( ( nat * list(A) ) > nat ) ).
tff(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set(product_prod(A,A)) * nat ) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( fun(B,A) > fun(list(B),list(B)) ) ).
tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( set(A) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).
tff(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).
tff(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).
tff(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : fun(list(A),set(A)) ).
tff(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).
tff(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).
tff(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).
tff(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).
tff(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( set(A) > set(list(A)) ) ).
tff(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).
tff(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > set(B) ) ).
tff(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) * list(B) ) > list(B) ) ).
tff(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(list(A),list(B))),fun(product_prod(fun(A,B),product_prod(list(A),list(B))),bool)) ).
tff(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).
tff(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( list(A) > fun(nat,A) ) ).
tff(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).
tff(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( fun(A,bool) > fun(list(A),product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
tff(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( ( nat * A ) > list(A) ) ).
tff(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).
tff(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).
tff(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).
tff(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).
tff(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).
tff(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Othose,type,
those:
!>[A: $tType] : ( list(option(A)) > option(list(A)) ) ).
tff(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
tff(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).
tff(sy_c_List_Oupt,type,
upt: ( nat * nat ) > list(nat) ).
tff(sy_c_List_Oupto,type,
upto: ( int * int ) > list(int) ).
tff(sy_c_List_Oupto__rel,type,
upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).
tff(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).
tff(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).
tff(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).
tff(sy_c_Map_Omap__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,option(C)) * fun(A,option(B)) * A ) > option(C) ) ).
tff(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > $o ) ).
tff(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).
tff(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).
tff(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).
tff(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).
tff(sy_c_Misc_OCODE__ABORT,type,
cODE_ABORT:
!>[A: $tType] : ( fun(product_unit,A) > A ) ).
tff(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( fun(A,bool) > option(A) ) ).
tff(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Misc_Obrk__rel,type,
brk_rel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod(bool,A),product_prod(bool,B))) ) ).
tff(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( set(A) > option(set(A)) ) ).
tff(sy_c_Misc_Ofilter__rev,type,
filter_rev:
!>[A: $tType] : fun(fun(A,bool),fun(list(A),list(A))) ).
tff(sy_c_Misc_Ofilter__rev__aux,type,
filter_rev_aux:
!>[A: $tType] : ( list(A) > fun(fun(A,bool),fun(list(A),list(A))) ) ).
tff(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).
tff(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).
tff(sy_c_Misc_Olist__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * list(A) * list(B) ) > $o ) ).
tff(sy_c_Misc_Olist__all__zip__rel,type,
list_all_zip_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool)) ).
tff(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( fun(B,set(A)) * list(B) ) > set(A) ) ).
tff(sy_c_Misc_Omap__mmupd,type,
map_mmupd:
!>[B: $tType,A: $tType] : ( ( fun(B,option(A)) * set(B) * A ) > fun(B,option(A)) ) ).
tff(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).
tff(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list(list(A)) * list(list(A)) ) > list(A) ) ).
tff(sy_c_Misc_Omerge__list__rel,type,
merge_list_rel:
!>[A: $tType] : fun(product_prod(list(list(A)),list(list(A))),fun(product_prod(list(list(A)),list(list(A))),bool)) ).
tff(sy_c_Misc_Omerge__rel,type,
merge_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_Misc_Omergesort,type,
mergesort:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(list(A),list(A)) ) ).
tff(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).
tff(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).
tff(sy_c_Misc_Omergesort__by__rel__split,type,
merges295452479951948502_split:
!>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),bool)) ).
tff(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).
tff(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),bool)) ).
tff(sy_c_Misc_Opartition__rev,type,
partition_rev:
!>[A: $tType] : ( ( fun(A,bool) * product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_Misc_Opartition__rev__rel,type,
partition_rev_rel:
!>[A: $tType] : fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),fun(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool)) ).
tff(sy_c_Misc_Oquicksort__by__rel,type,
quicksort_by_rel:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > fun(list(A),list(A)) ) ).
tff(sy_c_Misc_Oquicksort__by__rel__rel,type,
quicksort_by_rel_rel:
!>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool)) ).
tff(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),bool) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Misc_Oremove__rev,type,
remove_rev:
!>[A: $tType] : ( A > fun(list(A),list(A)) ) ).
tff(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Orevg__rel,type,
revg_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,option(A)) ) ).
tff(sy_c_Misc_Oslice,type,
slice:
!>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).
tff(sy_c_Misc_Osu__rel__fun,type,
su_rel_fun:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * fun(A,B) ) > $o ) ).
tff(sy_c_Misc_Othe__default,type,
the_default:
!>[A: $tType] : ( ( A * option(A) ) > A ) ).
tff(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).
tff(sy_c_Misc_Ozipf,type,
zipf:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * list(A) * list(B) ) > list(C) ) ).
tff(sy_c_Misc_Ozipf__rel,type,
zipf_rel:
!>[A: $tType,B: $tType,C: $tType] : fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),fun(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool)) ).
tff(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : fun(A,fun(multiset(A),multiset(A))) ).
tff(sy_c_Multiset_Ocomm__monoid__add_Osum__mset,type,
comm_monoid_sum_mset:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * multiset(A) ) > A ) ).
tff(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
comm_m7189776963980413722m_mset:
!>[A: $tType] : fun(multiset(A),A) ).
tff(sy_c_Multiset_Ocomm__monoid__mset,type,
comm_monoid_mset:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(multiset(A),A) ) ).
tff(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : fun(multiset(A),A) ).
tff(sy_c_Multiset_Ofilter__mset,type,
filter_mset:
!>[A: $tType] : ( ( fun(A,bool) * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * multiset(A) ) > B ) ).
tff(sy_c_Multiset_Oimage__mset,type,
image_mset:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(multiset(A),multiset(B)) ) ).
tff(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : ( ( multiset(A) * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).
tff(sy_c_Multiset_Oms__strict,type,
ms_strict: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).
tff(sy_c_Multiset_Oms__weak,type,
ms_weak: set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))) ).
tff(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( list(A) > multiset(A) ) ).
tff(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( set(B) > multiset(B) ) ).
tff(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).
tff(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(multiset(A),multiset(A))) ) ).
tff(sy_c_Multiset_Omulteqp__code,type,
multeqp_code:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Omultiset_OAbs__multiset,type,
abs_multiset:
!>[A: $tType] : fun(fun(A,nat),multiset(A)) ).
tff(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : fun(multiset(A),fun(A,nat)) ).
tff(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Omultp__code,type,
multp_code:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( fun(C,fun(B,bool)) > fun(fun(C,nat),fun(multiset(B),bool)) ) ).
tff(sy_c_Multiset_Opw__leq,type,
pw_leq: ( multiset(product_prod(nat,nat)) * multiset(product_prod(nat,nat)) ) > $o ).
tff(sy_c_Multiset_Orepeat__mset,type,
repeat_mset:
!>[A: $tType] : ( ( nat * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( ( nat * A ) > multiset(A) ) ).
tff(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( multiset(A) > set(A) ) ).
tff(sy_c_Multiset_Osize__multiset,type,
size_multiset:
!>[A: $tType] : ( ( fun(A,nat) * multiset(A) ) > nat ) ).
tff(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list(A) * list(A) ) > option(bool) ) ).
tff(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : ( ( multiset(A) * multiset(A) ) > $o ) ).
tff(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : fun(multiset(A),fun(multiset(A),bool)) ).
tff(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset(A) * multiset(A) ) > multiset(A) ) ).
tff(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( fun(A,nat) * multiset(A) * A ) > nat ) ).
tff(sy_c_Nat_OSuc,type,
suc: fun(nat,nat) ).
tff(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).
tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > T ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > list(nat) ).
tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: fun(nat,fun(nat,bool)) ).
tff(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: list(nat) > nat ).
tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: fun(list(nat),fun(list(nat),bool)) ).
tff(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > product_prod(nat,nat) ).
tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).
tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).
tff(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: fun(product_prod(nat,nat),nat) ).
tff(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > set(nat) ).
tff(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: set(nat) > nat ).
tff(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > sum_sum(nat,nat) ).
tff(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
tff(sy_c_Num_OBitM,type,
bitM: num > num ).
tff(sy_c_Num_Oinc,type,
inc: num > num ).
tff(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( ( num * num ) > A ) ).
tff(sy_c_Num_Onum_OBit0,type,
bit0: fun(num,num) ).
tff(sy_c_Num_Onum_OBit1,type,
bit1: fun(num,num) ).
tff(sy_c_Num_Onum_OOne,type,
one2: num ).
tff(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).
tff(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
tff(sy_c_Num_Onum__of__nat,type,
num_of_nat: fun(nat,num) ).
tff(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : fun(num,A) ).
tff(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
tff(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Num_Osqr,type,
sqr: num > num ).
tff(sy_c_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( sum_sum(A,nat) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OLeaf,type,
old_Leaf:
!>[A: $tType,B: $tType] : ( A > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_ONode,type,
old_Node:
!>[B: $tType,A: $tType] : set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))) ).
tff(sy_c_Old__Datatype_ONumb,type,
old_Numb:
!>[A: $tType,B: $tType] : ( nat > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OPush__Node,type,
old_Push_Node:
!>[B: $tType,A: $tType] : ( sum_sum(B,nat) > fun(old_node(A,B),old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OScons,type,
old_Scons:
!>[A: $tType,B: $tType] : ( ( set(old_node(A,B)) * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_Odprod,type,
old_dprod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).
tff(sy_c_Old__Datatype_Odsum,type,
old_dsum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).
tff(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( old_node(A,B) > nat ) ).
tff(sy_c_Old__Datatype_Onode_OAbs__Node,type,
old_Abs_Node:
!>[B: $tType,A: $tType] : ( product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) > old_node(A,B) ) ).
tff(sy_c_Option_Obind,type,
bind:
!>[A: $tType,B: $tType] : fun(option(A),fun(fun(A,option(B)),option(B))) ).
tff(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * option(A) * option(A) ) > option(A) ) ).
tff(sy_c_Option_Ois__none,type,
is_none:
!>[A: $tType] : ( option(A) > $o ) ).
tff(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : option(A) ).
tff(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : fun(A,option(A)) ).
tff(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : fun(B,fun(fun(A,B),fun(option(A),B))) ).
tff(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(option(A),option(Aa))) ).
tff(sy_c_Option_Ooption_Opred__option,type,
pred_option:
!>[A: $tType] : fun(fun(A,bool),fun(option(A),bool)) ).
tff(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : fun(C,fun(fun(A,C),fun(option(A),C))) ).
tff(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))) ).
tff(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : fun(option(A),set(A)) ).
tff(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( fun(A,nat) > fun(option(A),nat) ) ).
tff(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : fun(option(A),A) ).
tff(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( set(option(A)) > set(A) ) ).
tff(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).
tff(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A * A ) > A ) ).
tff(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A * A ) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > fun(A,A) ) ).
tff(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > fun(A,A) ) ).
tff(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > fun(set(A),A) ) ).
tff(sy_c_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > fun(A,fun(A,bool)) ) ).
tff(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( fun(set(C),B) > fun(set(fun(A,C)),fun(A,B)) ) ).
tff(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,bool)) > fun(fun(C,A),fun(fun(C,B),bool)) ) ).
tff(sy_c_Partial__Function_Oimg__lub,type,
partial_img_lub:
!>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( fun(A,C) * fun(D,B) * fun(set(C),D) ) > fun(set(A),B) ) ).
tff(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(C,fun(C,B)) ) > fun(A,fun(A,B)) ) ).
tff(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(set(A),A) ) > $o ) ).
tff(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : fun(A,fun(nat,A)) ).
tff(sy_c_Predicate_OSeq,type,
seq2:
!>[A: $tType] : ( fun(product_unit,seq(A)) > pred(A) ) ).
tff(sy_c_Predicate_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( pred(A) * fun(A,pred(B)) ) > pred(B) ) ).
tff(sy_c_Predicate_Oiterate__upto,type,
iterate_upto:
!>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > pred(A) ) ).
tff(sy_c_Predicate_Oiterate__upto__rel,type,
iterate_upto_rel:
!>[A: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool)) ).
tff(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : seq(A) ).
tff(sy_c_Predicate_Oseq_OInsert,type,
insert:
!>[A: $tType] : ( ( A * pred(A) ) > seq(A) ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).
tff(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
tff(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).
tff(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).
tff(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( ( T * T * bool ) > T ) ).
tff(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( ( T * T * bool ) > fun(T,bool) ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,bool) ) ).
tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( ( T * product_unit ) > fun(T,bool) ) ).
tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( ( T * product_unit ) > T ) ).
tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).
tff(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).
tff(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).
tff(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).
tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: fun(bool,product_unit) ).
tff(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: fun(product_unit,bool) ).
tff(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( ( A * product_unit ) > A ) ).
tff(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) ) > $o ) ).
tff(sy_c_Random_Oinc__shift,type,
inc_shift: ( code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).
tff(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).
tff(sy_c_Random_Olog,type,
log: ( code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Olog__rel,type,
log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),bool)) ).
tff(sy_c_Random_Ominus__shift,type,
minus_shift: ( code_natural * code_natural * code_natural ) > code_natural ).
tff(sy_c_Random_Onext,type,
next: fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list(product_prod(code_natural,A)) * code_natural ) > A ) ).
tff(sy_c_Random_Orange,type,
range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Osplit__seed,type,
split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).
tff(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))) * product_prod(code_natural,code_natural) ) > product_prod(pred(B),product_prod(code_natural,code_natural)) ) ).
tff(sy_c_Random__Pred_Oempty,type,
random_empty:
!>[A: $tType] : fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ).
tff(sy_c_Random__Pred_Oiterate__upto,type,
random_iterate_upto:
!>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(A),product_prod(code_natural,code_natural)) ) ).
tff(sy_c_Rat_OAbs__Rat,type,
abs_Rat: fun(product_prod(int,int),rat) ).
tff(sy_c_Rat_OFract,type,
fract: fun(int,fun(int,rat)) ).
tff(sy_c_Rat_OFrct,type,
frct: product_prod(int,int) > rat ).
tff(sy_c_Rat_ORep__Rat,type,
rep_Rat: fun(rat,product_prod(int,int)) ).
tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
tff(sy_c_Rat_Onormalize,type,
normalize: product_prod(int,int) > product_prod(int,int) ).
tff(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
tff(sy_c_Rat_Opcr__rat,type,
pcr_rat: fun(product_prod(int,int),fun(rat,bool)) ).
tff(sy_c_Rat_Opositive,type,
positive: rat > $o ).
tff(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod(int,int) ).
tff(sy_c_Rat_Oratrel,type,
ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).
tff(sy_c_Ref__Time_Oalloc,type,
ref_alloc:
!>[A: $tType] : ( ( A * heap_ext(product_unit) ) > product_prod(ref(A),heap_ext(product_unit)) ) ).
tff(sy_c_Ref__Time_Ochange,type,
ref_change:
!>[A: $tType] : ( ( fun(A,A) * ref(A) ) > heap_Time_Heap(A) ) ).
tff(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > A ) ).
tff(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ref(A) > heap_Time_Heap(A) ) ).
tff(sy_c_Ref__Time_Opresent,type,
ref_present:
!>[A: $tType] : ( ( heap_ext(product_unit) * ref(A) ) > $o ) ).
tff(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > heap_Time_Heap(ref(A)) ) ).
tff(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref(A) * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref(A) * A ) > heap_Time_Heap(product_unit) ) ).
tff(sy_c_Refine__Imp__Hol_Oassert_H,type,
refine_Imp_assert: bool > heap_Time_Heap(product_unit) ).
tff(sy_c_Refine__Imp__Hol_Orefines,type,
refine_Imp_refines:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $o ) ).
tff(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(A) ) ).
tff(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(A,bool) ) ).
tff(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).
tff(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,B)) * set(A) ) > set(B) ) ).
tff(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(B) ) ).
tff(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,bool) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).
tff(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).
tff(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) ) > fun(A,fun(C,bool)) ) ).
tff(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > bool ) ).
tff(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : fun(bool,A) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : fun(fun(A,bool),set(A)) ).
tff(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Set_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(B) ) ).
tff(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).
tff(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).
tff(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > fun(set(A),set(A)) ) ).
tff(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(B),set(A)) ) ).
tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).
tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool)) ).
tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_String_Ochar_OChar,type,
char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).
tff(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
tff(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).
tff(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(sum_sum(A,B),sum_sum(C,D)) ) ).
tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) ) > fun(sum_sum(A,B),C) ) ).
tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_Time__Reasoning_Ofails,type,
time_fails:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).
tff(sy_c_Time__Reasoning_Othe__heap,type,
time_the_heap:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_Time__Reasoning_Othe__res,type,
time_the_res:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > A ) ).
tff(sy_c_Time__Reasoning_Otime,type,
time_time:
!>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > nat ) ).
tff(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).
tff(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).
tff(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Typerep_Otyperep__class_Otyperep,type,
typerep_typerep:
!>[A: $tType] : ( itself(A) > typerep ) ).
tff(sy_c_Wellfounded_Oacc,type,
acc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,bool) ) ).
tff(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : set(product_prod(set(A),set(A))) ).
tff(sy_c_Wellfounded_Oless__than,type,
less_than: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),fun(set(A),bool)) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) ) > $o ) ).
tff(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).
tff(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).
tff(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).
tff(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) ) > fun(set(A),bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: ( bool * bool ) > bool ).
tff(sy_c_fdisj,type,
fdisj: ( bool * bool ) > bool ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * set(A) ) > bool ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_h,type,
h: heap_ext(product_unit) ).
tff(sy_v_xs,type,
xs: list(a) ).
% Relevant facts (6627)
tff(fact_0_fails__def,axiom,
! [A: $tType,M: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( time_fails(A,M,H)
<=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),M),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% fails_def
tff(fact_1_Heap__eqI,axiom,
! [A: $tType,F: heap_Time_Heap(A),G: heap_Time_Heap(A)] :
( ! [H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H2) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),G),H2)
=> ( F = G ) ) ).
% Heap_eqI
tff(fact_2_execute__raise,axiom,
! [A: $tType,S: list(char),X: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_raise(A,S)),X) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).
% execute_raise
tff(fact_3_execute__guard_I1_J,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( ~ pp(aa(heap_ext(product_unit),bool,P,H))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_guard(A,P,F)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% execute_guard(1)
tff(fact_4_execute__assert_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X2: A,H: heap_ext(product_unit)] :
( ~ pp(aa(A,bool,P,X2))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_assert(A,P,X2)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% execute_assert(2)
tff(fact_5_successI,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> heap_Time_success(A,F,H) ) ).
% successI
tff(fact_6_success__def,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( heap_Time_success(A,F,H)
<=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% success_def
tff(fact_7_Heap__ordE,axiom,
! [A: $tType,X2: heap_Time_Heap(A),Y: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),Y))
=> ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),X2),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),X2),H) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Y),H) ) ) ) ).
% Heap_ordE
tff(fact_8_Heap__ordI,axiom,
! [A: $tType,X2: heap_Time_Heap(A),Y: heap_Time_Heap(A)] :
( ! [H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),X2),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
| ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),X2),H2) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Y),H2) ) )
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),Y)) ) ).
% Heap_ordI
tff(fact_9_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,G)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).
% execute_bind(2)
tff(fact_10_the__default_Osimps_I2_J,axiom,
! [A: $tType,X2: A] : the_default(A,X2,none(A)) = X2 ).
% the_default.simps(2)
tff(fact_11_raise__bind,axiom,
! [B: $tType,A: $tType,E2: list(char),F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,heap_Time_raise(B,E2),F) = heap_Time_raise(A,E2) ).
% raise_bind
tff(fact_12_heap_Oleq__refl,axiom,
! [A: $tType,X2: heap_Time_Heap(A)] : pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),X2)) ).
% heap.leq_refl
tff(fact_13_heap_Oleq__trans,axiom,
! [A: $tType,X2: heap_Time_Heap(A),Y: heap_Time_Heap(A),Z2: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),Y))
=> ( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),Y),Z2))
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),Z2)) ) ) ).
% heap.leq_trans
tff(fact_14_heap_Oleq__antisym,axiom,
! [A: $tType,X2: heap_Time_Heap(A),Y: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),Y))
=> ( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),Y),X2))
=> ( X2 = Y ) ) ) ).
% heap.leq_antisym
tff(fact_15_assert__cong,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),P2: fun(A,bool),F: fun(A,heap_Time_Heap(B)),F2: fun(A,heap_Time_Heap(B)),X2: A] :
( ( P = P2 )
=> ( ! [X3: A] :
( pp(aa(A,bool,P2,X3))
=> ( aa(A,heap_Time_Heap(B),F,X3) = aa(A,heap_Time_Heap(B),F2,X3) ) )
=> ( heap_Time_bind(A,B,heap_Time_assert(A,P,X2),F) = heap_Time_bind(A,B,heap_Time_assert(A,P2,X2),F2) ) ) ) ).
% assert_cong
tff(fact_16_success__ifI,axiom,
! [A: $tType,C2: bool,T2: heap_Time_Heap(A),H: heap_ext(product_unit),E2: heap_Time_Heap(A)] :
( ( pp(C2)
=> heap_Time_success(A,T2,H) )
=> ( ( ~ pp(C2)
=> heap_Time_success(A,E2,H) )
=> heap_Time_success(A,if(heap_Time_Heap(A),C2,T2,E2),H) ) ) ).
% success_ifI
tff(fact_17_success__guardI,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( pp(aa(heap_ext(product_unit),bool,P,H))
=> heap_Time_success(A,heap_Time_guard(A,P,F),H) ) ).
% success_guardI
tff(fact_18_success__assertI,axiom,
! [A: $tType,P: fun(A,bool),X2: A,H: heap_ext(product_unit)] :
( pp(aa(A,bool,P,X2))
=> heap_Time_success(A,heap_Time_assert(A,P,X2),H) ) ).
% success_assertI
tff(fact_19_fails__bind,axiom,
! [A: $tType,B: $tType,M: heap_Time_Heap(B),F: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] :
( time_fails(A,heap_Time_bind(B,A,M,F),H)
<=> ( ~ time_fails(B,M,H)
=> time_fails(A,aa(B,heap_Time_Heap(A),F,time_the_res(B,M,H)),time_the_heap(B,M,H)) ) ) ).
% fails_bind
tff(fact_20_success__of__listI,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : heap_Time_success(array(A),array_of_list(A,Xs),H) ) ).
% success_of_listI
tff(fact_21_success__tapI,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_tap(A,F),H) ).
% success_tapI
tff(fact_22_success__heapI,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_heap(A,F),H) ).
% success_heapI
tff(fact_23_success__ureturnI,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X2),H) ).
% success_ureturnI
tff(fact_24_bind__ureturn,axiom,
! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_bind(A,A,F,heap_Time_ureturn(A)) = F ).
% bind_ureturn
tff(fact_25_ureturn__bind,axiom,
! [A: $tType,B: $tType,X2: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X2),F) = aa(B,heap_Time_Heap(A),F,X2) ).
% ureturn_bind
tff(fact_26_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),none(A)) = F1 ).
% option.simps(6)
tff(fact_27_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),X2: A,N: nat,G: fun(A,heap_Time_Heap(B))] :
( heap_Time_effect(A,F,H,H3,X2,N)
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X2),H3)
=> heap_Time_success(B,heap_Time_bind(A,B,F,G),H) ) ) ).
% success_bind_effectI
tff(fact_28_execute__guard_I2_J,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( pp(aa(heap_ext(product_unit),bool,P,H))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_guard(A,P,F)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) ) ) ).
% execute_guard(2)
tff(fact_29_combine__options__simps_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Y: option(A)] : combine_options(A,F,none(A),Y) = Y ).
% combine_options_simps(1)
tff(fact_30_combine__options__simps_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X2: option(A)] : combine_options(A,F,X2,none(A)) = X2 ).
% combine_options_simps(2)
tff(fact_31_option_Oinject,axiom,
! [A: $tType,X22: A,Y2: A] :
( ( aa(A,option(A),some(A),X22) = aa(A,option(A),some(A),Y2) )
<=> ( X22 = Y2 ) ) ).
% option.inject
tff(fact_32_not__None__eq,axiom,
! [A: $tType,X2: option(A)] :
( ( X2 != none(A) )
<=> ? [Y3: A] : X2 = aa(A,option(A),some(A),Y3) ) ).
% not_None_eq
tff(fact_33_not__Some__eq,axiom,
! [A: $tType,X2: option(A)] :
( ! [Y3: A] : X2 != aa(A,option(A),some(A),Y3)
<=> ( X2 = none(A) ) ) ).
% not_Some_eq
tff(fact_34_combine__options__simps_I3_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] : combine_options(A,F,aa(A,option(A),some(A),A3),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F,A3),B2)) ).
% combine_options_simps(3)
tff(fact_35_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X22: A] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),aa(A,option(A),some(A),X22)) = aa(A,C,F22,X22) ).
% option.simps(7)
tff(fact_36_effect__deterministic_I3_J,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),A3: A,N: nat,H4: heap_ext(product_unit),B2: A,N2: nat] :
( heap_Time_effect(A,F,H,H3,A3,N)
=> ( heap_Time_effect(A,F,H,H4,B2,N2)
=> ( N = N2 ) ) ) ).
% effect_deterministic(3)
tff(fact_37_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),A3: A,N: nat,H4: heap_ext(product_unit),B2: A,N2: nat] :
( heap_Time_effect(A,F,H,H3,A3,N)
=> ( heap_Time_effect(A,F,H,H4,B2,N2)
=> ( H3 = H4 ) ) ) ).
% effect_deterministic(2)
tff(fact_38_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),A3: A,N: nat,H4: heap_ext(product_unit),B2: A,N2: nat] :
( heap_Time_effect(A,F,H,H3,A3,N)
=> ( heap_Time_effect(A,F,H,H4,B2,N2)
=> ( A3 = B2 ) ) ) ).
% effect_deterministic(1)
tff(fact_39_effect__ifE,axiom,
! [A: $tType,C2: bool,T2: heap_Time_Heap(A),E2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,if(heap_Time_Heap(A),C2,T2,E2),H,H3,R,N)
=> ( ( pp(C2)
=> ~ heap_Time_effect(A,T2,H,H3,R,N) )
=> ~ ( ~ pp(C2)
=> ~ heap_Time_effect(A,E2,H,H3,R,N) ) ) ) ).
% effect_ifE
tff(fact_40_effect__ifI,axiom,
! [A: $tType,C2: bool,T2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat,E2: heap_Time_Heap(A)] :
( ( pp(C2)
=> heap_Time_effect(A,T2,H,H3,R,N) )
=> ( ( ~ pp(C2)
=> heap_Time_effect(A,E2,H,H3,R,N) )
=> heap_Time_effect(A,if(heap_Time_Heap(A),C2,T2,E2),H,H3,R,N) ) ) ).
% effect_ifI
tff(fact_41_combine__options__assoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X2: option(A),Y: option(A),Z2: option(A)] :
( ! [X3: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,X3),Y4)),Z3) = aa(A,A,aa(A,fun(A,A),F,X3),aa(A,A,aa(A,fun(A,A),F,Y4),Z3))
=> ( combine_options(A,F,combine_options(A,F,X2,Y),Z2) = combine_options(A,F,X2,combine_options(A,F,Y,Z2)) ) ) ).
% combine_options_assoc
tff(fact_42_combine__options__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X2: option(A),Y: option(A)] :
( ! [X3: A,Y4: A] : aa(A,A,aa(A,fun(A,A),F,X3),Y4) = aa(A,A,aa(A,fun(A,A),F,Y4),X3)
=> ( combine_options(A,F,X2,Y) = combine_options(A,F,Y,X2) ) ) ).
% combine_options_commute
tff(fact_43_combine__options__left__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Y: option(A),X2: option(A),Z2: option(A)] :
( ! [X3: A,Y4: A] : aa(A,A,aa(A,fun(A,A),F,X3),Y4) = aa(A,A,aa(A,fun(A,A),F,Y4),X3)
=> ( ! [X3: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,X3),Y4)),Z3) = aa(A,A,aa(A,fun(A,A),F,X3),aa(A,A,aa(A,fun(A,A),F,Y4),Z3))
=> ( combine_options(A,F,Y,combine_options(A,F,X2,Z2)) = combine_options(A,F,X2,combine_options(A,F,Y,Z2)) ) ) ) ).
% combine_options_left_commute
tff(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: fun(A,bool)] :
( pp(member(A,A3,aa(fun(A,bool),set(A),collect(A),P)))
<=> pp(aa(A,bool,P,A3)) ) ).
% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set(A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,bool),A4)) = A4 ).
% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X3: A] :
( pp(aa(A,bool,P,X3))
<=> pp(aa(A,bool,Q,X3)) )
=> ( aa(fun(A,bool),set(A),collect(A),P) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).
% Collect_cong
tff(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
% ext
tff(fact_48_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] : none(A) != aa(A,option(A),some(A),X22) ).
% option.distinct(1)
tff(fact_49_option_OdiscI,axiom,
! [A: $tType,Option: option(A),X22: A] :
( ( Option = aa(A,option(A),some(A),X22) )
=> ( Option != none(A) ) ) ).
% option.discI
tff(fact_50_option_Oexhaust,axiom,
! [A: $tType,Y: option(A)] :
( ( Y != none(A) )
=> ~ ! [X23: A] : Y != aa(A,option(A),some(A),X23) ) ).
% option.exhaust
tff(fact_51_split__option__ex,axiom,
! [A: $tType,P: fun(option(A),bool)] :
( ? [X_1: option(A)] : pp(aa(option(A),bool,P,X_1))
<=> ( pp(aa(option(A),bool,P,none(A)))
| ? [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).
% split_option_ex
tff(fact_52_split__option__all,axiom,
! [A: $tType,P: fun(option(A),bool)] :
( ! [X_1: option(A)] : pp(aa(option(A),bool,P,X_1))
<=> ( pp(aa(option(A),bool,P,none(A)))
& ! [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).
% split_option_all
tff(fact_53_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option(A),P: fun(option(A),fun(option(B),bool)),Y: option(B)] :
( ( ( X2 = none(A) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X2),Y)) )
=> ( ( ( Y = none(B) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X2),Y)) )
=> ( ! [A5: A,B3: B] :
( ( X2 = aa(A,option(A),some(A),A5) )
=> ( ( Y = aa(B,option(B),some(B),B3) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X2),Y)) ) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X2),Y)) ) ) ) ).
% combine_options_cases
tff(fact_54_success__effectE,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( heap_Time_success(A,C2,H)
=> ~ ! [R2: A,H5: heap_ext(product_unit),N3: nat] : ~ heap_Time_effect(A,C2,H,H5,R2,N3) ) ).
% success_effectE
tff(fact_55_effect__success,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,C2,H,H3,R,N)
=> heap_Time_success(A,C2,H) ) ).
% effect_success
tff(fact_56_effect__raiseE,axiom,
! [A: $tType,X2: list(char),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] : ~ heap_Time_effect(A,heap_Time_raise(A,X2),H,H3,R,N) ).
% effect_raiseE
tff(fact_57_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu: A,X2: A] : the_default(A,Uu,aa(A,option(A),some(A),X2)) = X2 ).
% the_default.simps(1)
tff(fact_58_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( ! [X3: A,H5: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X3),H5))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% Heap_cases
tff(fact_59_successE,axiom,
! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( heap_Time_success(A,F,H)
=> ~ ! [R2: A,H5: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),H5)) ) ).
% successE
tff(fact_60_execute__heap,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_heap(A,F)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F) ).
% execute_heap
tff(fact_61_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X2: A] : eps_Opt(A,aa(A,fun(A,bool),fequal(A),X2)) = aa(A,option(A),some(A),X2) ).
% some_opt_sym_eq_trivial
tff(fact_62_success__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A,H: heap_ext(product_unit)] : heap_Time_success(array(A),aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2),H) ) ).
% success_newI
tff(fact_63_success__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A),H: heap_ext(product_unit)] : heap_Time_success(array(A),array_make(A,N,F),H) ) ).
% success_makeI
tff(fact_64_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K2: set(A),M: fun(A,option(B)),V: B] :
( pp(member(A,K,K2))
=> ( aa(A,option(B),map_mmupd(A,B,M,K2,V),K) = aa(B,option(B),some(B),V) ) ) ).
% mmupd_in_upd
tff(fact_65_option_Ocollapse,axiom,
! [A: $tType,Option: option(A)] :
( ( Option != none(A) )
=> ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).
% option.collapse
tff(fact_66_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
( ! [X4: A,Y3: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3))
<=> ( V = none(product_prod(A,B)) ) ) ).
% not_Some_eq2
tff(fact_67_effect__tapE,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_tap(A,F),H,H3,R,N)
=> ~ ( ( H3 = H )
=> ( ( R = aa(heap_ext(product_unit),A,F,H) )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_tapE
tff(fact_68_effect__tapI,axiom,
! [A: $tType,H3: heap_ext(product_unit),H: heap_ext(product_unit),R: A,F: fun(heap_ext(product_unit),A)] :
( ( H3 = H )
=> ( ( R = aa(heap_ext(product_unit),A,F,H) )
=> heap_Time_effect(A,heap_Time_tap(A,F),H,H3,R,one_one(nat)) ) ) ).
% effect_tapI
tff(fact_69_effect__assertI,axiom,
! [A: $tType,P: fun(A,bool),X2: A,H3: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( pp(aa(A,bool,P,X2))
=> ( ( H3 = H )
=> ( ( R = X2 )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,heap_Time_assert(A,P,X2),H,H3,R,N) ) ) ) ) ).
% effect_assertI
tff(fact_70_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K2: set(A),M: fun(A,option(B)),V: B] :
( ~ pp(member(A,K,K2))
=> ( aa(A,option(B),map_mmupd(A,B,M,K2,V),K) = aa(A,option(B),M,K) ) ) ).
% mmupd_notin_upd
tff(fact_71_fun__comp__eq__conv,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(C,B),G: fun(A,C),Fg: fun(A,B)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = Fg )
<=> ! [X4: A] : aa(C,B,F,aa(A,C,G,X4)) = aa(A,B,Fg,X4) ) ).
% fun_comp_eq_conv
tff(fact_72_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X2: fun(A,B),Y: fun(A,B),F: fun(B,C)] :
( ( X2 = Y )
=> ( aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),X2) = aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),Y) ) ) ).
% comp_cong_right
tff(fact_73_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X2: fun(A,B),Y: fun(A,B),F: fun(C,A)] :
( ( X2 = Y )
=> ( aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),X2),F) = aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),Y),F) ) ) ).
% comp_cong_left
tff(fact_74_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S2: set(product_prod(A,B)),P: fun(A,fun(B,bool))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S2))
=> ( ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S2))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) )
=> ? [A5: A,B3: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3),S2))
& pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3)) ) ) ) ).
% bex2I
tff(fact_75_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X2: product_prod(fun(A,B),product_prod(A,A))] :
~ ! [F3: fun(A,B),A5: A,B3: A] : X2 != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F3),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3)) ).
% pairself.cases
tff(fact_76_fails__array__new,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A,H: heap_ext(product_unit)] : ~ time_fails(array(A),aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2),H) ) ).
% fails_array_new
tff(fact_77_timeFrame_Ocases,axiom,
! [A: $tType,X2: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( ! [N3: nat,R2: A,H2: heap_ext(product_unit),N4: nat] : X2 != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N3),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N4))))
=> ~ ! [N3: nat] : X2 != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N3),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).
% timeFrame.cases
tff(fact_78_execute__assert_I1_J,axiom,
! [A: $tType,P: fun(A,bool),X2: A,H: heap_ext(product_unit)] :
( pp(aa(A,bool,P,X2))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_assert(A,P,X2)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ) ).
% execute_assert(1)
tff(fact_79_execute__tap,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_tap(A,F)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,F,H)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ).
% execute_tap
tff(fact_80_option_Osel,axiom,
! [A: $tType,X22: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X22)) = X22 ).
% option.sel
tff(fact_81_option_Oexpand,axiom,
! [A: $tType,Option: option(A),Option2: option(A)] :
( ( ( Option = none(A) )
<=> ( Option2 = none(A) ) )
=> ( ( ( Option != none(A) )
=> ( ( Option2 != none(A) )
=> ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
tff(fact_82_effect__def,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,C2,H,H3,R,N)
<=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),N))) ) ) ).
% effect_def
tff(fact_83_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),R: A,H3: heap_ext(product_unit),N: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),N))) )
=> heap_Time_effect(A,C2,H,H3,R,N) ) ).
% effectI
tff(fact_84_map__mmupd__def,axiom,
! [A: $tType,B: $tType,K: B,K2: set(B),M: fun(B,option(A)),V: A] :
( ( pp(member(B,K,K2))
=> ( aa(B,option(A),map_mmupd(B,A,M,K2,V),K) = aa(A,option(A),some(A),V) ) )
& ( ~ pp(member(B,K,K2))
=> ( aa(B,option(A),map_mmupd(B,A,M,K2,V),K) = aa(B,option(A),M,K) ) ) ) ).
% map_mmupd_def
tff(fact_85_map__mmupdE,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),K2: set(B),V: A,K: B,X2: A] :
( ( aa(B,option(A),map_mmupd(B,A,M,K2,V),K) = aa(A,option(A),some(A),X2) )
=> ( ( ~ pp(member(B,K,K2))
=> ( aa(B,option(A),M,K) != aa(A,option(A),some(A),X2) ) )
=> ~ ( pp(member(B,K,K2))
=> ( X2 != V ) ) ) ) ).
% map_mmupdE
tff(fact_86_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),X2: A,H3: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),N))) )
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X2),H3)
=> heap_Time_success(B,heap_Time_bind(A,B,F,G),H) ) ) ).
% success_bind_executeI
tff(fact_87_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: fun(A,bool),X2: A] :
( ( eps_Opt(A,P) = aa(A,option(A),some(A),X2) )
=> pp(aa(A,bool,P,X2)) ) ).
% Eps_Opt_eq_Some_implies
tff(fact_88_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: fun(A,bool),X2: A] :
( ! [X5: A] :
( pp(aa(A,bool,P,X2))
=> ( pp(aa(A,bool,P,X5))
=> ( X5 = X2 ) ) )
=> ( ( eps_Opt(A,P) = aa(A,option(A),some(A),X2) )
<=> pp(aa(A,bool,P,X2)) ) ) ).
% Eps_Opt_eq_Some
tff(fact_89_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option(A)] :
( ( Option != none(A) )
=> ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).
% option.exhaust_sel
tff(fact_90_effect__assertE,axiom,
! [A: $tType,P: fun(A,bool),X2: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_assert(A,P,X2),H,H3,R,N)
=> ~ ( pp(aa(A,bool,P,X2))
=> ( ( R = X2 )
=> ( ( H3 = H )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_assertE
tff(fact_91_comp__apply,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),G: fun(C,B),X2: C] : aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X2) = aa(B,A,F,aa(C,B,G,X2)) ).
% comp_apply
tff(fact_92_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
<=> ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
tff(fact_93_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) )
<=> ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
tff(fact_94_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),X2: A,H3: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),N))) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,G)),H) = heap_Time_timeFrame(B,N,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),aa(A,heap_Time_Heap(B),G,X2)),H3)) ) ) ).
% execute_bind(1)
tff(fact_95_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A5: A,B3: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).
% prod_cases3
tff(fact_96_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A5: A,B3: B,C3: C,D2: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).
% prod_cases4
tff(fact_97_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A5: A,B3: B,C3: C,D2: D,E3: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B3),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E3)))) ).
% prod_cases5
tff(fact_98_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))))] :
~ ! [A5: A,B3: B,C3: C,D2: D,E3: E,F3: F4] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F4))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F4))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F4)))),B3),aa(product_prod(D,product_prod(E,F4)),product_prod(C,product_prod(D,product_prod(E,F4))),aa(C,fun(product_prod(D,product_prod(E,F4)),product_prod(C,product_prod(D,product_prod(E,F4)))),product_Pair(C,product_prod(D,product_prod(E,F4))),C3),aa(product_prod(E,F4),product_prod(D,product_prod(E,F4)),aa(D,fun(product_prod(E,F4),product_prod(D,product_prod(E,F4))),product_Pair(D,product_prod(E,F4)),D2),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),E3),F3))))) ).
% prod_cases6
tff(fact_99_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))))] :
~ ! [A5: A,B3: B,C3: C,D2: D,E3: E,F3: F4,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F4,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F4,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),C3),aa(product_prod(E,product_prod(F4,G2)),product_prod(D,product_prod(E,product_prod(F4,G2))),aa(D,fun(product_prod(E,product_prod(F4,G2)),product_prod(D,product_prod(E,product_prod(F4,G2)))),product_Pair(D,product_prod(E,product_prod(F4,G2))),D2),aa(product_prod(F4,G2),product_prod(E,product_prod(F4,G2)),aa(E,fun(product_prod(F4,G2),product_prod(E,product_prod(F4,G2))),product_Pair(E,product_prod(F4,G2)),E3),aa(G2,product_prod(F4,G2),aa(F4,fun(G2,product_prod(F4,G2)),product_Pair(F4,G2),F3),G3)))))) ).
% prod_cases7
tff(fact_100_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),bool),X2: product_prod(A,product_prod(B,C))] :
( ! [A5: A,B3: B,C3: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P,X2)) ) ).
% prod_induct3
tff(fact_101_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool),X2: product_prod(A,product_prod(B,product_prod(C,D)))] :
( ! [A5: A,B3: B,C3: C,D2: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P,X2)) ) ).
% prod_induct4
tff(fact_102_timeFrame_Osimps_I2_J,axiom,
! [A: $tType,N: nat] : heap_Time_timeFrame(A,N,none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).
% timeFrame.simps(2)
tff(fact_103_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) )
=> ~ ( ( A3 = A6 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
tff(fact_104_prod__cases,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool),P3: product_prod(A,B)] :
( ! [A5: A,B3: B] : pp(aa(product_prod(A,B),bool,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)))
=> pp(aa(product_prod(A,B),bool,P,P3)) ) ).
% prod_cases
tff(fact_105_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod(A,B)] :
? [X3: A,Y4: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4) ).
% surj_pair
tff(fact_106_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A5: A,B3: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) ).
% old.prod.exhaust
tff(fact_107_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X2: A] :
( ( one_one(A) = X2 )
<=> ( X2 = one_one(A) ) ) ) ).
% one_reorient
tff(fact_108_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(A,B),V: A] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = C2 )
=> ( aa(C,B,A3,aa(A,C,B2,V)) = aa(A,B,C2,V) ) ) ).
% comp_eq_dest_lhs
tff(fact_109_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(D,B),D3: fun(A,D)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),C2),D3) )
=> ! [V2: A] : aa(C,B,A3,aa(A,C,B2,V2)) = aa(D,B,C2,aa(A,D,D3,V2)) ) ).
% comp_eq_elim
tff(fact_110_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(D,B),D3: fun(A,D),V: A] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),C2),D3) )
=> ( aa(C,B,A3,aa(A,C,B2,V)) = aa(D,B,C2,aa(A,D,D3,V)) ) ) ).
% comp_eq_dest
tff(fact_111_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: fun(D,B),G: fun(C,D),H: fun(A,C)] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aa(fun(C,D),fun(C,B),aa(fun(D,B),fun(fun(C,D),fun(C,B)),comp(D,B,C),F),G)),H) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),F),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),G),H)) ).
% comp_assoc
tff(fact_112_comp__def,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B),X: A] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G),X) = aa(B,C,F,aa(A,B,G,X)) ).
% comp_def
tff(fact_113_prod__induct7,axiom,
! [G2: $tType,F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),bool),X2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))))] :
( ! [A5: A,B3: B,C3: C,D2: D,E3: E,F3: F4,G3: G2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F4,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F4,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F4,G2)))),C3),aa(product_prod(E,product_prod(F4,G2)),product_prod(D,product_prod(E,product_prod(F4,G2))),aa(D,fun(product_prod(E,product_prod(F4,G2)),product_prod(D,product_prod(E,product_prod(F4,G2)))),product_Pair(D,product_prod(E,product_prod(F4,G2))),D2),aa(product_prod(F4,G2),product_prod(E,product_prod(F4,G2)),aa(E,fun(product_prod(F4,G2),product_prod(E,product_prod(F4,G2))),product_Pair(E,product_prod(F4,G2)),E3),aa(G2,product_prod(F4,G2),aa(F4,fun(G2,product_prod(F4,G2)),product_Pair(F4,G2),F3),G3))))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F4,G2)))))),bool,P,X2)) ) ).
% prod_induct7
tff(fact_114_prod__induct6,axiom,
! [F4: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),bool),X2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))))] :
( ! [A5: A,B3: B,C3: C,D2: D,E3: E,F3: F4] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F4))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F4))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F4)))),B3),aa(product_prod(D,product_prod(E,F4)),product_prod(C,product_prod(D,product_prod(E,F4))),aa(C,fun(product_prod(D,product_prod(E,F4)),product_prod(C,product_prod(D,product_prod(E,F4)))),product_Pair(C,product_prod(D,product_prod(E,F4))),C3),aa(product_prod(E,F4),product_prod(D,product_prod(E,F4)),aa(D,fun(product_prod(E,F4),product_prod(D,product_prod(E,F4))),product_Pair(D,product_prod(E,F4)),D2),aa(F4,product_prod(E,F4),aa(E,fun(F4,product_prod(E,F4)),product_Pair(E,F4),E3),F3)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F4))))),bool,P,X2)) ) ).
% prod_induct6
tff(fact_115_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool),X2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
( ! [A5: A,B3: B,C3: C,D2: D,E3: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B3),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E3))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,X2)) ) ).
% prod_induct5
tff(fact_116_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: fun(A,fun(B,T)),A3: A,B2: B] : product_rec_prod(A,B,T,F1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = aa(B,T,aa(A,fun(B,T),F1,A3),B2) ).
% old.prod.rec
tff(fact_117_timeFrame_Oelims,axiom,
! [A: $tType,X2: nat,Xa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( ( heap_Time_timeFrame(A,X2,Xa) = Y )
=> ( ! [R2: A,H2: heap_ext(product_unit),N4: nat] :
( ( Xa = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N4))) )
=> ( Y != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),N4)))) ) )
=> ~ ( ( Xa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( Y != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).
% timeFrame.elims
tff(fact_118_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),X2: A,H3: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B)),Y: B,H4: heap_ext(product_unit),N2: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),X2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H3),N))) )
=> ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),aa(A,heap_Time_Heap(B),G,X2)),H3) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),N2))) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,G)),H) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)))) ) ) ) ).
% execute_bind_eq_SomeI
tff(fact_119_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).
% uncurry_apply
tff(fact_120_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R: A,H: heap_ext(product_unit),N2: nat] : heap_Time_timeFrame(A,N,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),N2)))) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)))) ).
% timeFrame.simps(1)
tff(fact_121_heap__def,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_heap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F)) ).
% heap_def
tff(fact_122_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C2,A3),B2) ).
% internal_case_prod_conv
tff(fact_123_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
( pp(aa(B,bool,P,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
<=> ( ( ( Option = none(A) )
=> pp(aa(B,bool,P,F1)) )
& ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
=> pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).
% option.split_sel
tff(fact_124_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F22: fun(A,B),Option: option(A)] :
( pp(aa(B,bool,P,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
<=> ~ ( ( ( Option = none(A) )
& ~ pp(aa(B,bool,P,F1)) )
| ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
& ~ pp(aa(B,bool,P,aa(A,B,F22,aa(option(A),A,the2(A),Option)))) ) ) ) ).
% option.split_sel_asm
tff(fact_125_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
~ ! [F3: fun(nat,fun(A,A)),A5: nat,B3: nat,Acc: A] : X2 != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))) ).
% fold_atLeastAtMost_nat.cases
tff(fact_126_one__natural_Orsp,axiom,
one_one(nat) = one_one(nat) ).
% one_natural.rsp
tff(fact_127_timeFrame_Opelims,axiom,
! [A: $tType,X2: nat,Xa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( ( heap_Time_timeFrame(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X2),Xa)))
=> ( ! [R2: A,H2: heap_ext(product_unit),N4: nat] :
( ( Xa = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N4))) )
=> ( ( Y = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),N4)))) )
=> ~ pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X2),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N4)))))) ) )
=> ~ ( ( Xa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ~ pp(aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),X2),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ) ) ) ) ) ).
% timeFrame.pelims
tff(fact_128_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B2: A,A3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% add_right_cancel
tff(fact_129_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
<=> ( B2 = C2 ) ) ) ).
% add_left_cancel
tff(fact_130_Heap_Oinject,axiom,
! [A: $tType,X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Ya: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( ( aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Ya) )
<=> ( X2 = Ya ) ) ).
% Heap.inject
tff(fact_131_Heap__execute,axiom,
! [A: $tType,F: heap_Time_Heap(A)] : aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F)) = F ).
% Heap_execute
tff(fact_132_timeFrame__assoc,axiom,
! [A: $tType,N: nat,N2: nat,F: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,N,heap_Time_timeFrame(A,N2,F)) = heap_Time_timeFrame(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2),F) ).
% timeFrame_assoc
tff(fact_133_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [B2: A,A3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
tff(fact_134_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
tff(fact_135_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% ab_semigroup_add_class.add.left_commute
tff(fact_136_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ).
% ab_semigroup_add_class.add.commute
tff(fact_137_add_Oright__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% add.right_cancel
tff(fact_138_add_Oleft__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
<=> ( B2 = C2 ) ) ) ).
% add.left_cancel
tff(fact_139_add_Oassoc,axiom,
! [A: $tType] :
( semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% add.assoc
tff(fact_140_group__cancel_Oadd2,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [B5: A,K: A,B2: A,A3: A] :
( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% group_cancel.add2
tff(fact_141_group__cancel_Oadd1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).
% group_cancel.add1
tff(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
tff(fact_143_Heap_Oexhaust,axiom,
! [A: $tType,Y: heap_Time_Heap(A)] :
~ ! [X3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : Y != aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X3) ).
% Heap.exhaust
tff(fact_144_execute_Osimps,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),F)) = F ).
% execute.simps
tff(fact_145_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X22: A] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),aa(A,option(A),some(A),X22)) = aa(A,B,F22,X22) ).
% option.simps(5)
tff(fact_146_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),none(A)) = F1 ).
% option.simps(4)
tff(fact_147_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit),H4: heap_ext(product_unit),R3: A,N: nat] :
( heap_Time_effect(A,heap_Time_bind(B,A,F,G),H,H4,R3,N)
=> ~ ! [H5: heap_ext(product_unit),R2: B,N1: nat] :
( heap_Time_effect(B,F,H,H5,R2,N1)
=> ! [N22: nat] :
( heap_Time_effect(A,aa(B,heap_Time_Heap(A),G,R2),H5,H4,R3,N22)
=> ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),N22) ) ) ) ) ).
% effect_bindE
tff(fact_148_effect__bindI,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat,G: fun(A,heap_Time_Heap(B)),H4: heap_ext(product_unit),R3: B,N2: nat] :
( heap_Time_effect(A,F,H,H3,R,N)
=> ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),G,R),H3,H4,R3,N2)
=> heap_Time_effect(B,heap_Time_bind(A,B,F,G),H,H4,R3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)) ) ) ).
% effect_bindI
tff(fact_149_option_Ocase__eq__if,axiom,
! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
( ( ( Option = none(A) )
=> ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = F1 ) )
& ( ( Option != none(A) )
=> ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).
% option.case_eq_if
tff(fact_150_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : uncurry(A,B,C,product_curry(A,B,C,F)) = F ).
% uncurry_curry_id
tff(fact_151_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : product_curry(A,B,C,uncurry(A,B,C,F)) = F ).
% curry_uncurry_id
tff(fact_152_accp_Ocases,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A] :
( pp(aa(A,bool,accp(A,R),A3))
=> ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,Y5),A3))
=> pp(aa(A,bool,accp(A,R),Y5)) ) ) ).
% accp.cases
tff(fact_153_accp_Osimps,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A] :
( pp(aa(A,bool,accp(A,R),A3))
<=> ? [X4: A] :
( ( A3 = X4 )
& ! [Xa2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,Xa2),X4))
=> pp(aa(A,bool,accp(A,R),Xa2)) ) ) ) ).
% accp.simps
tff(fact_154_accpI,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X2: A] :
( ! [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,Y4),X2))
=> pp(aa(A,bool,accp(A,R),Y4)) )
=> pp(aa(A,bool,accp(A,R),X2)) ) ).
% accpI
tff(fact_155_accp__induct,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A,P: fun(A,bool)] :
( pp(aa(A,bool,accp(A,R),A3))
=> ( ! [X3: A] :
( pp(aa(A,bool,accp(A,R),X3))
=> ( ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,Y5),X3))
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) ) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% accp_induct
tff(fact_156_accp__downward,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),B2: A,A3: A] :
( pp(aa(A,bool,accp(A,R),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R,A3),B2))
=> pp(aa(A,bool,accp(A,R),A3)) ) ) ).
% accp_downward
tff(fact_157_not__accp__down,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),X2: A] :
( ~ pp(aa(A,bool,accp(A,R4),X2))
=> ~ ! [Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R4,Z3),X2))
=> pp(aa(A,bool,accp(A,R4),Z3)) ) ) ).
% not_accp_down
tff(fact_158_accp__induct__rule,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A,P: fun(A,bool)] :
( pp(aa(A,bool,accp(A,R),A3))
=> ( ! [X3: A] :
( pp(aa(A,bool,accp(A,R),X3))
=> ( ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R,Y5),X3))
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) ) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% accp_induct_rule
tff(fact_159_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: fun(B,C),F: fun(A,B),V: fun(D,A)] : aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),G),aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),V)) = aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% fun.map_comp
tff(fact_160_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: fun(B,A),G: fun(C,B),X2: C,F2: fun(D,A),G4: fun(E,D),X6: E] :
( ( aa(B,A,F,aa(C,B,G,X2)) = aa(D,A,F2,aa(E,D,G4,X6)) )
=> ( aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X2) = aa(E,A,aa(fun(E,D),fun(E,A),aa(fun(D,A),fun(fun(E,D),fun(E,A)),comp(D,A,E),F2),G4),X6) ) ) ).
% comp_cong
tff(fact_161_curryI,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
( pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A3),B2)) ) ).
% curryI
tff(fact_162_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(B,C),A),A3: B,B2: C] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),A3),B2) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).
% curry_conv
tff(fact_163_curryE,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A3),B2))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).
% curryE
tff(fact_164_curryD,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A3),B2))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).
% curryD
tff(fact_165_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))))] :
( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32)))))
=> ( ! [F3: fun(nat,fun(A,A)),A5: nat,B3: nat,Acc: A] :
( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A5),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc)))))
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B3),A5))
=> pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),one_one(nat))),B3),aa(A,A,aa(nat,fun(A,A),F3,A5),Acc))) )
=> pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F3),A5),B3),Acc)) ) )
=> pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,A0),A1),A22),A32)) ) ) ).
% fold_atLeastAtMost_nat.pinduct
tff(fact_166_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4)),lex_prod(A,B,R,S)))
<=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A6),R))
| ( ( A3 = A6 )
& pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B4),S)) ) ) ) ).
% in_lex_prod
tff(fact_167_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: fun(C,B),H: fun(A,C),R1: fun(D,B),R22: fun(A,D),F: fun(B,E),L: fun(D,E)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),H) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),R1),R22) )
=> ( ( aa(fun(D,B),fun(D,E),aa(fun(B,E),fun(fun(D,B),fun(D,E)),comp(B,E,D),F),R1) = L )
=> ( aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),aa(fun(C,B),fun(C,E),aa(fun(B,E),fun(fun(C,B),fun(C,E)),comp(B,E,C),F),G)),H) = aa(fun(A,D),fun(A,E),aa(fun(D,E),fun(fun(A,D),fun(A,E)),comp(D,E,A),L),R22) ) ) ) ).
% rewriteR_comp_comp2
tff(fact_168_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: fun(C,B),G: fun(A,C),L1: fun(D,B),L2: fun(A,D),H: fun(E,A),R: fun(E,D)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),L1),L2) )
=> ( ( aa(fun(E,A),fun(E,D),aa(fun(A,D),fun(fun(E,A),fun(E,D)),comp(A,D,E),L2),H) = R )
=> ( aa(fun(E,C),fun(E,B),aa(fun(C,B),fun(fun(E,C),fun(E,B)),comp(C,B,E),F),aa(fun(E,A),fun(E,C),aa(fun(A,C),fun(fun(E,A),fun(E,C)),comp(A,C,E),G),H)) = aa(fun(E,D),fun(E,B),aa(fun(D,B),fun(fun(E,D),fun(E,B)),comp(D,B,E),L1),R) ) ) ) ).
% rewriteL_comp_comp2
tff(fact_169_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: fun(C,B),H: fun(A,C),R: fun(A,B),F: fun(B,D)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),H) = R )
=> ( aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),F),G)),H) = aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),R) ) ) ).
% rewriteR_comp_comp
tff(fact_170_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,B),G: fun(A,C),L: fun(A,B),H: fun(D,A)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = L )
=> ( aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),F),aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),G),H)) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),L),H) ) ) ).
% rewriteL_comp_comp
tff(fact_171_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M2: fun(B,A),G: fun(C,B),X2: C,N5: fun(D,A),H: fun(C,D),F: fun(A,E)] :
( ( aa(B,A,M2,aa(C,B,G,X2)) = aa(D,A,N5,aa(C,D,H,X2)) )
=> ( aa(C,E,aa(fun(C,B),fun(C,E),aa(fun(B,E),fun(fun(C,B),fun(C,E)),comp(B,E,C),aa(fun(B,A),fun(B,E),aa(fun(A,E),fun(fun(B,A),fun(B,E)),comp(A,E,B),F),M2)),G),X2) = aa(C,E,aa(fun(C,D),fun(C,E),aa(fun(D,E),fun(fun(C,D),fun(C,E)),comp(D,E,C),aa(fun(D,A),fun(D,E),aa(fun(A,E),fun(fun(D,A),fun(D,E)),comp(A,E,D),F),N5)),H),X2) ) ) ).
% type_copy_map_cong0
tff(fact_172_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),X2: C,H: fun(D,A),K: fun(C,D)] :
( ( aa(B,A,F,aa(C,B,G,X2)) = aa(D,A,H,aa(C,D,K,X2)) )
=> ( aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X2) = aa(C,A,aa(fun(C,D),fun(C,A),aa(fun(D,A),fun(fun(C,D),fun(C,A)),comp(D,A,C),H),K),X2) ) ) ).
% comp_apply_eq
tff(fact_173_effect__returnE,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X2),H,H3,R,N)
=> ~ ( ( R = X2 )
=> ( ( H3 = H )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_returnE
tff(fact_174_effect__returnI,axiom,
! [A: $tType,H: heap_ext(product_unit),H3: heap_ext(product_unit),X2: A] :
( ( H = H3 )
=> heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X2),H,H3,X2,one_one(nat)) ) ).
% effect_returnI
tff(fact_175_add_Oright__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).
% add.right_commute
tff(fact_176_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_cancel_right
tff(fact_177_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_cancel_left
tff(fact_178_fold__if__return,axiom,
! [A: $tType,B2: bool,C2: A,D3: A] :
( ( pp(B2)
=> ( aa(A,heap_Time_Heap(A),heap_Time_return(A),C2) = aa(A,heap_Time_Heap(A),heap_Time_return(A),if(A,B2,C2,D3)) ) )
& ( ~ pp(B2)
=> ( aa(A,heap_Time_Heap(A),heap_Time_return(A),D3) = aa(A,heap_Time_Heap(A),heap_Time_return(A),if(A,B2,C2,D3)) ) ) ) ).
% fold_if_return
tff(fact_179_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_imp_less_right
tff(fact_180_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% add_less_imp_less_left
tff(fact_181_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).
% add_strict_right_mono
tff(fact_182_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% add_strict_left_mono
tff(fact_183_add__strict__mono,axiom,
! [A: $tType] :
( strict9044650504122735259up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_strict_mono
tff(fact_184_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
& ( K = L ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(1)
tff(fact_185_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(2)
tff(fact_186_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(5)
tff(fact_187_success__returnI,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X2),H) ).
% success_returnI
tff(fact_188_fails__return,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit)] : ~ time_fails(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X2),H) ).
% fails_return
tff(fact_189_add_Oright__assoc,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% add.right_assoc
tff(fact_190_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% nat_add_left_cancel_less
tff(fact_191_less__option__Some,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A,Y: A] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),aa(A,option(A),some(A),X2)),aa(A,option(A),some(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y)) ) ) ).
% less_option_Some
tff(fact_192_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X2: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X2,Xa,Xb,Xc) = Y )
=> ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = Xc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = set_fo6178422350223883121st_nat(A,X2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X2,Xa),Xc)) ) ) )
=> ~ pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),X2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
tff(fact_193_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2)))))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = Acc2 ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2)) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
tff(fact_194_in__measure,axiom,
! [A: $tType,X2: A,Y: A,F: fun(A,nat)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measure(A,F)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y))) ) ).
% in_measure
tff(fact_195_less__add__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))) ) ).
% less_add_one
tff(fact_196_add__mono1,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A)))) ) ) ).
% add_mono1
tff(fact_197_lift__collapse,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B] : aa(B,heap_Time_Heap(A),heap_Time_lift(B,A,F),X2) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,F,X2)) ).
% lift_collapse
tff(fact_198_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).
% add_lessD1
tff(fact_199_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).
% add_less_mono
tff(fact_200_not__add__less1,axiom,
! [I: nat,J: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I)) ).
% not_add_less1
tff(fact_201_less__option__None,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A)] : ~ pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),X2),none(A))) ) ).
% less_option_None
tff(fact_202_less__option__None__Some__code,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),aa(A,option(A),some(A),X2))) ) ).
% less_option_None_Some_code
tff(fact_203_less__option__None__Some,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),aa(A,option(A),some(A),X2))) ) ).
% less_option_None_Some
tff(fact_204_less__option__None__is__Some,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),none(A)),X2))
=> ? [Z3: A] : X2 = aa(A,option(A),some(A),Z3) ) ) ).
% less_option_None_is_Some
tff(fact_205_lift__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : heap_Time_lift(A,B,F) = aa(fun(A,B),fun(A,heap_Time_Heap(B)),aa(fun(B,heap_Time_Heap(B)),fun(fun(A,B),fun(A,heap_Time_Heap(B))),comp(B,heap_Time_Heap(B),A),heap_Time_return(B)),F) ).
% lift_def
tff(fact_206_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType,B2: nat,A3: nat,F: fun(nat,fun(A,A)),Acc2: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = Acc2 ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2)) ) ) ) ).
% fold_atLeastAtMost_nat.simps
tff(fact_207_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X2: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X2,Xa,Xb,Xc) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = Xc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
=> ( Y = set_fo6178422350223883121st_nat(A,X2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X2,Xa),Xc)) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
tff(fact_208_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
% less_add_eq_less
tff(fact_209_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).
% trans_less_add2
tff(fact_210_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).
% trans_less_add1
tff(fact_211_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).
% add_less_mono1
tff(fact_212_not__add__less2,axiom,
! [J: nat,I: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I)) ).
% not_add_less2
tff(fact_213_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).
% less_numeral_extra(4)
tff(fact_214_mlex__iff,axiom,
! [A: $tType,X2: A,Y: A,F: fun(A,nat),R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),mlex_prod(A,F,R4)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y)))
| ( ( aa(A,nat,F,X2) = aa(A,nat,F,Y) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R4)) ) ) ) ).
% mlex_iff
tff(fact_215_mlex__less,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y: A,R4: set(product_prod(A,A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),mlex_prod(A,F,R4))) ) ).
% mlex_less
tff(fact_216_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X2: fun(A,B),Xa: product_prod(A,A),Y: product_prod(B,B)] :
( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X2),Xa) = Y )
=> ( pp(aa(product_prod(fun(A,B),product_prod(A,A)),bool,accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B)),aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),X2),Xa)))
=> ~ ! [A5: A,B3: A] :
( ( Xa = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3) )
=> ( ( Y = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X2,A5)),aa(A,B,X2,B3)) )
=> ~ pp(aa(product_prod(fun(A,B),product_prod(A,A)),bool,accp(product_prod(fun(A,B),product_prod(A,A)),pairself_rel(A,B)),aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),X2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3)))) ) ) ) ) ).
% pairself.pelims
tff(fact_217_execute__return,axiom,
! [A: $tType,X2: A] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),aa(A,heap_Time_Heap(A),heap_Time_return(A),X2)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aa(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X2)) ).
% execute_return
tff(fact_218_same__fstI,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),X2: A,Y6: B,Y: B,R4: fun(A,set(product_prod(B,B)))] :
( pp(aa(A,bool,P,X2))
=> ( pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y6),Y),aa(A,set(product_prod(B,B)),R4,X2)))
=> pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),same_fst(A,B,P,R4))) ) ) ).
% same_fstI
tff(fact_219_zero__less__two,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ).
% zero_less_two
tff(fact_220_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% is_num_normalize(1)
tff(fact_221_add_Osafe__commute,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [X2: A,Y: A,A3: A,B2: A] :
( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ) ) ).
% add.safe_commute
tff(fact_222_bijective__def,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B))] :
( bijective(A,B,R4)
<=> ( ! [X4: A,Y3: B,Z4: B] :
( ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R4))
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4),R4)) )
=> ( Y3 = Z4 ) )
& ! [X4: A,Y3: A,Z4: B] :
( ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4),R4))
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z4),R4)) )
=> ( X4 = Y3 ) ) ) ) ).
% bijective_def
tff(fact_223_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R4: set(product_prod(A,B)),S3: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S),R4))
=> ( ( S3 = S )
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S3),R4)) ) ) ).
% ssubst_Pair_rhs
tff(fact_224_add__is__0,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
% add_is_0
tff(fact_225_Nat_Oadd__0__right,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).
% Nat.add_0_right
tff(fact_226_timeFrame__zero,axiom,
! [A: $tType,H: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,zero_zero(nat),H) = H ).
% timeFrame_zero
tff(fact_227_Heap__Time__Monad_Obind__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: heap_Time_Heap(A),G: fun(A,heap_Time_Heap(C)),K: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,heap_Time_bind(A,C,F,G),K) = heap_Time_bind(A,B,F,aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_ab(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),G),K)) ).
% Heap_Time_Monad.bind_bind
tff(fact_228_not__gr__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
<=> ( N = zero_zero(A) ) ) ) ).
% not_gr_zero
tff(fact_229_add__0,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).
% add_0
tff(fact_230_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X2: A,Y: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y) )
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_231_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X2: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y) = zero_zero(A) )
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_232_add__cancel__right__right,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_right_right
tff(fact_233_add__cancel__right__left,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_right_left
tff(fact_234_add__cancel__left__right,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = A3 )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_left_right
tff(fact_235_add__cancel__left__left,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = A3 )
<=> ( B2 = zero_zero(A) ) ) ) ).
% add_cancel_left_left
tff(fact_236_double__zero__sym,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% double_zero_sym
tff(fact_237_add_Oright__neutral,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).
% add.right_neutral
tff(fact_238_add__gr__0,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
% add_gr_0
tff(fact_239_less__one,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
<=> ( N = zero_zero(nat) ) ) ).
% less_one
tff(fact_240_some__opt__eq__trivial,axiom,
! [A: $tType,X2: A] : eps_Opt(A,aTP_Lamp_ac(A,fun(A,bool),X2)) = aa(A,option(A),some(A),X2) ).
% some_opt_eq_trivial
tff(fact_241_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
tff(fact_242_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
tff(fact_243_less__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).
% less_add_same_cancel2
tff(fact_244_less__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).
% less_add_same_cancel1
tff(fact_245_add__less__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% add_less_same_cancel2
tff(fact_246_add__less__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% add_less_same_cancel1
tff(fact_247_some__opt__false__trivial,axiom,
! [A: $tType] : eps_Opt(A,aTP_Lamp_ad(A,bool)) = none(A) ).
% some_opt_false_trivial
tff(fact_248_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
% zero_reorient
tff(fact_249_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: fun(D,A)] : aa(fun(D,A),fun(D,A),aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),aTP_Lamp_ae(A,A)),T2) = T2 ).
% fun.map_ident
tff(fact_250_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option(A)] :
( ( Option != none(A) )
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fFalse),aTP_Lamp_af(A,bool)),Option)) ) ).
% option.disc_eq_case(2)
tff(fact_251_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option(A)] :
( ( Option = none(A) )
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fTrue),aTP_Lamp_ad(A,bool)),Option)) ) ).
% option.disc_eq_case(1)
tff(fact_252_option_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)) = aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),case_option(C,A),aa(B,C,H,F1)),aa(fun(A,B),fun(A,C),aTP_Lamp_ag(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22)),Option) ).
% option.case_distrib
tff(fact_253_effect__LetI,axiom,
! [B: $tType,A: $tType,X2: A,T2: A,F: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: B,N: nat] :
( ( X2 = T2 )
=> ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,X2),H,H3,R,N)
=> heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,T2),H,H3,R,N) ) ) ).
% effect_LetI
tff(fact_254_distrib__if__bind,axiom,
! [A: $tType,B: $tType,B2: bool,C2: heap_Time_Heap(B),D3: heap_Time_Heap(B),F: fun(B,heap_Time_Heap(A))] :
( ( pp(B2)
=> ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C2,D3),F) = heap_Time_bind(B,A,C2,F) ) )
& ( ~ pp(B2)
=> ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C2,D3),F) = heap_Time_bind(B,A,D3,F) ) ) ) ).
% distrib_if_bind
tff(fact_255_success__LetI,axiom,
! [A: $tType,B: $tType,X2: A,T2: A,F: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
( ( X2 = T2 )
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),F,X2),H)
=> heap_Time_success(B,aa(A,heap_Time_Heap(B),F,T2),H) ) ) ).
% success_LetI
tff(fact_256_curry__K,axiom,
! [B: $tType,A: $tType,C: $tType,C2: C,X: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,aTP_Lamp_ah(C,fun(product_prod(A,B),C),C2)),X),Xa3) = C2 ).
% curry_K
tff(fact_257_case__optionE,axiom,
! [A: $tType,P: bool,Q: fun(A,bool),X2: option(A)] :
( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),P),Q),X2))
=> ( ( ( X2 = none(A) )
=> ~ pp(P) )
=> ~ ! [Y4: A] :
( ( X2 = aa(A,option(A),some(A),Y4) )
=> ~ pp(aa(A,bool,Q,Y4)) ) ) ) ).
% case_optionE
tff(fact_258_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X: fun(product_prod(A,B),C),Xa3: A,Xb2: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X),Xa3),Xb2) = aa(product_prod(A,B),C,X,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa3),Xb2)) ).
% curry_def
tff(fact_259_combine__options__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X2: option(A),Y: option(A)] : combine_options(A,F,X2,Y) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),Y),aa(option(A),fun(A,option(A)),aTP_Lamp_aj(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F),Y)),X2) ).
% combine_options_def
tff(fact_260_less__option__def,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A),Y: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less(option(A)),X2),Y))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fFalse),aTP_Lamp_al(option(A),fun(A,bool),X2)),Y)) ) ) ).
% less_option_def
tff(fact_261_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
<=> ( N != zero_zero(A) ) ) ) ).
% zero_less_iff_neq_zero
tff(fact_262_gr__implies__not__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [M: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
=> ( N != zero_zero(A) ) ) ) ).
% gr_implies_not_zero
tff(fact_263_not__less__zero,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).
% not_less_zero
tff(fact_264_gr__zeroI,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( ( N != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).
% gr_zeroI
tff(fact_265_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).
% add.group_left_neutral
tff(fact_266_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).
% add.comm_neutral
tff(fact_267_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).
% comm_monoid_add_class.add_0
tff(fact_268_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
% zero_neq_one
tff(fact_269_array__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A] : aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2) = array_make(A,N,aTP_Lamp_am(A,fun(nat,A),X2)) ) ).
% array_make
tff(fact_270_plus__nat_Oadd__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).
% plus_nat.add_0
tff(fact_271_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
=> ( N = zero_zero(nat) ) ) ).
% add_eq_self_zero
tff(fact_272_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).
% less_numeral_extra(1)
tff(fact_273_ureturn__def,axiom,
! [A: $tType,X2: A] : aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X2) = heap_Time_heap(A,aTP_Lamp_an(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X2)) ).
% ureturn_def
tff(fact_274_raise__def,axiom,
! [A: $tType,S: list(char)] : heap_Time_raise(A,S) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_ao(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% raise_def
tff(fact_275_bind__lift,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(B),G: fun(B,A)] : heap_Time_bind(B,A,F,heap_Time_lift(B,A,G)) = heap_Time_bind(B,A,F,aTP_Lamp_ap(fun(B,A),fun(B,heap_Time_Heap(A)),G)) ).
% bind_lift
tff(fact_276_guard__def,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(A,P,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aq(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),P),F)) ).
% guard_def
tff(fact_277_pos__add__strict,axiom,
! [A: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).
% pos_add_strict
tff(fact_278_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ ! [C3: A] :
( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
=> ( C3 = zero_zero(A) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
tff(fact_279_add__pos__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_pos_pos
tff(fact_280_add__neg__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_neg_neg
tff(fact_281_add__less__zeroD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).
% add_less_zeroD
tff(fact_282_zero__less__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).
% zero_less_one
tff(fact_283_not__one__less__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).
% not_one_less_zero
tff(fact_284_execute__ureturn,axiom,
! [A: $tType,X2: A] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X2)) = aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),some(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_an(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X2)) ).
% execute_ureturn
tff(fact_285_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ? [K3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K3))
& ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K3) = J ) ) ) ).
% less_imp_add_positive
tff(fact_286_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,B2: A] : aa(product_prod(A,A),product_prod(B,B),pairself(A,B,F),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,A3)),aa(A,B,F,B2)) ).
% pairself.simps
tff(fact_287_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X2: fun(A,B),Xa: product_prod(A,A),Y: product_prod(B,B)] :
( ( aa(product_prod(A,A),product_prod(B,B),pairself(A,B,X2),Xa) = Y )
=> ~ ! [A5: A,B3: A] :
( ( Xa = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3) )
=> ( Y != aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,X2,A5)),aa(A,B,X2,B3)) ) ) ) ).
% pairself.elims
tff(fact_288_return__def,axiom,
! [A: $tType,X2: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),X2) = heap_Time_heap(A,aTP_Lamp_aa(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X2)) ).
% return_def
tff(fact_289_effect__ureturnI,axiom,
! [A: $tType,H: heap_ext(product_unit),H3: heap_ext(product_unit),X2: A] :
( ( H = H3 )
=> heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X2),H,H3,X2,zero_zero(nat)) ) ).
% effect_ureturnI
tff(fact_290_effect__ureturnE,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X2),H,H3,R,N)
=> ~ ( ( R = X2 )
=> ( ( H3 = H )
=> ( N != zero_zero(nat) ) ) ) ) ).
% effect_ureturnE
tff(fact_291_tap__def,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_ar(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F)) ).
% tap_def
tff(fact_292_double__eq__0__iff,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% double_eq_0_iff
tff(fact_293_return__bind,axiom,
! [A: $tType,B: $tType,X2: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X2),F) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_as(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X2),F)) ).
% return_bind
tff(fact_294_bind__return,axiom,
! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_bind(A,A,F,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_at(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F)) ).
% bind_return
tff(fact_295_add__0__iff,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [B2: A,A3: A] :
( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% add_0_iff
tff(fact_296_verit__sum__simplify,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).
% verit_sum_simplify
tff(fact_297_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_inc(A,zero_zero(A)) = one_one(A) ) ) ).
% dbl_inc_simps(2)
tff(fact_298_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: bool,Q: fun(A,bool),X2: option(A),R4: fun(B,bool),F: B,G: fun(A,B)] :
( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),P),Q),X2))
=> ( ( pp(P)
=> pp(aa(B,bool,R4,F)) )
=> ( ! [Q2: A] :
( pp(aa(A,bool,Q,Q2))
=> pp(aa(B,bool,R4,aa(A,B,G,Q2))) )
=> pp(aa(B,bool,R4,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F),G),X2))) ) ) ) ).
% disjE_realizer2
tff(fact_299_set__to__map__def,axiom,
! [A: $tType,B: $tType,S2: set(product_prod(B,A)),K: B] : aa(B,option(A),set_to_map(B,A,S2),K) = eps_Opt(A,aa(B,fun(A,bool),aTP_Lamp_au(set(product_prod(B,A)),fun(B,fun(A,bool)),S2),K)) ).
% set_to_map_def
tff(fact_300_K__record__comp,axiom,
! [A: $tType,C: $tType,B: $tType,C2: B,F: fun(A,C),X: A] : aa(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aTP_Lamp_av(B,fun(C,B),C2)),F),X) = C2 ).
% K_record_comp
tff(fact_301_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( ! [X4: A,Xa2: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2),R4))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa2),S2)) )
<=> ( R4 = S2 ) ) ).
% pred_equals_eq2
tff(fact_302_dbl__inc__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X2: A] : neg_numeral_dbl_inc(A,X2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),X2)),one_one(A)) ) ).
% dbl_inc_def
tff(fact_303_zero__natural_Orsp,axiom,
zero_zero(nat) = zero_zero(nat) ).
% zero_natural.rsp
tff(fact_304_fails__wait,axiom,
! [X2: nat,H: heap_ext(product_unit)] : ~ time_fails(product_unit,heap_Time_wait(X2),H) ).
% fails_wait
tff(fact_305_Euclid__induct,axiom,
! [P: fun(nat,fun(nat,bool)),A3: nat,B2: nat] :
( ! [A5: nat,B3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),B3))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B3),A5)) )
=> ( ! [A5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),zero_zero(nat)))
=> ( ! [A5: nat,B3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),B3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B3))) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A3),B2)) ) ) ) ).
% Euclid_induct
tff(fact_306_lookup__chain,axiom,
! [B: $tType,A: $tType] :
( heap(B)
=> ! [R: ref(B),F: heap_Time_Heap(A)] : heap_Time_bind(B,A,ref_lookup(B,R),aTP_Lamp_aw(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),F)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_at(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F)) ) ).
% lookup_chain
tff(fact_307_Ints__odd__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(member(A,A3,ring_1_Ints(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% Ints_odd_less_0
tff(fact_308_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(nat,num,num_of_nat,M)),aa(nat,num,num_of_nat,N)) ) ) ) ).
% num_of_nat_plus_distrib
tff(fact_309_divides__aux__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Q3: A,R: A] :
( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R))
<=> ( R = zero_zero(A) ) ) ) ).
% divides_aux_eq
tff(fact_310_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).
% sgn_pos
tff(fact_311_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X2)))) ) ).
% abs_add_one_gt_zero
tff(fact_312_map__option__case,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Y: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),Y) = aa(option(B),option(A),aa(fun(B,option(A)),fun(option(B),option(A)),aa(option(A),fun(fun(B,option(A)),fun(option(B),option(A))),case_option(option(A),B),none(A)),aTP_Lamp_ax(fun(B,A),fun(B,option(A)),F)),Y) ).
% map_option_case
tff(fact_313_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).
% add_neg_numeral_special(7)
tff(fact_314_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).
% add_neg_numeral_special(8)
tff(fact_315_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).
% dbl_dec_simps(3)
tff(fact_316_neg__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) )
<=> ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
tff(fact_317_add_Oinverse__inverse,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).
% add.inverse_inverse
tff(fact_318_abs__idempotent,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_idempotent
tff(fact_319_neg__equal__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( aa(A,A,uminus_uminus(A),A3) = A3 )
<=> ( A3 = zero_zero(A) ) ) ) ).
% neg_equal_zero
tff(fact_320_equal__neg__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( ( A3 = aa(A,A,uminus_uminus(A),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% equal_neg_zero
tff(fact_321_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] :
( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% neg_equal_0_iff_equal
tff(fact_322_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
<=> ( zero_zero(A) = A3 ) ) ) ).
% neg_0_equal_iff_equal
tff(fact_323_add_Oinverse__neutral,axiom,
! [A: $tType] :
( group_add(A)
=> ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).
% add.inverse_neutral
tff(fact_324_neg__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% neg_less_iff_less
tff(fact_325_minus__add__distrib,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) ) ).
% minus_add_distrib
tff(fact_326_minus__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = B2 ) ).
% minus_add_cancel
tff(fact_327_add__minus__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2)) = B2 ) ).
% add_minus_cancel
tff(fact_328_abs__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).
% abs_zero
tff(fact_329_abs__eq__0,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_eq_0
tff(fact_330_abs__0__eq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_0_eq
tff(fact_331_abs__add__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_add_abs
tff(fact_332_abs__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_1
tff(fact_333_abs__minus__cancel,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_minus_cancel
tff(fact_334_sgn__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).
% sgn_1
tff(fact_335_map__option__eq__Some,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Xo: option(B),Y: A] :
( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),Xo) = aa(A,option(A),some(A),Y) )
<=> ? [Z4: B] :
( ( Xo = aa(B,option(B),some(B),Z4) )
& ( aa(B,A,F,Z4) = Y ) ) ) ).
% map_option_eq_Some
tff(fact_336_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: option(B)] :
( ( none(A) = aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),X2) )
<=> ( X2 = none(B) ) ) ).
% None_eq_map_option_iff
tff(fact_337_map__option__is__None,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Opt: option(B)] :
( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),Opt) = none(A) )
<=> ( Opt = none(B) ) ) ).
% map_option_is_None
tff(fact_338_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: option(A)] :
( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),A3) = none(B) )
<=> ( A3 = none(A) ) ) ).
% option.map_disc_iff
tff(fact_339_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% neg_less_0_iff_less
tff(fact_340_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% neg_0_less_iff_less
tff(fact_341_neg__less__pos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% neg_less_pos
tff(fact_342_less__neg__neg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% less_neg_neg
tff(fact_343_ab__left__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).
% ab_left_minus
tff(fact_344_add_Oright__inverse,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).
% add.right_inverse
tff(fact_345_zero__less__abs__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)))
<=> ( A3 != zero_zero(A) ) ) ) ).
% zero_less_abs_iff
tff(fact_346_abs__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).
% abs_neg_one
tff(fact_347_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F: fun(C,B),X2: option(C)] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),G),H),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F),X2)) = aa(option(C),A,aa(fun(C,A),fun(option(C),A),aa(A,fun(fun(C,A),fun(option(C),A)),case_option(A,C),G),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),H),F)),X2) ).
% case_map_option
tff(fact_348_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% dbl_inc_simps(4)
tff(fact_349_abs__sgn__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).
% abs_sgn_eq_1
tff(fact_350_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% dbl_dec_simps(2)
tff(fact_351_sgn__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% sgn_neg
tff(fact_352_minus__equation__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,uminus_uminus(A),A3) = B2 )
<=> ( aa(A,A,uminus_uminus(A),B2) = A3 ) ) ) ).
% minus_equation_iff
tff(fact_353_equation__minus__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),B2) )
<=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% equation_minus_iff
tff(fact_354_same__sgn__abs__add,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [B2: A,A3: A] :
( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
=> ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).
% same_sgn_abs_add
tff(fact_355_sgn__minus__1,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% sgn_minus_1
tff(fact_356_option_Omap__ident,axiom,
! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_ae(A,A)),T2) = T2 ).
% option.map_ident
tff(fact_357_abs__of__neg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_neg
tff(fact_358_abs__sgn__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( ( A3 = zero_zero(A) )
=> ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = zero_zero(A) ) )
& ( ( A3 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ) ).
% abs_sgn_eq
tff(fact_359_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_one
tff(fact_360_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),Option: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Option)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),Option) ).
% map_option.compositionality
tff(fact_361_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),V)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% option.map_comp
tff(fact_362_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),aa(fun(option(B),option(C)),fun(fun(option(A),option(B)),fun(option(A),option(C))),comp(option(B),option(C),option(A)),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G)) = aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).
% map_option.comp
tff(fact_363_minus__less__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).
% minus_less_iff
tff(fact_364_less__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% less_minus_iff
tff(fact_365_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).
% is_num_normalize(8)
tff(fact_366_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).
% add.inverse_distrib_swap
tff(fact_367_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,K: A,A3: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).
% group_cancel.neg1
tff(fact_368_one__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% one_neq_neg_one
tff(fact_369_map__option__cong,axiom,
! [B: $tType,A: $tType,X2: option(A),Y: option(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Y )
=> ( ! [A5: A] :
( ( Y = aa(A,option(A),some(A),A5) )
=> ( aa(A,B,F,A5) = aa(A,B,G,A5) ) )
=> ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Y) ) ) ) ).
% map_option_cong
tff(fact_370_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X22: A] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),aa(A,option(A),some(A),X22)) = aa(B,option(B),some(B),aa(A,B,F,X22)) ).
% option.simps(9)
tff(fact_371_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),none(A)) = none(B) ).
% option.simps(8)
tff(fact_372_same__sgn__sgn__add,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [B2: A,A3: A] :
( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
=> ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).
% same_sgn_sgn_add
tff(fact_373_Ints__add,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A,B2: A] :
( pp(member(A,A3,ring_1_Ints(A)))
=> ( pp(member(A,B2,ring_1_Ints(A)))
=> pp(member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),ring_1_Ints(A))) ) ) ) ).
% Ints_add
tff(fact_374_success__lookupI,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H: heap_ext(product_unit)] : heap_Time_success(A,ref_lookup(A,R),H) ) ).
% success_lookupI
tff(fact_375_Ints__1,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(member(A,one_one(A),ring_1_Ints(A))) ) ).
% Ints_1
tff(fact_376_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( pp(member(A,X2,ring_1_Ints(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X2)),one_one(A)))
=> ( X2 = zero_zero(A) ) ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_377_sgn__if,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( ( ( X2 = zero_zero(A) )
=> ( aa(A,A,sgn_sgn(A),X2) = zero_zero(A) ) )
& ( ( X2 != zero_zero(A) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X2))
=> ( aa(A,A,sgn_sgn(A),X2) = one_one(A) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X2))
=> ( aa(A,A,sgn_sgn(A),X2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).
% sgn_if
tff(fact_378_sgn__1__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% sgn_1_neg
tff(fact_379_abs__not__less__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))) ) ).
% abs_not_less_zero
tff(fact_380_abs__of__pos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).
% abs_of_pos
tff(fact_381_option_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,C),F: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),aa(fun(option(B),C),fun(fun(option(A),option(B)),fun(option(A),C)),comp(option(B),C,option(A)),aa(fun(B,C),fun(option(B),C),aa(C,fun(fun(B,C),fun(option(B),C)),rec_option(C,B),G),Ga)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F)) = aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),G),aa(fun(A,B),fun(A,C),aTP_Lamp_ag(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F)) ).
% option.rec_o_map
tff(fact_382_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,uminus_uminus(A),A3) = B2 )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).
% neg_eq_iff_add_eq_0
tff(fact_383_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),B2) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).
% eq_neg_iff_add_eq_0
tff(fact_384_add_Oinverse__unique,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),A3) = B2 ) ) ) ).
% add.inverse_unique
tff(fact_385_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).
% ab_group_add_class.ab_left_minus
tff(fact_386_add__eq__0__iff,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
<=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% add_eq_0_iff
tff(fact_387_zero__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ( zero_zero(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% zero_neq_neg_one
tff(fact_388_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% less_minus_one_simps(4)
tff(fact_389_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).
% less_minus_one_simps(2)
tff(fact_390_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A3: option(A),F: fun(A,B)] :
( ( A3 != none(A) )
=> ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),A3)) = aa(A,B,F,aa(option(A),A,the2(A),A3)) ) ) ).
% option.map_sel
tff(fact_391_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [A3: A] :
( pp(member(A,A3,ring_1_Ints(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ) ).
% Ints_double_eq_0_iff
tff(fact_392_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% less_minus_one_simps(3)
tff(fact_393_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).
% less_minus_one_simps(1)
tff(fact_394_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% sgn_1_pos
tff(fact_395_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [A3: A] :
( pp(member(A,A3,ring_1_Ints(A)))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).
% Ints_odd_nonzero
tff(fact_396_map__option__o__empty,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,B),X: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),aTP_Lamp_ay(A,option(C))),X) = none(B) ).
% map_option_o_empty
tff(fact_397_execute__lookup,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),ref_lookup(A,R)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),ref_get(A,H,R)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ).
% execute_lookup
tff(fact_398_change__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R: ref(A)] : ref_change(A,F,R) = heap_Time_bind(A,A,ref_lookup(A,R),aa(ref(A),fun(A,heap_Time_Heap(A)),aTP_Lamp_ba(fun(A,A),fun(ref(A),fun(A,heap_Time_Heap(A))),F),R)) ) ).
% change_def
tff(fact_399_fails__assert_H,axiom,
! [P: bool,H: heap_ext(product_unit)] :
( time_fails(product_unit,refine_Imp_assert(P),H)
<=> ~ pp(P) ) ).
% fails_assert'
tff(fact_400_effect__lookupI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H3: heap_ext(product_unit),H: heap_ext(product_unit),X2: A,R: ref(A),N: nat] :
( ( H3 = H )
=> ( ( X2 = ref_get(A,H,R) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,ref_lookup(A,R),H,H3,X2,N) ) ) ) ) ).
% effect_lookupI
tff(fact_401_effect__lookupE,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),X2: A,N: nat] :
( heap_Time_effect(A,ref_lookup(A,R),H,H3,X2,N)
=> ~ ( ( H3 = H )
=> ( ( X2 = ref_get(A,H,R) )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_lookupE
tff(fact_402_add_Ogroup__axioms,axiom,
! [A: $tType] :
( group_add(A)
=> group(A,plus_plus(A),zero_zero(A),uminus_uminus(A)) ) ).
% add.group_axioms
tff(fact_403_num__of__nat__double,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = aa(num,num,bit0,aa(nat,num,num_of_nat,N)) ) ) ).
% num_of_nat_double
tff(fact_404_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( pp(member(A,X2,ring_1_Ints(A)))
=> ( ( X2 != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X2))) ) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_405_lookup__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A)] : ref_lookup(A,R) = heap_Time_tap(A,aTP_Lamp_bb(ref(A),fun(heap_ext(product_unit),A),R)) ) ).
% lookup_def
tff(fact_406_map__conv__bind__option,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),X2) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X2),aa(fun(B,A),fun(B,option(A)),aa(fun(A,option(A)),fun(fun(B,A),fun(B,option(A))),comp(A,option(A),B),some(A)),F)) ).
% map_conv_bind_option
tff(fact_407_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% nat_add_left_cancel_le
tff(fact_408_less__eq__option__None__code,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),none(A)),X2)) ) ).
% less_eq_option_None_code
tff(fact_409_bind__assoc,axiom,
! [C: $tType,A: $tType,B: $tType,X2: option(C),F: fun(C,option(B)),G: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind(C,B),X2),F)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X2),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_bc(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),F),G)) ).
% bind_assoc
tff(fact_410_le__zero__eq,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
<=> ( N = zero_zero(A) ) ) ) ).
% le_zero_eq
tff(fact_411_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_cancel_left
tff(fact_412_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_cancel_right
tff(fact_413_neg__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% neg_le_iff_le
tff(fact_414_less__eq__option__Some,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A,Y: A] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),X2)),aa(A,option(A),some(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y)) ) ) ).
% less_eq_option_Some
tff(fact_415_less__eq__option__Some__None,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ~ pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),X2)),none(A))) ) ).
% less_eq_option_Some_None
tff(fact_416_bind__runit,axiom,
! [A: $tType,X2: option(A)] : aa(fun(A,option(A)),option(A),aa(option(A),fun(fun(A,option(A)),option(A)),bind(A,A),X2),some(A)) = X2 ).
% bind_runit
tff(fact_417_bind__rzero,axiom,
! [B: $tType,A: $tType,X2: option(B)] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X2),aTP_Lamp_bd(B,option(A))) = none(A) ).
% bind_rzero
tff(fact_418_add__le__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% add_le_same_cancel1
tff(fact_419_add__le__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% add_le_same_cancel2
tff(fact_420_le__add__same__cancel1,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).
% le_add_same_cancel1
tff(fact_421_le__add__same__cancel2,axiom,
! [A: $tType] :
( ordere1937475149494474687imp_le(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).
% le_add_same_cancel2
tff(fact_422_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
tff(fact_423_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
tff(fact_424_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% neg_0_le_iff_le
tff(fact_425_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% neg_le_0_iff_le
tff(fact_426_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% less_eq_neg_nonpos
tff(fact_427_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% neg_less_eq_nonneg
tff(fact_428_abs__of__nonneg,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).
% abs_of_nonneg
tff(fact_429_abs__le__self__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% abs_le_self_iff
tff(fact_430_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)))
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_le_zero_iff
tff(fact_431_abs__of__nonpos,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).
% abs_of_nonpos
tff(fact_432_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( A3 = B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
=> ( ( C2 = D3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),D3)) ) ) ) ) ).
% ord_eq_le_eq_trans
tff(fact_433_bind__option__cong__code,axiom,
! [B: $tType,A: $tType,X2: option(A),Y: option(A),F: fun(A,option(B))] :
( ( X2 = Y )
=> ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X2),F) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),F) ) ) ).
% bind_option_cong_code
tff(fact_434_group_Oleft__cancel,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A,C2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,A3),C2) )
<=> ( B2 = C2 ) ) ) ).
% group.left_cancel
tff(fact_435_group_Oleft__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A3)),A3) = Z2 ) ) ).
% group.left_inverse
tff(fact_436_group_Oright__cancel,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),B2: A,A3: A,C2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,B2),A3) = aa(A,A,aa(A,fun(A,A),F,C2),A3) )
<=> ( B2 = C2 ) ) ) ).
% group.right_cancel
tff(fact_437_group_Oright__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,Inverse,A3)) = Z2 ) ) ).
% group.right_inverse
tff(fact_438_group_Oinverse__unique,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F,Z2,Inverse)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
=> ( aa(A,A,Inverse,A3) = B2 ) ) ) ).
% group.inverse_unique
tff(fact_439_group_Oinverse__inverse,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).
% group.inverse_inverse
tff(fact_440_group_Oinverse__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,Z2) = Z2 ) ) ).
% group.inverse_neutral
tff(fact_441_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).
% group.group_left_neutral
tff(fact_442_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A),A3: A,B2: A] :
( group(A,F,Z2,Inverse)
=> ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).
% group.inverse_distrib_swap
tff(fact_443_accp__subset,axiom,
! [A: $tType,R12: fun(A,fun(A,bool)),R23: fun(A,fun(A,bool))] :
( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R12),R23))
=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),accp(A,R23)),accp(A,R12))) ) ).
% accp_subset
tff(fact_444_le__some__optE,axiom,
! [A: $tType] :
( preorder(A)
=> ! [M: A,X2: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),X2))
=> ~ ! [M3: A] :
( ( X2 = aa(A,option(A),some(A),M3) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),M3)) ) ) ) ).
% le_some_optE
tff(fact_445_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R4)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R4),S2)) ) ).
% pred_subset_eq2
tff(fact_446_less__eq__option__def,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A),Y: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),X2),Y))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fTrue),aTP_Lamp_bf(option(A),fun(A,bool),Y)),X2)) ) ) ).
% less_eq_option_def
tff(fact_447_zero__le,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2)) ) ).
% zero_le
tff(fact_448_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
& ( K = L ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(3)
tff(fact_449_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(2)
tff(fact_450_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_semiring(1)
tff(fact_451_add__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_mono
tff(fact_452_add__left__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% add_left_mono
tff(fact_453_less__eqE,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ~ ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ) ).
% less_eqE
tff(fact_454_add__right__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ).
% add_right_mono
tff(fact_455_le__iff__add,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).
% le_iff_add
tff(fact_456_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_imp_le_left
tff(fact_457_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% add_le_imp_le_right
tff(fact_458_accp__subset__induct,axiom,
! [A: $tType,D4: fun(A,bool),R4: fun(A,fun(A,bool)),X2: A,P: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),D4),accp(A,R4)))
=> ( ! [X3: A,Z3: A] :
( pp(aa(A,bool,D4,X3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,Z3),X3))
=> pp(aa(A,bool,D4,Z3)) ) )
=> ( pp(aa(A,bool,D4,X2))
=> ( ! [X3: A] :
( pp(aa(A,bool,D4,X3))
=> ( ! [Z5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R4,Z5),X3))
=> pp(aa(A,bool,P,Z5)) )
=> pp(aa(A,bool,P,X3)) ) )
=> pp(aa(A,bool,P,X2)) ) ) ) ) ).
% accp_subset_induct
tff(fact_459_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).
% le_numeral_extra(4)
tff(fact_460_subrelI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( ! [X3: A,Y4: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4),R))
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4),S)) )
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R),S)) ) ).
% subrelI
tff(fact_461_le__minus__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% le_minus_iff
tff(fact_462_minus__le__iff,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A3)) ) ) ).
% minus_le_iff
tff(fact_463_le__imp__neg__le,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% le_imp_neg_le
tff(fact_464_exists__leI,axiom,
! [N: nat,P: fun(nat,bool)] :
( ( ! [N6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N6),N))
=> ~ pp(aa(nat,bool,P,N6)) )
=> pp(aa(nat,bool,P,N)) )
=> ? [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N))
& pp(aa(nat,bool,P,N4)) ) ) ).
% exists_leI
tff(fact_465_nat__le__iff__add,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
<=> ? [K4: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K4) ) ).
% nat_le_iff_add
tff(fact_466_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).
% trans_le_add2
tff(fact_467_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).
% trans_le_add1
tff(fact_468_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).
% add_le_mono1
tff(fact_469_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).
% add_le_mono
tff(fact_470_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
=> ? [N3: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N3) ) ).
% le_Suc_ex
tff(fact_471_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).
% add_leD2
tff(fact_472_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% add_leD1
tff(fact_473_le__add2,axiom,
! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ).
% le_add2
tff(fact_474_le__add1,axiom,
! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).
% le_add1
tff(fact_475_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
=> ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).
% add_leE
tff(fact_476_abs__le__D1,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% abs_le_D1
tff(fact_477_abs__ge__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3))) ) ).
% abs_ge_self
tff(fact_478_less__eq__option__None,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),none(A)),X2)) ) ).
% less_eq_option_None
tff(fact_479_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),ord_less_eq(option(A)),X2),none(A)))
=> ( X2 = none(A) ) ) ) ).
% less_eq_option_None_is_None
tff(fact_480_bind_Obind__lunit,axiom,
! [B: $tType,A: $tType,X2: A,F: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),aa(A,option(A),some(A),X2)),F) = aa(A,option(B),F,X2) ).
% bind.bind_lunit
tff(fact_481_Option_Obind__cong,axiom,
! [B: $tType,A: $tType,X2: option(A),Y: option(A),F: fun(A,option(B)),G: fun(A,option(B))] :
( ( X2 = Y )
=> ( ! [A5: A] :
( ( Y = aa(A,option(A),some(A),A5) )
=> ( aa(A,option(B),F,A5) = aa(A,option(B),G,A5) ) )
=> ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X2),F) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),G) ) ) ) ).
% Option.bind_cong
tff(fact_482_bind__eq__Some__conv,axiom,
! [B: $tType,A: $tType,F: option(B),G: fun(B,option(A)),X2: A] :
( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),F),G) = aa(A,option(A),some(A),X2) )
<=> ? [Y3: B] :
( ( F = aa(B,option(B),some(B),Y3) )
& ( aa(B,option(A),G,Y3) = aa(A,option(A),some(A),X2) ) ) ) ).
% bind_eq_Some_conv
tff(fact_483_bind_Obind__lzero,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),none(A)),F) = none(B) ).
% bind.bind_lzero
tff(fact_484_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),X2: option(C),G: fun(C,option(B))] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind(C,B),X2),G)) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X2),aa(fun(C,option(B)),fun(C,option(A)),aa(fun(option(B),option(A)),fun(fun(C,option(B)),fun(C,option(A))),comp(option(B),option(A),C),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F)),G)) ).
% map_option_bind
tff(fact_485_add__decreasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2)) ) ) ) ).
% add_decreasing
tff(fact_486_add__increasing,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).
% add_increasing
tff(fact_487_add__decreasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2)) ) ) ) ).
% add_decreasing2
tff(fact_488_add__increasing2,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).
% add_increasing2
tff(fact_489_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_nonneg_nonneg
tff(fact_490_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_nonpos_nonpos
tff(fact_491_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y) = zero_zero(A) )
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_492_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y) = zero_zero(A) )
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_493_not__one__le__zero,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).
% not_one_le_zero
tff(fact_494_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).
% linordered_nonzero_semiring_class.zero_le_one
tff(fact_495_zero__less__one__class_Ozero__le__one,axiom,
! [A: $tType] :
( zero_less_one(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).
% zero_less_one_class.zero_le_one
tff(fact_496_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(4)
tff(fact_497_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [I: A,J: A,K: A,L: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),J))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).
% add_mono_thms_linordered_field(3)
tff(fact_498_add__le__less__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_le_less_mono
tff(fact_499_add__less__le__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D3))) ) ) ) ).
% add_less_le_mono
tff(fact_500_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).
% le_minus_one_simps(2)
tff(fact_501_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% le_minus_one_simps(4)
tff(fact_502_abs__ge__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))) ) ).
% abs_ge_zero
tff(fact_503_abs__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).
% abs_triangle_ineq
tff(fact_504_abs__leI,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2)) ) ) ) ).
% abs_leI
tff(fact_505_abs__le__D2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ).
% abs_le_D2
tff(fact_506_abs__le__iff,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ) ) ).
% abs_le_iff
tff(fact_507_abs__ge__minus__self,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3))) ) ).
% abs_ge_minus_self
tff(fact_508_mono__nat__linear__lb,axiom,
! [F: fun(nat,nat),M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F,M4)),aa(nat,nat,F,N3))) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F,M)),K)),aa(nat,nat,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).
% mono_nat_linear_lb
tff(fact_509_nat__geq__1__eq__neqz,axiom,
! [X2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),X2))
<=> ( X2 != zero_zero(nat) ) ) ).
% nat_geq_1_eq_neqz
tff(fact_510_bind__split__asm,axiom,
! [A: $tType,B: $tType,P: fun(option(A),bool),M: option(B),F: fun(B,option(A))] :
( pp(aa(option(A),bool,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),M),F)))
<=> ~ ( ( ( M = none(B) )
& ~ pp(aa(option(A),bool,P,none(A))) )
| ? [X4: B] :
( ( M = aa(B,option(B),some(B),X4) )
& ~ pp(aa(option(A),bool,P,aa(B,option(A),F,X4))) ) ) ) ).
% bind_split_asm
tff(fact_511_bind__split,axiom,
! [A: $tType,B: $tType,P: fun(option(A),bool),M: option(B),F: fun(B,option(A))] :
( pp(aa(option(A),bool,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),M),F)))
<=> ( ( ( M = none(B) )
=> pp(aa(option(A),bool,P,none(A))) )
& ! [V3: B] :
( ( M = aa(B,option(B),some(B),V3) )
=> pp(aa(option(A),bool,P,aa(B,option(A),F,V3))) ) ) ) ).
% bind_split
tff(fact_512_mlex__leq,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y: A,R4: set(product_prod(A,A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),mlex_prod(A,F,R4))) ) ) ).
% mlex_leq
tff(fact_513_bind__eq__None__conv,axiom,
! [B: $tType,A: $tType,A3: option(B),F: fun(B,option(A))] :
( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),A3),F) = none(A) )
<=> ( ( A3 = none(B) )
| ( aa(B,option(A),F,aa(option(B),B,the2(B),A3)) = none(A) ) ) ) ).
% bind_eq_None_conv
tff(fact_514_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X2: option(C),G: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F),X2)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind(C,A),X2),aa(fun(C,B),fun(C,option(A)),aa(fun(B,option(A)),fun(fun(C,B),fun(C,option(A))),comp(B,option(A),C),G),F)) ).
% bind_map_option
tff(fact_515_add__neg__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_neg_nonpos
tff(fact_516_add__nonneg__pos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_nonneg_pos
tff(fact_517_add__nonpos__neg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),zero_zero(A))) ) ) ) ).
% add_nonpos_neg
tff(fact_518_add__pos__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% add_pos_nonneg
tff(fact_519_add__strict__increasing,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).
% add_strict_increasing
tff(fact_520_add__strict__increasing2,axiom,
! [A: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2))) ) ) ) ).
% add_strict_increasing2
tff(fact_521_discrete,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B2)) ) ) ).
% discrete
tff(fact_522_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).
% le_minus_one_simps(1)
tff(fact_523_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% le_minus_one_simps(3)
tff(fact_524_dense__eq0__I,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs(A)
& dense_linorder(A) )
=> ! [X2: A] :
( ! [E3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),E3)) )
=> ( X2 = zero_zero(A) ) ) ) ).
% dense_eq0_I
tff(fact_525_abs__minus__le__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A))) ) ).
% abs_minus_le_zero
tff(fact_526_success__changeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R: ref(A),H: heap_ext(product_unit)] : heap_Time_success(A,ref_change(A,F,R),H) ) ).
% success_changeI
tff(fact_527_semiring__norm_I6_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% semiring_norm(6)
tff(fact_528_field__le__epsilon,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,Y: A] :
( ! [E3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y)) ) ) ).
% field_le_epsilon
tff(fact_529_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: fun(A,bool),K: A,F: fun(A,nat),N: nat] :
( pp(aa(A,bool,P,K))
=> ( ! [X3: A] :
( pp(aa(A,bool,P,X3))
=> ? [Y5: A] :
( pp(aa(A,bool,P,Y5))
& ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,Y5)),aa(A,nat,F,X3))) ) )
=> ? [Y4: A] :
( pp(aa(A,bool,P,Y4))
& ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,Y4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,K)),N))) ) ) ) ).
% ex_has_greatest_nat_lemma
tff(fact_530_map__comp__def,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,option(C)),G: fun(A,option(B)),X: A] : map_comp(B,C,A,F,G,X) = aa(option(B),option(C),aa(fun(B,option(C)),fun(option(B),option(C)),aa(option(C),fun(fun(B,option(C)),fun(option(B),option(C))),case_option(option(C),B),none(C)),F),aa(A,option(B),G,X)) ).
% map_comp_def
tff(fact_531_convex__bound__lt,axiom,
! [A: $tType] :
( linord715952674999750819strict(A)
=> ! [X2: A,A3: A,Y: A,U: A,V: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_532_map__comp__empty_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,M: fun(C,option(B)),X: A] : map_comp(C,B,A,M,aTP_Lamp_ay(A,option(C)),X) = none(B) ).
% map_comp_empty(1)
tff(fact_533_map__comp__empty_I2_J,axiom,
! [C: $tType,B: $tType,D: $tType,M: fun(C,option(B)),X: C] : map_comp(B,D,C,aTP_Lamp_bg(B,option(D)),M,X) = none(D) ).
% map_comp_empty(2)
tff(fact_534_map__comp__None__iff,axiom,
! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K: C] :
( ( map_comp(B,A,C,M1,M22,K) = none(A) )
<=> ( ( aa(C,option(B),M22,K) = none(B) )
| ? [K5: B] :
( ( aa(C,option(B),M22,K) = aa(B,option(B),some(B),K5) )
& ( aa(B,option(A),M1,K5) = none(A) ) ) ) ) ).
% map_comp_None_iff
tff(fact_535_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( pp(member(A,X2,ring_1_Ints(A)))
=> ( pp(member(A,Y,ring_1_Ints(A)))
=> ( ( X2 = Y )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Y))),one_one(A))) ) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_536_nat0__intermed__int__val,axiom,
! [N: nat,F: fun(nat,int),K: int] :
( ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F,I2)))),one_one(int))) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F,N)))
=> ? [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
& ( aa(nat,int,F,I2) = K ) ) ) ) ) ).
% nat0_intermed_int_val
tff(fact_537_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).
% diff_diff_left
tff(fact_538_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
<=> ( ( M = one_one(nat) )
& ( N = one_one(nat) ) ) ) ).
% nat_1_eq_mult_iff
tff(fact_539_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
<=> ( ( M = one_one(nat) )
& ( N = one_one(nat) ) ) ) ).
% nat_mult_eq_1_iff
tff(fact_540_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).
% cancel_comm_monoid_add_class.diff_cancel
tff(fact_541_diff__zero,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).
% diff_zero
tff(fact_542_zero__diff,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% zero_diff
tff(fact_543_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).
% diff_0_right
tff(fact_544_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).
% diff_self
tff(fact_545_mult_Oright__neutral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).
% mult.right_neutral
tff(fact_546_mult__1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).
% mult_1
tff(fact_547_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).
% add_diff_cancel_right'
tff(fact_548_add__diff__cancel__right,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).
% add_diff_cancel_right
tff(fact_549_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),A3) = B2 ) ).
% add_diff_cancel_left'
tff(fact_550_add__diff__cancel__left,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).
% add_diff_cancel_left
tff(fact_551_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ).
% diff_add_cancel
tff(fact_552_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).
% add_diff_cancel
tff(fact_553_minus__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).
% minus_diff_eq
tff(fact_554_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).
% Nat.add_diff_assoc
tff(fact_555_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).
% Nat.add_diff_assoc2
tff(fact_556_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).
% Nat.diff_diff_right
tff(fact_557_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z2)) ) ).
% zle_add1_eq_le
tff(fact_558_map__comp__simps_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,M22: fun(B,option(A)),K: B,K6: A,M1: fun(A,option(C))] :
( ( aa(B,option(A),M22,K) = aa(A,option(A),some(A),K6) )
=> ( map_comp(A,C,B,M1,M22,K) = aa(A,option(C),M1,K6) ) ) ).
% map_comp_simps(2)
tff(fact_559_map__comp__simps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,M22: fun(B,option(A)),K: B,M1: fun(A,option(C))] :
( ( aa(B,option(A),M22,K) = none(A) )
=> ( map_comp(A,C,B,M1,M22,K) = none(C) ) ) ).
% map_comp_simps(1)
tff(fact_560_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% diff_ge_0_iff_ge
tff(fact_561_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% zero_comp_diff_simps(1)
tff(fact_562_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% diff_gt_0_iff_gt
tff(fact_563_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% zero_comp_diff_simps(2)
tff(fact_564_le__add__diff__inverse,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).
% le_add_diff_inverse
tff(fact_565_le__add__diff__inverse2,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ) ) ).
% le_add_diff_inverse2
tff(fact_566_mult__cancel__left1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,B2: A] :
( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
<=> ( ( C2 = zero_zero(A) )
| ( B2 = one_one(A) ) ) ) ) ).
% mult_cancel_left1
tff(fact_567_mult__cancel__left2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = C2 )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = one_one(A) ) ) ) ) ).
% mult_cancel_left2
tff(fact_568_mult__cancel__right1,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [C2: A,B2: A] :
( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
<=> ( ( C2 = zero_zero(A) )
| ( B2 = one_one(A) ) ) ) ) ).
% mult_cancel_right1
tff(fact_569_mult__cancel__right2,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A3: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = C2 )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = one_one(A) ) ) ) ) ).
% mult_cancel_right2
tff(fact_570_diff__add__zero,axiom,
! [A: $tType] :
( comm_monoid_diff(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).
% diff_add_zero
tff(fact_571_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).
% diff_numeral_special(9)
tff(fact_572_diff__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).
% diff_0
tff(fact_573_mult__minus1,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z2) = aa(A,A,uminus_uminus(A),Z2) ) ).
% mult_minus1
tff(fact_574_mult__minus1__right,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z2) ) ).
% mult_minus1_right
tff(fact_575_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).
% uminus_add_conv_diff
tff(fact_576_diff__minus__eq__add,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).
% diff_minus_eq_add
tff(fact_577_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).
% diff_numeral_special(12)
tff(fact_578_le__imp__0__less,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2))) ) ).
% le_imp_0_less
tff(fact_579_q__pos__lemma,axiom,
! [B4: int,Q4: int,R3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R3)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q4)) ) ) ) ).
% q_pos_lemma
tff(fact_580_zdiv__mono2__lemma,axiom,
! [B2: int,Q3: int,R: int,B4: int,Q4: int,R3: int] :
( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R3) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R3)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B4),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q3),Q4)) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
tff(fact_581_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q3: int,R: int,B4: int,Q4: int,R3: int] :
( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R3) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R3)),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B4),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q3)) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
tff(fact_582_unique__quotient__lemma,axiom,
! [B2: int,Q4: int,R3: int,Q3: int,R: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q3)) ) ) ) ) ).
% unique_quotient_lemma
tff(fact_583_unique__quotient__lemma__neg,axiom,
! [B2: int,Q4: int,R3: int,Q3: int,R: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q3),Q4)) ) ) ) ) ).
% unique_quotient_lemma_neg
tff(fact_584_int__ge__induct,axiom,
! [K: int,I: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I))
=> ( pp(aa(int,bool,P,K))
=> ( ! [I2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
=> ( pp(aa(int,bool,P,I2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_ge_induct
tff(fact_585_int__induct,axiom,
! [P: fun(int,bool),K: int,I: int] :
( pp(aa(int,bool,P,K))
=> ( ! [I2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
=> ( pp(aa(int,bool,P,I2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
=> ( ! [I2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K))
=> ( pp(aa(int,bool,P,I2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_induct
tff(fact_586_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z2)) ) ).
% zless_imp_add1_zle
tff(fact_587_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2)) ) ).
% add1_zle_eq
tff(fact_588_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M) = N ) ) ) ).
% nat_eq_add_iff1
tff(fact_589_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
<=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N) ) ) ) ).
% nat_eq_add_iff2
tff(fact_590_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N)) ) ) ).
% nat_le_add_iff1
tff(fact_591_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N))) ) ) ).
% nat_le_add_iff2
tff(fact_592_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N) ) ) ).
% nat_diff_add_eq1
tff(fact_593_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N)) ) ) ).
% nat_diff_add_eq2
tff(fact_594_bounded__Max__nat,axiom,
! [P: fun(nat,bool),X2: nat,M2: nat] :
( pp(aa(nat,bool,P,X2))
=> ( ! [X3: nat] :
( pp(aa(nat,bool,P,X3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M2)) )
=> ~ ! [M4: nat] :
( pp(aa(nat,bool,P,M4))
=> ~ ! [X: nat] :
( pp(aa(nat,bool,P,X))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),M4)) ) ) ) ) ).
% bounded_Max_nat
tff(fact_595_subset__Collect__conv,axiom,
! [A: $tType,S2: set(A),P: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(fun(A,bool),set(A),collect(A),P)))
<=> ! [X4: A] :
( pp(member(A,X4,S2))
=> pp(aa(A,bool,P,X4)) ) ) ).
% subset_Collect_conv
tff(fact_596_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_597_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% ab_semigroup_mult_class.mult.left_commute
tff(fact_598_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ).
% ab_semigroup_mult_class.mult.commute
tff(fact_599_mult_Oassoc,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% mult.assoc
tff(fact_600_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
=> ( ( A3 = B2 )
<=> ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
tff(fact_601_one__integer_Orsp,axiom,
one_one(int) = one_one(int) ).
% one_integer.rsp
tff(fact_602_int__gr__induct,axiom,
! [K: int,I: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I))
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
=> ( ! [I2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2))
=> ( pp(aa(int,bool,P,I2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_gr_induct
tff(fact_603_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2))
| ( W = Z2 ) ) ) ).
% zless_add1_eq
tff(fact_604_odd__nonzero,axiom,
! [Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2) != zero_zero(int) ).
% odd_nonzero
tff(fact_605_odd__less__0__iff,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)),Z2)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int))) ) ).
% odd_less_0_iff
tff(fact_606_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E2)),C2) = D3 ) ) ) ).
% eq_add_iff1
tff(fact_607_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
<=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E2)),D3) ) ) ) ).
% eq_add_iff2
tff(fact_608_square__diff__square__factored,axiom,
! [A: $tType] :
( comm_ring(A)
=> ! [X2: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Y)) ) ).
% square_diff_square_factored
tff(fact_609_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).
% left_add_mult_distrib
tff(fact_610_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N)) ) ) ).
% nat_less_add_iff1
tff(fact_611_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)),U)),N))) ) ) ).
% nat_less_add_iff2
tff(fact_612_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E2)),C2)),D3)) ) ) ).
% le_add_iff1
tff(fact_613_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E2)),D3))) ) ) ).
% le_add_iff2
tff(fact_614_less__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E2)),C2)),D3)) ) ) ).
% less_add_iff1
tff(fact_615_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E2)),D3))) ) ) ).
% less_add_iff2
tff(fact_616_square__diff__one__factored,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),one_one(A))) ) ).
% square_diff_one_factored
tff(fact_617_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3)) ) ) ) ).
% diff_eq_diff_less_eq
tff(fact_618_diff__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).
% diff_right_mono
tff(fact_619_diff__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).
% diff_left_mono
tff(fact_620_diff__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).
% diff_mono
tff(fact_621_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) ) ) ) ).
% eq_iff_diff_eq_0
tff(fact_622_diff__strict__right__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).
% diff_strict_right_mono
tff(fact_623_diff__strict__left__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2))) ) ) ).
% diff_strict_left_mono
tff(fact_624_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3)) ) ) ) ).
% diff_eq_diff_less
tff(fact_625_diff__strict__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))) ) ) ) ).
% diff_strict_mono
tff(fact_626_diff__diff__eq,axiom,
! [A: $tType] :
( cancel2418104881723323429up_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).
% diff_diff_eq
tff(fact_627_add__implies__diff,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [C2: A,B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A3 )
=> ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).
% add_implies_diff
tff(fact_628_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) ) ).
% diff_add_eq_diff_diff_swap
tff(fact_629_diff__add__eq,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).
% diff_add_eq
tff(fact_630_diff__diff__eq2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).
% diff_diff_eq2
tff(fact_631_add__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).
% add_diff_eq
tff(fact_632_eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C2 ) ) ) ).
% eq_diff_eq
tff(fact_633_diff__eq__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = C2 )
<=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).
% diff_eq_eq
tff(fact_634_group__cancel_Osub1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).
% group_cancel.sub1
tff(fact_635_minus__diff__commute,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).
% minus_diff_commute
tff(fact_636_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% ring_class.ring_distribs(2)
tff(fact_637_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).
% ring_class.ring_distribs(1)
tff(fact_638_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% comm_semiring_class.distrib
tff(fact_639_distrib__left,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).
% distrib_left
tff(fact_640_distrib__right,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% distrib_right
tff(fact_641_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,E2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E2)),C2) ) ).
% combine_common_factor
tff(fact_642_crossproduct__eq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [W: A,Y: A,X2: A,Z2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y)) )
<=> ( ( W = X2 )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
tff(fact_643_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( A3 != B2 )
& ( C2 != D3 ) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% crossproduct_noteq
tff(fact_644_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).
% comm_monoid_mult_class.mult_1
tff(fact_645_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).
% mult.comm_neutral
tff(fact_646_abs__minus__commute,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ).
% abs_minus_commute
tff(fact_647_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).
% Nat.diff_cancel
tff(fact_648_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).
% diff_cancel2
tff(fact_649_diff__add__inverse,axiom,
! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).
% diff_add_inverse
tff(fact_650_diff__add__inverse2,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).
% diff_add_inverse2
tff(fact_651_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ).
% le_diff_conv
tff(fact_652_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ) ).
% Nat.le_diff_conv2
tff(fact_653_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).
% Nat.diff_add_assoc
tff(fact_654_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I) ) ) ).
% Nat.diff_add_assoc2
tff(fact_655_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K )
<=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).
% Nat.le_imp_diff_is_add
tff(fact_656_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).
% add_mult_distrib
tff(fact_657_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).
% add_mult_distrib2
tff(fact_658_mlex__leI,axiom,
! [A3: nat,A6: nat,B2: nat,B4: nat,N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),A6))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),B4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N5)),B4))) ) ) ).
% mlex_leI
tff(fact_659_nat__mult__1,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).
% nat_mult_1
tff(fact_660_nat__mult__1__right,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).
% nat_mult_1_right
tff(fact_661_mult__eq__if,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
& ( ( M != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).
% mult_eq_if
tff(fact_662_lambda__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aTP_Lamp_bh(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).
% lambda_one
tff(fact_663_is__num__normalize_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X2: A,Y: A] :
( neg_numeral_is_num(A,X2)
=> ( neg_numeral_is_num(A,Y)
=> neg_numeral_is_num(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)) ) ) ) ).
% is_num_normalize(6)
tff(fact_664_is__num__add__commute,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X2: A,Y: A] :
( neg_numeral_is_num(A,X2)
=> ( neg_numeral_is_num(A,Y)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),X2) ) ) ) ) ).
% is_num_add_commute
tff(fact_665_is__num__add__left__commute,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X2: A,Y: A,Z2: A] :
( neg_numeral_is_num(A,X2)
=> ( neg_numeral_is_num(A,Y)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Z2)) ) ) ) ) ).
% is_num_add_left_commute
tff(fact_666_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> neg_numeral_is_num(A,one_one(A)) ) ).
% is_num_normalize(4)
tff(fact_667_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).
% le_iff_diff_le_0
tff(fact_668_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),zero_zero(A))) ) ) ).
% less_iff_diff_less_0
tff(fact_669_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) = C2 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_670_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_671_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_672_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_673_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A3) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_676_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B2)) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_677_le__add__diff,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A3))) ) ) ).
% le_add_diff
tff(fact_678_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_679_le__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2)) ) ) ).
% le_diff_eq
tff(fact_680_diff__le__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% diff_le_eq
tff(fact_681_add__le__imp__le__diff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [I: A,K: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K))) ) ) ).
% add_le_imp_le_diff
tff(fact_682_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [I: A,K: A,N: A,J: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K)),J)) ) ) ) ) ) ).
% add_le_add_imp_diff_le
tff(fact_683_less__diff__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2)) ) ) ).
% less_diff_eq
tff(fact_684_diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).
% diff_less_eq
tff(fact_685_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
tff(fact_686_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_687_diff__conv__add__uminus,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% diff_conv_add_uminus
tff(fact_688_group__cancel_Osub2,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [B5: A,K: A,B2: A,A3: A] :
( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).
% group_cancel.sub2
tff(fact_689_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)))) ) ).
% abs_triangle_ineq2_sym
tff(fact_690_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).
% abs_triangle_ineq3
tff(fact_691_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))) ) ).
% abs_triangle_ineq2
tff(fact_692_add__scale__eq__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [R: A,A3: A,B2: A,C2: A,D3: A] :
( ( R != zero_zero(A) )
=> ( ( ( A3 = B2 )
& ( C2 != D3 ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R),D3)) ) ) ) ) ).
% add_scale_eq_noteq
tff(fact_693_less__1__mult,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N))) ) ) ) ).
% less_1_mult
tff(fact_694_square__eq__1__iff,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [X2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2) = one_one(A) )
<=> ( ( X2 = one_one(A) )
| ( X2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% square_eq_1_iff
tff(fact_695_diff__add__0,axiom,
! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).
% diff_add_0
tff(fact_696_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)) ) ).
% less_diff_conv
tff(fact_697_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = M ) ) ).
% add_diff_inverse_nat
tff(fact_698_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ) ).
% less_diff_conv2
tff(fact_699_mlex__snd__decrI,axiom,
! [A3: nat,A6: nat,B2: nat,B4: nat,N5: nat] :
( ( A3 = A6 )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),B4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N5)),B4))) ) ) ).
% mlex_snd_decrI
tff(fact_700_mlex__fst__decrI,axiom,
! [A3: nat,A6: nat,B2: nat,N5: nat,B4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A6))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B4),N5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N5)),B4))) ) ) ) ).
% mlex_fst_decrI
tff(fact_701_mlex__bound,axiom,
! [A3: nat,A4: nat,B2: nat,N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),A4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),N5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N5)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N5))) ) ) ).
% mlex_bound
tff(fact_702_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
=> ( ( N = one_one(nat) )
| ( M = zero_zero(nat) ) ) ) ).
% mult_eq_self_implies_10
tff(fact_703_map__comp__Some__iff,axiom,
! [C: $tType,B: $tType,A: $tType,M1: fun(B,option(A)),M22: fun(C,option(B)),K: C,V: A] :
( ( map_comp(B,A,C,M1,M22,K) = aa(A,option(A),some(A),V) )
<=> ? [K5: B] :
( ( aa(C,option(B),M22,K) = aa(B,option(B),some(B),K5) )
& ( aa(B,option(A),M1,K5) = aa(A,option(A),some(A),V) ) ) ) ).
% map_comp_Some_iff
tff(fact_704_field__le__mult__one__interval,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,Y: A] :
( ! [Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),X2)),Y)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y)) ) ) ).
% field_le_mult_one_interval
tff(fact_705_abs__diff__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,A3: A,R: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3))),R))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).
% abs_diff_le_iff
tff(fact_706_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D3)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3))))) ) ).
% abs_diff_triangle_ineq
tff(fact_707_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)))) ) ).
% abs_triangle_ineq4
tff(fact_708_abs__diff__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,A3: A,R: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3))),R))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).
% abs_diff_less_iff
tff(fact_709_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X2: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).
% sum_squares_ge_zero
tff(fact_710_mult__left__le,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3)) ) ) ) ).
% mult_left_le
tff(fact_711_mult__le__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ) ).
% mult_le_one
tff(fact_712_mult__right__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y)),X2)) ) ) ) ) ).
% mult_right_le_one_le
tff(fact_713_mult__left__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X2)),X2)) ) ) ) ) ).
% mult_left_le_one_le
tff(fact_714_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X2: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).
% not_sum_squares_lt_zero
tff(fact_715_nat__diff__split,axiom,
! [P: fun(nat,bool),A3: nat,B2: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> pp(aa(nat,bool,P,zero_zero(nat))) )
& ! [D5: nat] :
( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
=> pp(aa(nat,bool,P,D5)) ) ) ) ).
% nat_diff_split
tff(fact_716_nat__diff__split__asm,axiom,
! [P: fun(nat,bool),A3: nat,B2: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)))
<=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
& ~ pp(aa(nat,bool,P,zero_zero(nat))) )
| ? [D5: nat] :
( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
& ~ pp(aa(nat,bool,P,D5)) ) ) ) ).
% nat_diff_split_asm
tff(fact_717_dbl__dec__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X2: A] : neg_numeral_dbl_dec(A,X2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),X2)),one_one(A)) ) ).
% dbl_dec_def
tff(fact_718_mult__le__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).
% mult_le_cancel_left1
tff(fact_719_mult__le__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).
% mult_le_cancel_left2
tff(fact_720_mult__le__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).
% mult_le_cancel_right1
tff(fact_721_mult__le__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3)) ) ) ) ) ).
% mult_le_cancel_right2
tff(fact_722_mult__less__cancel__left1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).
% mult_less_cancel_left1
tff(fact_723_mult__less__cancel__left2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).
% mult_less_cancel_left2
tff(fact_724_mult__less__cancel__right1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).
% mult_less_cancel_right1
tff(fact_725_mult__less__cancel__right2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) )
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3)) ) ) ) ) ).
% mult_less_cancel_right2
tff(fact_726_convex__bound__le,axiom,
! [A: $tType] :
( linord6961819062388156250ring_1(A)
=> ! [X2: A,A3: A,Y: A,U: A,V: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_le
tff(fact_727_is__num_Ocases,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
=> ( ( A3 != one_one(A) )
=> ( ! [X3: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X3) )
=> ~ neg_numeral_is_num(A,X3) )
=> ~ ! [X3: A,Y4: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y4) )
=> ( neg_numeral_is_num(A,X3)
=> ~ neg_numeral_is_num(A,Y4) ) ) ) ) ) ) ).
% is_num.cases
tff(fact_728_is__num_Osimps,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
<=> ( ( A3 = one_one(A) )
| ? [X4: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X4) )
& neg_numeral_is_num(A,X4) )
| ? [X4: A,Y3: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y3) )
& neg_numeral_is_num(A,X4)
& neg_numeral_is_num(A,Y3) ) ) ) ) ).
% is_num.simps
tff(fact_729_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
<=> ( ( X2 != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_730_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_731_incr__lemma,axiom,
! [D3: int,Z2: int,X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),Z2))),one_one(int))),D3)))) ) ).
% incr_lemma
tff(fact_732_decr__lemma,axiom,
! [D3: int,X2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),Z2))),one_one(int))),D3))),Z2)) ) ).
% decr_lemma
tff(fact_733_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X2: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_734_frac__unique__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,A3: A] :
( ( archimedean_frac(A,X2) = A3 )
<=> ( pp(member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),A3),ring_1_Ints(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A))) ) ) ) ).
% frac_unique_iff
tff(fact_735_scaling__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [U: A,V: A,R: A,S: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),S))),V)) ) ) ) ) ).
% scaling_mono
tff(fact_736_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% divide_le_eq_1_neg
tff(fact_737_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% divide_le_eq_1_pos
tff(fact_738_div__by__1,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).
% div_by_1
tff(fact_739_div__self,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).
% div_self
tff(fact_740_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% zero_eq_1_divide_iff
tff(fact_741_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% one_divide_eq_0_iff
tff(fact_742_eq__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) )
<=> ( ( A3 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
tff(fact_743_divide__eq__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = one_one(A) )
<=> ( ( A3 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
tff(fact_744_divide__self__if,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( ( A3 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = zero_zero(A) ) )
& ( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ) ).
% divide_self_if
tff(fact_745_divide__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).
% divide_self
tff(fact_746_one__eq__divide__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) )
<=> ( ( B2 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
tff(fact_747_divide__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
<=> ( ( B2 != zero_zero(A) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
tff(fact_748_divide__minus1,axiom,
! [A: $tType] :
( field(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X2) ) ).
% divide_minus1
tff(fact_749_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% zero_le_divide_1_iff
tff(fact_750_divide__le__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% divide_le_0_1_iff
tff(fact_751_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% zero_less_divide_1_iff
tff(fact_752_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% less_divide_eq_1_pos
tff(fact_753_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% less_divide_eq_1_neg
tff(fact_754_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% divide_less_eq_1_pos
tff(fact_755_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% divide_less_eq_1_neg
tff(fact_756_divide__less__0__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% divide_less_0_1_iff
tff(fact_757_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).
% nonzero_divide_mult_cancel_right
tff(fact_758_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).
% nonzero_divide_mult_cancel_left
tff(fact_759_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% le_divide_eq_1_pos
tff(fact_760_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% le_divide_eq_1_neg
tff(fact_761_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).
% int_distrib(2)
tff(fact_762_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).
% int_distrib(1)
tff(fact_763_plus__int__code_I2_J,axiom,
! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).
% plus_int_code(2)
tff(fact_764_plus__int__code_I1_J,axiom,
! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).
% plus_int_code(1)
tff(fact_765_zero__integer_Orsp,axiom,
zero_zero(int) = zero_zero(int) ).
% zero_integer.rsp
tff(fact_766_add__divide__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).
% add_divide_distrib
tff(fact_767_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q3 ) ) ) ) ).
% int_div_pos_eq
tff(fact_768_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2) = Q3 ) ) ) ) ).
% int_div_neg_eq
tff(fact_769_split__zdiv,axiom,
! [P: fun(int,bool),N: int,K: int] :
( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)))
<=> ( ( ( K = zero_zero(int) )
=> pp(aa(int,bool,P,zero_zero(int))) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ! [I3: int,J2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),K))
& ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J2) ) )
=> pp(aa(int,bool,P,I3)) ) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ! [I3: int,J2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),zero_zero(int)))
& ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J2) ) )
=> pp(aa(int,bool,P,I3)) ) ) ) ) ).
% split_zdiv
tff(fact_770_frac__lt__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X2)),one_one(A))) ) ).
% frac_lt_1
tff(fact_771_frac__1__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),one_one(A))) = archimedean_frac(A,X2) ) ).
% frac_1_eq
tff(fact_772_right__inverse__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
<=> ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
tff(fact_773_less__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% less_divide_eq_1
tff(fact_774_divide__less__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) )
| ( A3 = zero_zero(A) ) ) ) ) ).
% divide_less_eq_1
tff(fact_775_divide__add__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X2),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).
% divide_add_eq_iff
tff(fact_776_add__divide__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Z2)),Y)),Z2) ) ) ) ).
% add_divide_eq_iff
tff(fact_777_add__num__frac,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X2: A] :
( ( Y != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X2),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).
% add_num_frac
tff(fact_778_add__frac__num,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,X2: A,Z2: A] :
( ( Y != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X2),Y)),Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).
% add_frac_num
tff(fact_779_add__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X2: A,W: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X2),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X2),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% add_frac_eq
tff(fact_780_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = A3 ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2)),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(1)
tff(fact_781_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = B2 ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(2)
tff(fact_782_less__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))))) ) ) ).
% less_half_sum
tff(fact_783_gt__half__sum,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2)) ) ) ).
% gt_half_sum
tff(fact_784_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( ( B2 != zero_zero(A) )
& ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).
% divide_eq_minus_1_iff
tff(fact_785_div__pos__geq,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).
% div_pos_geq
tff(fact_786_frac__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( ( archimedean_frac(A,X2) = X2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),one_one(A))) ) ) ) ).
% frac_eq
tff(fact_787_frac__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X2)),archimedean_frac(A,Y))),one_one(A)))
=> ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X2)),archimedean_frac(A,Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X2)),archimedean_frac(A,Y))),one_one(A)))
=> ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X2)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).
% frac_add
tff(fact_788_incr__mult__lemma,axiom,
! [D3: int,P: fun(int,bool),K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D3))
=> ( ! [X3: int] :
( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3))) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X: int] :
( pp(aa(int,bool,P,X))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K),D3)))) ) ) ) ) ).
% incr_mult_lemma
tff(fact_789_divide__le__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),one_one(A)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
| ( A3 = zero_zero(A) ) ) ) ) ).
% divide_le_eq_1
tff(fact_790_le__divide__eq__1,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) )
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% le_divide_eq_1
tff(fact_791_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X2),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).
% minus_divide_add_eq_iff
tff(fact_792_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = B2 ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(3)
tff(fact_793_frac__neg,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( ( pp(member(A,X2,ring_1_Ints(A)))
=> ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X2)) = zero_zero(A) ) )
& ( ~ pp(member(A,X2,ring_1_Ints(A)))
=> ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X2)) ) ) ) ) ).
% frac_neg
tff(fact_794_div__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).
% div_mult_self4
tff(fact_795_div__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).
% div_mult_self3
tff(fact_796_div__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).
% div_mult_self2
tff(fact_797_div__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).
% div_mult_self1
tff(fact_798_div__minus1__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).
% div_minus1_right
tff(fact_799_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% div_pos_neg_trivial
tff(fact_800_bits__div__by__1,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).
% bits_div_by_1
tff(fact_801_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))))) ) ).
% dividend_less_times_div
tff(fact_802_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)))) ) ).
% dividend_less_div_times
tff(fact_803_split__div,axiom,
! [P: fun(nat,bool),M: nat,N: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
<=> ( ( ( N = zero_zero(nat) )
=> pp(aa(nat,bool,P,zero_zero(nat))) )
& ( ( N != zero_zero(nat) )
=> ! [I3: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
=> ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J2) )
=> pp(aa(nat,bool,P,I3)) ) ) ) ) ) ).
% split_div
tff(fact_804_div__less__dividend,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M)) ) ) ).
% div_less_dividend
tff(fact_805_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = M )
<=> ( N = one_one(nat) ) ) ) ).
% div_eq_dividend_iff
tff(fact_806_div__add__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).
% div_add_self1
tff(fact_807_div__add__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).
% div_add_self2
tff(fact_808_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [I2: int,J3: int] :
( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J3)))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J3))
=> pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J3)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I2),J3)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% upto.pinduct
tff(fact_809_eucl__rel__int_Osimps,axiom,
! [A1: int,A22: int,A32: product_prod(int,int)] :
( eucl_rel_int(A1,A22,A32)
<=> ( ? [K4: int] :
( ( A1 = K4 )
& ( A22 = zero_zero(int) )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K4) ) )
| ? [L3: int,K4: int,Q5: int] :
( ( A1 = K4 )
& ( A22 = L3 )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
& ( L3 != zero_zero(int) )
& ( K4 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L3) ) )
| ? [R5: int,L3: int,K4: int,Q5: int] :
( ( A1 = K4 )
& ( A22 = L3 )
& ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5) )
& ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L3) )
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L3)))
& ( K4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L3)),R5) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_810_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod(int,int)] :
( eucl_rel_int(A1,A22,A32)
=> ( ( ( A22 = zero_zero(int) )
=> ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
=> ( ! [Q2: int] :
( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int)) )
=> ( ( A22 != zero_zero(int) )
=> ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q2),A22) ) ) )
=> ~ ! [R2: int,Q2: int] :
( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2) )
=> ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),A22) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),A22)))
=> ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),A22)),R2) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_811_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
<=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),L)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
=> ( Q3 = zero_zero(int) ) ) ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_812_power_Opower__eq__if,axiom,
! [A: $tType,M: nat,One: A,Times: fun(A,fun(A,A)),P3: A] :
( ( ( M = zero_zero(nat) )
=> ( power2(A,One,Times,P3,M) = One ) )
& ( ( M != zero_zero(nat) )
=> ( power2(A,One,Times,P3,M) = aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ).
% power.power_eq_if
tff(fact_813_split__neg__lemma,axiom,
! [K: int,P: fun(int,fun(int,bool)),N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K)))
<=> ! [I3: int,J2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),zero_zero(int)))
& ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J2) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J2)) ) ) ) ).
% split_neg_lemma
tff(fact_814_split__pos__lemma,axiom,
! [K: int,P: fun(int,fun(int,bool)),N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K)))
<=> ! [I3: int,J2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),K))
& ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J2) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J2)) ) ) ) ).
% split_pos_lemma
tff(fact_815_mod__add__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_add_self2
tff(fact_816_mod__add__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_add_self1
tff(fact_817_mod__by__1,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),one_one(A)) = zero_zero(A) ) ).
% mod_by_1
tff(fact_818_bits__mod__by__1,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),one_one(A)) = zero_zero(A) ) ).
% bits_mod_by_1
tff(fact_819_mod__mult__self1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_mult_self1
tff(fact_820_mod__mult__self2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_mult_self2
tff(fact_821_mod__mult__self3,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_mult_self3
tff(fact_822_mod__mult__self4,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A3)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_mult_self4
tff(fact_823_mod__minus1__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).
% mod_minus1_right
tff(fact_824_mod__int__unique,axiom,
! [K: int,L: int,Q3: int,R: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = R ) ) ).
% mod_int_unique
tff(fact_825_unique__quotient,axiom,
! [A3: int,B2: int,Q3: int,R: int,Q4: int,R3: int] :
( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R3))
=> ( Q3 = Q4 ) ) ) ).
% unique_quotient
tff(fact_826_unique__remainder,axiom,
! [A3: int,B2: int,Q3: int,R: int,Q4: int,R3: int] :
( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R3))
=> ( R = R3 ) ) ) ).
% unique_remainder
tff(fact_827_full__exhaustive__int_H_Ocases,axiom,
! [X2: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
~ ! [F3: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: int,I2: int] : X2 != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int)),aa(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int))),product_Pair(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(int,int)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2)) ).
% full_exhaustive_int'.cases
tff(fact_828_exhaustive__int_H_Ocases,axiom,
! [X2: product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
~ ! [F3: fun(int,option(product_prod(bool,list(code_term)))),D2: int,I2: int] : X2 != aa(product_prod(int,int),product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int)),aa(fun(int,option(product_prod(bool,list(code_term)))),fun(product_prod(int,int),product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))),product_Pair(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2)) ).
% exhaustive_int'.cases
tff(fact_829_small__lazy_H_Ocases,axiom,
! [X2: product_prod(int,int)] :
~ ! [D2: int,I2: int] : X2 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2) ).
% small_lazy'.cases
tff(fact_830_eucl__rel__int,axiom,
! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))) ).
% eucl_rel_int
tff(fact_831_mod__add__right__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).
% mod_add_right_eq
tff(fact_832_mod__add__left__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).
% mod_add_left_eq
tff(fact_833_mod__add__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,A6: A,B2: A,B4: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A6),C2) )
=> ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B4),C2) )
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B4)),C2) ) ) ) ) ).
% mod_add_cong
tff(fact_834_mod__add__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).
% mod_add_eq
tff(fact_835_eucl__rel__int__by0,axiom,
! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).
% eucl_rel_int_by0
tff(fact_836_div__int__unique,axiom,
! [K: int,L: int,Q3: int,R: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q3 ) ) ).
% div_int_unique
tff(fact_837_mod__eqE,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2) )
=> ~ ! [D2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ).
% mod_eqE
tff(fact_838_div__add1__eq,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2)) ) ).
% div_add1_eq
tff(fact_839_mult__div__mod__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) = A3 ) ).
% mult_div_mod_eq
tff(fact_840_mod__mult__div__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))) = A3 ) ).
% mod_mult_div_eq
tff(fact_841_mod__div__mult__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)) = A3 ) ).
% mod_div_mult_eq
tff(fact_842_div__mult__mod__eq,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) = A3 ) ).
% div_mult_mod_eq
tff(fact_843_mod__div__decomp,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) ) ).
% mod_div_decomp
tff(fact_844_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).
% cancel_div_mod_rules(1)
tff(fact_845_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).
% cancel_div_mod_rules(2)
tff(fact_846_div__mult1__eq,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2)) ) ).
% div_mult1_eq
tff(fact_847_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L != zero_zero(int) )
=> ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).
% eucl_rel_int_dividesI
tff(fact_848_div__mod__decomp__int,axiom,
! [A4: int,N: int] : A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),N)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A4),N)) ).
% div_mod_decomp_int
tff(fact_849_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int)))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).
% mod_pos_neg_trivial
tff(fact_850_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_851_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),B2))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) = R ) ) ) ) ).
% int_mod_pos_eq
tff(fact_852_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) = R ) ) ) ) ).
% int_mod_neg_eq
tff(fact_853_split__zmod,axiom,
! [P: fun(int,bool),N: int,K: int] :
( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K)))
<=> ( ( ( K = zero_zero(int) )
=> pp(aa(int,bool,P,N)) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ! [I3: int,J2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J2),K))
& ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J2) ) )
=> pp(aa(int,bool,P,J2)) ) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ! [I3: int,J2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J2),zero_zero(int)))
& ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J2) ) )
=> pp(aa(int,bool,P,J2)) ) ) ) ) ).
% split_zmod
tff(fact_854_zmod__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),C2))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2)) ) ) ).
% zmod_zmult2_eq
tff(fact_855_zminus1__lemma,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> ( ( B2 != zero_zero(int) )
=> eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int)),aa(int,int,uminus_uminus(int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).
% zminus1_lemma
tff(fact_856_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q3: int] :
( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),L)))
=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R)) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_857_verit__le__mono__div__int,axiom,
! [A4: int,B5: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A4),B5))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B5),N)),zero_zero(int)),one_one(int),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N))) ) ) ).
% verit_le_mono_div_int
tff(fact_858_bezw__0,axiom,
! [X2: nat] : bezw(X2,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).
% bezw_0
tff(fact_859_verit__le__mono__div,axiom,
! [A4: nat,B5: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A4),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B5),N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),N))) ) ) ).
% verit_le_mono_div
tff(fact_860_zero__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X2)) ) ) ).
% zero_le_ceiling
tff(fact_861_ceiling__less__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X2)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).
% ceiling_less_zero
tff(fact_862_floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X2)),archimedean_frac(A,Y))),one_one(A)))
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),archim6421214686448440834_floor(A,Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X2)),archimedean_frac(A,Y))),one_one(A)))
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).
% floor_add
tff(fact_863_cpmi,axiom,
! [D4: int,P: fun(int,bool),P2: fun(int,bool),B5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( ? [Z5: int] :
! [X3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z5))
=> ( pp(aa(int,bool,P,X3))
<=> pp(aa(int,bool,P2,X3)) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,B5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
=> ( ! [X3: int,K3: int] :
( pp(aa(int,bool,P2,X3))
<=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D4)))) )
=> ( ? [X_1: int] : pp(aa(int,bool,P,X_1))
<=> ( ? [X4: int] :
( pp(member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& pp(aa(int,bool,P2,X4)) )
| ? [X4: int] :
( pp(member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& ? [Xa2: int] :
( pp(member(int,Xa2,B5))
& pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa2),X4))) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_864_cppi,axiom,
! [D4: int,P: fun(int,bool),P2: fun(int,bool),A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( ? [Z5: int] :
! [X3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),X3))
=> ( pp(aa(int,bool,P,X3))
<=> pp(aa(int,bool,P2,X3)) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A4))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
=> ( ! [X3: int,K3: int] :
( pp(aa(int,bool,P2,X3))
<=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D4)))) )
=> ( ? [X_1: int] : pp(aa(int,bool,P,X_1))
<=> ( ? [X4: int] :
( pp(member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& pp(aa(int,bool,P2,X4)) )
| ? [X4: int] :
( pp(member(int,X4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& ? [Xa2: int] :
( pp(member(int,Xa2,A4))
& pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X4))) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_865_bset_I6_J,axiom,
! [D4: int,B5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),T2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),T2)) ) ) ) ).
% bset(6)
tff(fact_866_bset_I8_J,axiom,
! [D4: int,T2: int,B5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int)),B5))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4))) ) ) ) ) ).
% bset(8)
tff(fact_867_aset_I6_J,axiom,
! [D4: int,T2: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int)),A4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),T2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),T2)) ) ) ) ) ).
% aset(6)
tff(fact_868_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,L: A,U: A] :
( pp(member(A,I,set_or1337092689740270186AtMost(A,L,U)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),U)) ) ) ) ).
% atLeastAtMost_iff
tff(fact_869_Icc__eq__Icc,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,H: A,L4: A,H3: A] :
( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L4,H3) )
<=> ( ( ( L = L4 )
& ( H = H3 ) )
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L4),H3)) ) ) ) ) ).
% Icc_eq_Icc
tff(fact_870_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
<=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_871_floor__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).
% floor_one
tff(fact_872_ceiling__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
% ceiling_one
tff(fact_873_floor__add2,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Y: A] :
( ( pp(member(A,X2,ring_1_Ints(A)))
| pp(member(A,Y,ring_1_Ints(A))) )
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),archim6421214686448440834_floor(A,Y)) ) ) ) ).
% floor_add2
tff(fact_874_zero__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X2)) ) ) ).
% zero_less_floor
tff(fact_875_floor__le__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X2)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),one_one(A))) ) ) ).
% floor_le_zero
tff(fact_876_one__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X2)) ) ) ).
% one_le_floor
tff(fact_877_floor__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X2)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),one_one(A))) ) ) ).
% floor_less_one
tff(fact_878_ceiling__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X2)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),one_one(A))) ) ) ).
% ceiling_le_one
tff(fact_879_one__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X2)) ) ) ).
% one_less_ceiling
tff(fact_880_ceiling__add__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X2)),one_one(int)) ) ).
% ceiling_add_one
tff(fact_881_floor__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X2)),one_one(int)) ) ).
% floor_diff_one
tff(fact_882_ceiling__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X2)),one_one(int)) ) ).
% ceiling_diff_one
tff(fact_883_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
<=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3))
& ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D3)) ) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_884_nat__mod__eq__iff,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),N) )
<=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).
% nat_mod_eq_iff
tff(fact_885_le__floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)))) ) ).
% le_floor_add
tff(fact_886_bset_I1_J,axiom,
! [D4: int,B5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,B5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,B5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X3))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X))
& pp(aa(int,bool,Q,X)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)))
& pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4))) ) ) ) ) ) ).
% bset(1)
tff(fact_887_bset_I2_J,axiom,
! [D4: int,B5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,B5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,B5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X3))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) )
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X))
| pp(aa(int,bool,Q,X)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)))
| pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4))) ) ) ) ) ) ).
% bset(2)
tff(fact_888_aset_I1_J,axiom,
! [D4: int,A4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A4))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A4))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X3))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X))
& pp(aa(int,bool,Q,X)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)))
& pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4))) ) ) ) ) ) ).
% aset(1)
tff(fact_889_aset_I2_J,axiom,
! [D4: int,A4: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A4))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
=> ( ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A4))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,Q,X3))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( ( pp(aa(int,bool,P,X))
| pp(aa(int,bool,Q,X)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)))
| pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4))) ) ) ) ) ) ).
% aset(2)
tff(fact_890_nat__mod__eq__lemma,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),N) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X2))
=> ? [Q2: nat] : X2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ) ) ).
% nat_mod_eq_lemma
tff(fact_891_mod__eq__nat2E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q3) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ~ ! [S4: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S4)) ) ) ).
% mod_eq_nat2E
tff(fact_892_mod__eq__nat1E,axiom,
! [M: nat,Q3: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q3) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ~ ! [S4: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S4)) ) ) ).
% mod_eq_nat1E
tff(fact_893_ceiling__add__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X2)),archimedean_ceiling(A,Y)))) ) ).
% ceiling_add_le
tff(fact_894_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),Q3))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) ).
% mod_mult2_eq
tff(fact_895_div__mod__decomp,axiom,
! [A4: nat,N: nat] : A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),N)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A4),N)) ).
% div_mod_decomp
tff(fact_896_one__add__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),one_one(A))) ) ).
% one_add_floor
tff(fact_897_split__mod,axiom,
! [P: fun(nat,bool),M: nat,N: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)))
<=> ( ( ( N = zero_zero(nat) )
=> pp(aa(nat,bool,P,M)) )
& ( ( N != zero_zero(nat) )
=> ! [I3: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
=> ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J2) )
=> pp(aa(nat,bool,P,J2)) ) ) ) ) ) ).
% split_mod
tff(fact_898_bset_I3_J,axiom,
! [D4: int,T2: int,B5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int)),B5))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( X = T2 )
=> ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4) = T2 ) ) ) ) ) ).
% bset(3)
tff(fact_899_bset_I4_J,axiom,
! [D4: int,T2: int,B5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,T2,B5))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ( X != T2 )
=> ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4) != T2 ) ) ) ) ) ).
% bset(4)
tff(fact_900_bset_I5_J,axiom,
! [D4: int,B5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),T2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),T2)) ) ) ) ).
% bset(5)
tff(fact_901_bset_I7_J,axiom,
! [D4: int,T2: int,B5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,T2,B5))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4))) ) ) ) ) ).
% bset(7)
tff(fact_902_aset_I3_J,axiom,
! [D4: int,T2: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int)),A4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( ( X = T2 )
=> ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4) = T2 ) ) ) ) ) ).
% aset(3)
tff(fact_903_aset_I4_J,axiom,
! [D4: int,T2: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,T2,A4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( ( X != T2 )
=> ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4) != T2 ) ) ) ) ) ).
% aset(4)
tff(fact_904_aset_I5_J,axiom,
! [D4: int,T2: int,A4: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,T2,A4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),T2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),T2)) ) ) ) ) ).
% aset(5)
tff(fact_905_aset_I7_J,axiom,
! [D4: int,A4: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4))) ) ) ) ).
% aset(7)
tff(fact_906_aset_I8_J,axiom,
! [D4: int,A4: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4))) ) ) ) ).
% aset(8)
tff(fact_907_ceiling__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)),P3)) ) ) ).
% ceiling_divide_lower
tff(fact_908_floor__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q3: A,P3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3))) ) ) ).
% floor_divide_upper
tff(fact_909_power__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ) ) ) ).
% power_decreasing_iff
tff(fact_910_divmod__nat__def,axiom,
! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) ).
% divmod_nat_def
tff(fact_911_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X2: B,Y: B,A3: A,B2: A,C2: A] :
( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X2),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).
% distrib_left_NO_MATCH
tff(fact_912_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X2: B,Y: B,C2: A,A3: A,B2: A] :
( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X2),Y),C2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% distrib_right_NO_MATCH
tff(fact_913_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))),M) = one_one(nat) ) ) ).
% Suc_times_mod_eq
tff(fact_914_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel(A)
& field(B) )
=> ! [X2: B,B2: A,A3: A] :
( nO_MATCH(B,A,X2,B2)
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ) ).
% div_add_self1_no_field
tff(fact_915_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel(A)
& field(B) )
=> ! [X2: B,B2: A,A3: A] :
( nO_MATCH(B,A,X2,B2)
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ) ).
% div_add_self2_no_field
tff(fact_916_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).
% power_one
tff(fact_917_add__Suc__right,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).
% add_Suc_right
tff(fact_918_power__one__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),one_one(nat)) = A3 ) ).
% power_one_right
tff(fact_919_power__inject__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) )
<=> ( M = N ) ) ) ) ).
% power_inject_exp
tff(fact_920_mult__Suc__right,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).
% mult_Suc_right
tff(fact_921_diff__Suc__1,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),one_one(nat)) = N ).
% diff_Suc_1
tff(fact_922_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( aa(int,A,ring_1_of_int(A),Z2) = one_one(A) )
<=> ( Z2 = one_one(int) ) ) ) ).
% of_int_eq_1_iff
tff(fact_923_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).
% of_int_1
tff(fact_924_of__int__add,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_add
tff(fact_925_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X2: nat,Y: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Y)) ) ) ) ).
% power_strict_increasing_iff
tff(fact_926_left__minus__one__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A3)) = A3 ) ).
% left_minus_one_mult_self
tff(fact_927_minus__one__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).
% minus_one_mult_self
tff(fact_928_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).
% diff_Suc_diff_eq2
tff(fact_929_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).
% diff_Suc_diff_eq1
tff(fact_930_Suc__diff,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
=> ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).
% Suc_diff
tff(fact_931_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),M))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).
% Suc_mod_mult_self4
tff(fact_932_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).
% Suc_mod_mult_self3
tff(fact_933_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).
% Suc_mod_mult_self2
tff(fact_934_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).
% Suc_mod_mult_self1
tff(fact_935_ceiling__add__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Z2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X2)),Z2) ) ).
% ceiling_add_of_int
tff(fact_936_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ) ).
% power_strict_decreasing_iff
tff(fact_937_power__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X2: nat,Y: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),X2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Y)) ) ) ) ).
% power_increasing_iff
tff(fact_938_Suc__diff__1,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N ) ) ).
% Suc_diff_1
tff(fact_939_of__int__1__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2)) ) ) ).
% of_int_1_le_iff
tff(fact_940_of__int__le__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),one_one(int))) ) ) ).
% of_int_le_1_iff
tff(fact_941_of__int__less__1__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),one_one(int))) ) ) ).
% of_int_less_1_iff
tff(fact_942_of__int__1__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ) ).
% of_int_1_less_iff
tff(fact_943_power__gt1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)))) ) ) ).
% power_gt1
tff(fact_944_add__Suc__shift,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).
% add_Suc_shift
tff(fact_945_add__Suc,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).
% add_Suc
tff(fact_946_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A3: nat] :
( ( A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A3) )
=> ( aa(nat,nat,suc,A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A3)) ) ) ).
% nat_arith.suc1
tff(fact_947_power__Suc__le__self,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3)) ) ) ) ).
% power_Suc_le_self
tff(fact_948_power__Suc__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).
% power_Suc_less_one
tff(fact_949_power__minus_H,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: A,N: nat] :
( nO_MATCH(A,A,one_one(A),X2)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)) ) ) ) ).
% power_minus'
tff(fact_950_one__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).
% one_le_power
tff(fact_951_left__right__inverse__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X2: A,Y: A,N: nat] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y) = one_one(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).
% left_right_inverse_power
tff(fact_952_power__0,axiom,
! [A: $tType] :
( power(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),zero_zero(nat)) = one_one(A) ) ).
% power_0
tff(fact_953_power__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_one_over
tff(fact_954_power__add,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_add
tff(fact_955_nat__compl__induct_H,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N3))
=> pp(aa(nat,bool,P,Nn)) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
% nat_compl_induct'
tff(fact_956_nat__compl__induct,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Nn),N3))
=> pp(aa(nat,bool,P,Nn)) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
% nat_compl_induct
tff(fact_957_one__is__add,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
<=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).
% one_is_add
tff(fact_958_add__is__1,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).
% add_is_1
tff(fact_959_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),aa(nat,nat,suc,A3))) )
<=> ( B2 = A3 ) ) ).
% nat_in_between_eq(2)
tff(fact_960_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(nat,nat,suc,A3))) )
<=> ( B2 = aa(nat,nat,suc,A3) ) ) ).
% nat_in_between_eq(1)
tff(fact_961_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).
% less_imp_Suc_add
tff(fact_962_less__iff__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
<=> ? [K4: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K4)) ) ).
% less_iff_Suc_add
tff(fact_963_less__add__Suc2,axiom,
! [I: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)))) ).
% less_add_Suc2
tff(fact_964_less__add__Suc1,axiom,
! [I: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)))) ).
% less_add_Suc1
tff(fact_965_less__natE,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ~ ! [Q2: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ) ).
% less_natE
tff(fact_966_Suc__to__right,axiom,
! [N: nat,M: nat] :
( ( aa(nat,nat,suc,N) = M )
=> ( N = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).
% Suc_to_right
tff(fact_967_One__nat__def,axiom,
one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).
% One_nat_def
tff(fact_968_mult__Suc,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).
% mult_Suc
tff(fact_969_Suc__eq__plus1__left,axiom,
! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).
% Suc_eq_plus1_left
tff(fact_970_plus__1__eq__Suc,axiom,
aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).
% plus_1_eq_Suc
tff(fact_971_Suc__eq__plus1,axiom,
! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).
% Suc_eq_plus1
tff(fact_972_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N) ).
% diff_Suc_eq_diff_pred
tff(fact_973_power__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))) ) ) ) ).
% power_le_one
tff(fact_974_power__less__power__Suc,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).
% power_less_power_Suc
tff(fact_975_power__gt1__lemma,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).
% power_gt1_lemma
tff(fact_976_power__0__left,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).
% power_0_left
tff(fact_977_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N5: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N5))) ) ) ) ).
% power_strict_increasing
tff(fact_978_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).
% power_less_imp_less_exp
tff(fact_979_power__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% power_minus
tff(fact_980_power__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N5: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N5))) ) ) ) ).
% power_increasing
tff(fact_981_floor__add__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),Z2) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(int,A,ring_1_of_int(A),Z2))) ) ).
% floor_add_int
tff(fact_982_int__add__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X2: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),archim6421214686448440834_floor(A,X2)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),X2)) ) ).
% int_add_floor
tff(fact_983_nat__induct__non__zero,axiom,
! [N: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( pp(aa(nat,bool,P,one_one(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) )
=> pp(aa(nat,bool,P,N)) ) ) ) ).
% nat_induct_non_zero
tff(fact_984_of__int__leD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X2))
=> ( ( N = zero_zero(int) )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X2)) ) ) ) ).
% of_int_leD
tff(fact_985_of__int__lessD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X2))
=> ( ( N = zero_zero(int) )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X2)) ) ) ) ).
% of_int_lessD
tff(fact_986_floor__exists,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X2: A] :
? [Z3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int))))) ) ) ).
% floor_exists
tff(fact_987_floor__exists1,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X2: A] :
? [X3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int)))))
& ! [Y5: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y5)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y5),one_one(int))))) )
=> ( Y5 = X3 ) ) ) ) ).
% floor_exists1
tff(fact_988_power__Suc__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).
% power_Suc_less
tff(fact_989_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N5: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_strict_decreasing
tff(fact_990_power__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N5: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_decreasing
tff(fact_991_power__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
% power_le_imp_le_exp
tff(fact_992_self__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).
% self_le_power
tff(fact_993_one__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).
% one_less_power
tff(fact_994_nz__le__conv__less,axiom,
! [K: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),aa(nat,nat,suc,zero_zero(nat)))),M)) ) ) ).
% nz_le_conv_less
tff(fact_995_ceiling__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( ( ( X2 = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X2)) )
=> ( archimedean_ceiling(A,X2) = archim6421214686448440834_floor(A,X2) ) )
& ( ( X2 != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X2)) )
=> ( archimedean_ceiling(A,X2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),one_one(int)) ) ) ) ) ).
% ceiling_altdef
tff(fact_996_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).
% Suc_diff_eq_diff_pred
tff(fact_997_Suc__pred_H,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).
% Suc_pred'
tff(fact_998_add__eq__if,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
& ( ( M != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).
% add_eq_if
tff(fact_999_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),M))
=> ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).
% Suc_n_minus_m_eq
tff(fact_1000_floor__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))))
=> ( archim6421214686448440834_floor(A,X2) = Z2 ) ) ) ) ).
% floor_unique
tff(fact_1001_floor__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,A3: int] :
( ( archim6421214686448440834_floor(A,X2) = A3 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A)))) ) ) ) ).
% floor_eq_iff
tff(fact_1002_floor__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T2: A] :
( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T2)))
<=> ! [I3: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I3)),T2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A)))) )
=> pp(aa(int,bool,P,I3)) ) ) ) ).
% floor_split
tff(fact_1003_ceiling__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X2))),one_one(A))),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X2)))) ) ) ).
% ceiling_correct
tff(fact_1004_ceiling__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(int,A,ring_1_of_int(A),Z2)))
=> ( archimedean_ceiling(A,X2) = Z2 ) ) ) ) ).
% ceiling_unique
tff(fact_1005_ceiling__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,A3: int] :
( ( archimedean_ceiling(A,X2) = A3 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).
% ceiling_eq_iff
tff(fact_1006_ceiling__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T2: A] :
( pp(aa(int,bool,P,archimedean_ceiling(A,T2)))
<=> ! [I3: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))),T2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I3))) )
=> pp(aa(int,bool,P,I3)) ) ) ) ).
% ceiling_split
tff(fact_1007_less__floor__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archim6421214686448440834_floor(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X2)) ) ) ).
% less_floor_iff
tff(fact_1008_floor__le__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X2)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).
% floor_le_iff
tff(fact_1009_ceiling__less__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X2)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A)))) ) ) ).
% ceiling_less_iff
tff(fact_1010_le__ceiling__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archimedean_ceiling(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z2)),one_one(A))),X2)) ) ) ).
% le_ceiling_iff
tff(fact_1011_floor__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X2))),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X2)),one_one(int))))) ) ) ).
% floor_correct
tff(fact_1012_power__diff__power__eq,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,N: nat,M: nat] :
( ( A3 != zero_zero(A) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ) ) ).
% power_diff_power_eq
tff(fact_1013_power__eq__if,axiom,
! [A: $tType] :
( power(A)
=> ! [M: nat,P3: A] :
( ( ( M = zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = one_one(A) ) )
& ( ( M != zero_zero(nat) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).
% power_eq_if
tff(fact_1014_power__minus__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).
% power_minus_mult
tff(fact_1015_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1016_option_Osize__gen_I2_J,axiom,
! [A: $tType,X2: fun(A,nat),X22: A] : aa(option(A),nat,size_option(A,X2),aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X2,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% option.size_gen(2)
tff(fact_1017_Heap_Osize__gen,axiom,
! [A: $tType,Xa: fun(A,nat),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),nat,heap_Time_size_Heap(A,Xa),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).
% Heap.size_gen
tff(fact_1018_num_Osize__gen_I2_J,axiom,
! [X22: num] : size_num(aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% num.size_gen(2)
tff(fact_1019_normalize__negative,axiom,
! [Q3: int,P3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q3),zero_zero(int)))
=> ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).
% normalize_negative
tff(fact_1020_option_Osize__gen_I1_J,axiom,
! [A: $tType,X2: fun(A,nat)] : aa(option(A),nat,size_option(A,X2),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).
% option.size_gen(1)
tff(fact_1021_Heap_Osize_I2_J,axiom,
! [A: $tType,X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).
% Heap.size(2)
tff(fact_1022_num_Osize_I5_J,axiom,
! [X22: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% num.size(5)
tff(fact_1023_option_Osize_I3_J,axiom,
! [A: $tType] : aa(option(A),nat,size_size(option(A)),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).
% option.size(3)
tff(fact_1024_normalize__denom__zero,axiom,
! [P3: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).
% normalize_denom_zero
tff(fact_1025_option_Osize__neq,axiom,
! [A: $tType,X2: option(A)] : aa(option(A),nat,size_size(option(A)),X2) != zero_zero(nat) ).
% option.size_neq
tff(fact_1026_Heap_Osize__neq,axiom,
! [A: $tType,X2: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),X2) != zero_zero(nat) ).
% Heap.size_neq
tff(fact_1027_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),aa(fun(option(B),nat),fun(fun(option(A),option(B)),fun(option(A),nat)),comp(option(B),nat,option(A)),size_option(B,F)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G)) = size_option(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% option.size_gen_o_map
tff(fact_1028_normalize__denom__pos,axiom,
! [R: product_prod(int,int),P3: int,Q3: int] :
( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3)) ) ).
% normalize_denom_pos
tff(fact_1029_normalize__crossproduct,axiom,
! [Q3: int,S: int,P3: int,R: int] :
( ( Q3 != zero_zero(int) )
=> ( ( S != zero_zero(int) )
=> ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R),S)) )
=> ( aa(int,int,aa(int,fun(int,int),times_times(int),P3),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q3) ) ) ) ) ).
% normalize_crossproduct
tff(fact_1030_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).
% option.size(4)
tff(fact_1031_divmod__nat__if,axiom,
! [N: nat,M: nat] :
( ( ( ( N = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
=> ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) )
& ( ~ ( ( N = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
=> ( divmod_nat(M,N) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bi(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).
% divmod_nat_if
tff(fact_1032_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M: nat,N: nat,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(nat,nat,suc,N))) ) ) ) ).
% sum_gp_multiplied
tff(fact_1033_pair__lessI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
=> pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_less)) ) ) ).
% pair_lessI2
tff(fact_1034_pair__less__iff1,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Z2)),fun_pair_less))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),Z2)) ) ).
% pair_less_iff1
tff(fact_1035_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N))),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).
% Suc_times_numeral_mod_eq
tff(fact_1036_one__int_Otransfer,axiom,
pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),one_one(int))) ).
% one_int.transfer
tff(fact_1037_less__than__iff,axiom,
! [X2: nat,Y: nat] :
( pp(member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y),less_than))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Y)) ) ).
% less_than_iff
tff(fact_1038_one__int__def,axiom,
one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))) ).
% one_int_def
tff(fact_1039_sum__telescope_H_H,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M: nat,N: nat,F: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,N)),aa(nat,A,F,M)) ) ) ) ).
% sum_telescope''
tff(fact_1040_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).
% case_prod_conv
tff(fact_1041_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),product_curry(A,B,C,F)) = F ).
% case_prod_curry
tff(fact_1042_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : product_curry(A,B,C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)) = F ).
% curry_case_prod
tff(fact_1043_distrib__right__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).
% distrib_right_numeral
tff(fact_1044_distrib__left__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& semiring(A) )
=> ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).
% distrib_left_numeral
tff(fact_1045_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).
% add_neg_numeral_simps(3)
tff(fact_1046_add__numeral__left,axiom,
! [A: $tType] :
( numeral(A)
=> ! [V: num,W: num,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z2) ) ).
% add_numeral_left
tff(fact_1047_numeral__plus__numeral,axiom,
! [A: $tType] :
( numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).
% numeral_plus_numeral
tff(fact_1048_semiring__norm_I168_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y) ) ).
% semiring_norm(168)
tff(fact_1049_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).
% diff_numeral_simps(3)
tff(fact_1050_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).
% diff_numeral_simps(2)
tff(fact_1051_ceiling__add__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X2)),aa(num,int,numeral_numeral(int),V)) ) ).
% ceiling_add_numeral
tff(fact_1052_power__add__numeral,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).
% power_add_numeral
tff(fact_1053_power__add__numeral2,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).
% power_add_numeral2
tff(fact_1054_numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X2)) ) ) ).
% numeral_less_floor
tff(fact_1055_floor__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X2)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).
% floor_le_numeral
tff(fact_1056_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [N: nat,M: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).
% sum.cl_ivl_Suc
tff(fact_1057_ceiling__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X2)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).
% ceiling_less_numeral
tff(fact_1058_numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X2)) ) ) ).
% numeral_le_ceiling
tff(fact_1059_neg__numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X2)) ) ) ).
% neg_numeral_less_floor
tff(fact_1060_floor__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).
% floor_le_neg_numeral
tff(fact_1061_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).
% ceiling_less_neg_numeral
tff(fact_1062_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X2: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X2)) ) ) ).
% neg_numeral_le_ceiling
tff(fact_1063_nested__case__prod__simp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),X2: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F),X2),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_bk(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y)),X2) ).
% nested_case_prod_simp
tff(fact_1064_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(C,D,H,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod)) = aa(product_prod(A,B),D,aa(fun(A,fun(B,D)),fun(product_prod(A,B),D),product_case_prod(A,B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_bl(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),H),F)),Prod) ).
% prod.case_distrib
tff(fact_1065_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_1066_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.atLeastAtMost_shift_bounds
tff(fact_1067_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P3: product_prod(A,B)] :
( ! [X3: A,Y4: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4) = Q3 )
=> ( aa(B,C,aa(A,fun(B,C),F,X3),Y4) = aa(B,C,aa(A,fun(B,C),G,X3),Y4) ) )
=> ( ( P3 = Q3 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q3) ) ) ) ).
% split_cong
tff(fact_1068_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(B,C)),X1: A,X22: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = aa(B,C,aa(A,fun(B,C),F,X1),X22) ).
% old.prod.case
tff(fact_1069_uminus__int_Oabs__eq,axiom,
! [X2: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat)))),X2)) ).
% uminus_int.abs_eq
tff(fact_1070_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P3) = P3 ).
% case_prod_Pair_iden
tff(fact_1071_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.shift_bounds_cl_Suc_ivl
tff(fact_1072_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.shift_bounds_cl_nat_ivl
tff(fact_1073_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
( ! [X3: A,Y4: B] : aa(B,C,aa(A,fun(B,C),F,X3),Y4) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) = G ) ) ).
% cond_case_prod_eta
tff(fact_1074_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_bp(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).
% case_prod_eta
tff(fact_1075_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: fun(A,bool),P: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
( pp(aa(A,bool,Q,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P),Z2)))
=> ~ ! [X3: B,Y4: C] :
( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y4) )
=> ~ pp(aa(A,bool,Q,aa(C,A,aa(B,fun(C,A),P,X3),Y4))) ) ) ).
% case_prodE2
tff(fact_1076_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y4: nat] : Z2 != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),Y4)) ).
% eq_Abs_Integ
tff(fact_1077_one__le__numeral,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).
% one_le_numeral
tff(fact_1078_not__numeral__less__one,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).
% not_numeral_less_one
tff(fact_1079_one__plus__numeral__commute,axiom,
! [A: $tType] :
( numeral(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X2)),one_one(A)) ) ).
% one_plus_numeral_commute
tff(fact_1080_numeral__Bit0,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).
% numeral_Bit0
tff(fact_1081_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).
% one_neq_neg_numeral
tff(fact_1082_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% numeral_neq_neg_one
tff(fact_1083_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : uncurry(A,B,C,F) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) ).
% uncurry_def
tff(fact_1084_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] : produc5280177257484947105e_prod(A,B,C) = product_case_prod(A,B,C) ).
% internal_case_prod_def
tff(fact_1085_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).
% sum.atLeastAtMost_rev
tff(fact_1086_numeral__code_I2_J,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).
% numeral_code(2)
tff(fact_1087_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(nat,A),K: nat] :
( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).
% sum_shift_lb_Suc0_0
tff(fact_1088_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% sum.atLeast0_atMost_Suc
tff(fact_1089_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).
% sum.atLeast0_atMost_Suc_shift
tff(fact_1090_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).
% sum.atLeast_Suc_atMost
tff(fact_1091_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% sum.nat_ivl_Suc'
tff(fact_1092_neg__numeral__le__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).
% neg_numeral_le_one
tff(fact_1093_neg__one__le__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).
% neg_one_le_numeral
tff(fact_1094_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% neg_numeral_le_neg_one
tff(fact_1095_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% not_numeral_le_neg_one
tff(fact_1096_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).
% not_one_le_neg_numeral
tff(fact_1097_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).
% not_neg_one_less_neg_numeral
tff(fact_1098_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).
% not_one_less_neg_numeral
tff(fact_1099_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).
% not_numeral_less_neg_one
tff(fact_1100_neg__one__less__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).
% neg_one_less_numeral
tff(fact_1101_neg__numeral__less__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).
% neg_numeral_less_one
tff(fact_1102_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_1103_sum__Suc__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M: nat,N: nat,F: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_br(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,aa(nat,nat,suc,N))),aa(nat,A,F,M)) ) ) ) ).
% sum_Suc_diff
tff(fact_1104_sum__power__add,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,M: nat,I4: set(nat)] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bs(A,fun(nat,fun(nat,A)),X2),M)),I4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),I4)) ) ).
% sum_power_add
tff(fact_1105_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_bt(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.atLeast_atMost_pred_shift
tff(fact_1106_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_bu(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,zero_zero(A)) ) ).
% sum_atLeastAtMost_code
tff(fact_1107_sum_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A),P3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).
% sum.ub_add_nat
tff(fact_1108_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).
% sum.atLeastAtMost_shift_0
tff(fact_1109_zero__int__def,axiom,
zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))) ).
% zero_int_def
tff(fact_1110_pair__lessI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_less)) ) ).
% pair_lessI1
tff(fact_1111_zero__int_Otransfer,axiom,
pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),zero_zero(int))) ).
% zero_int.transfer
tff(fact_1112_sum__natinterval__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M: nat,N: nat,F: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,M)),aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).
% sum_natinterval_diff
tff(fact_1113_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( linordered_idom(B)
=> ! [I4: set(A),X2: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
( ! [I2: A] :
( pp(member(A,I2,I4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X2,I2))) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X2),I4) = one_one(B) )
=> ( ! [I2: A] :
( pp(member(A,I2,I4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A3,I2)),B2))),Delta)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_bw(fun(A,B),fun(fun(A,B),fun(A,B)),X2),A3)),I4)),B2))),Delta)) ) ) ) ) ).
% convex_sum_bound_le
tff(fact_1114_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add(B)
& comm_monoid_add(A) )
=> ! [H: fun(B,A),G: fun(C,B),A4: set(C)] :
( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
=> ( ! [X3: B,Y4: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y4))
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),H),G)),A4) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4)) ) ) ) ) ).
% sum_comp_morphism
tff(fact_1115_sum__gp,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [N: nat,M: nat,X2: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( ( ( X2 = one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)) ) )
& ( ( X2 != one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)) ) ) ) ) ) ) ).
% sum_gp
tff(fact_1116_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ) ).
% divmod_digit_1(1)
tff(fact_1117_sum__gp__offset,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X2: A,M: nat,N: nat] :
( ( ( X2 = one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
& ( ( X2 != one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),M)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(nat,nat,suc,N))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)) ) ) ) ) ).
% sum_gp_offset
tff(fact_1118_Sum__Icc__nat,axiom,
! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bx(nat,nat)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).
% Sum_Icc_nat
tff(fact_1119_arith__series__nat,axiom,
! [A3: nat,D3: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_by(nat,fun(nat,fun(nat,nat)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D3)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).
% arith_series_nat
tff(fact_1120_mod__double__modulus,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: A,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
=> ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X2),M) )
| ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X2),M)),M) ) ) ) ) ) ).
% mod_double_modulus
tff(fact_1121_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),H: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bz(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A4)) ) ).
% sum.distrib
tff(fact_1122_case__prodI,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,bool)),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A3),B2))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).
% case_prodI
tff(fact_1123_case__prodI2,axiom,
! [B: $tType,A: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,bool))] :
( ! [A5: A,B3: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C2,A5),B3)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C2),P3)) ) ).
% case_prodI2
tff(fact_1124_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),A3: B,B2: C] :
( pp(member(A,Z2,aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B2)))
=> pp(member(A,Z2,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)))) ) ).
% mem_case_prodI
tff(fact_1125_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z2: C,C2: fun(A,fun(B,set(C)))] :
( ! [A5: A,B3: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) )
=> pp(member(C,Z2,aa(B,set(C),aa(A,fun(B,set(C)),C2,A5),B3))) )
=> pp(member(C,Z2,aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C2),P3))) ) ).
% mem_case_prodI2
tff(fact_1126_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool))),X2: C] :
( ! [A5: A,B3: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) = P3 )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A5),B3),X2)) )
=> pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C2),P3),X2)) ) ).
% case_prodI2'
tff(fact_1127_one__eq__numeral__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: num] :
( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
<=> ( one2 = N ) ) ) ).
% one_eq_numeral_iff
tff(fact_1128_numeral__eq__one__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: num] :
( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
<=> ( N = one2 ) ) ) ).
% numeral_eq_one_iff
tff(fact_1129_of__nat__add,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_add
tff(fact_1130_of__nat__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,semiring_1_of_nat(A),one_one(nat)) = one_one(A) ) ) ).
% of_nat_1
tff(fact_1131_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
<=> ( N = one_one(nat) ) ) ) ).
% of_nat_1_eq_iff
tff(fact_1132_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
<=> ( N = one_one(nat) ) ) ) ).
% of_nat_eq_1_iff
tff(fact_1133_semiring__norm_I2_J,axiom,
aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = aa(num,num,bit0,one2) ).
% semiring_norm(2)
tff(fact_1134_numeral__le__one__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).
% numeral_le_one_iff
tff(fact_1135_one__less__numeral__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).
% one_less_numeral_iff
tff(fact_1136_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
tff(fact_1137_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] :
( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
<=> ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
tff(fact_1138_of__nat__Suc,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).
% of_nat_Suc
tff(fact_1139_Suc__numeral,axiom,
! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).
% Suc_numeral
tff(fact_1140_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))))
<=> ( M != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
tff(fact_1141_one__add__one,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).
% one_add_one
tff(fact_1142_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))))
<=> ( M != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
tff(fact_1143_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% bits_one_mod_two_eq_one
tff(fact_1144_one__mod__two__eq__one,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% one_mod_two_eq_one
tff(fact_1145_add__2__eq__Suc,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
% add_2_eq_Suc
tff(fact_1146_add__2__eq__Suc_H,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
% add_2_eq_Suc'
tff(fact_1147_Suc__1,axiom,
aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).
% Suc_1
tff(fact_1148_numeral__plus__one,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).
% numeral_plus_one
tff(fact_1149_one__plus__numeral,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).
% one_plus_numeral
tff(fact_1150_add__self__div__2,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = M ).
% add_self_div_2
tff(fact_1151_bits__1__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).
% bits_1_div_2
tff(fact_1152_one__div__two__eq__zero,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).
% one_div_two_eq_zero
tff(fact_1153_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).
% add_neg_numeral_special(9)
tff(fact_1154_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).
% diff_numeral_special(11)
tff(fact_1155_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).
% diff_numeral_special(10)
tff(fact_1156_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_1157_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_1158_minus__1__div__2__eq,axiom,
! [A: $tType] :
( euclid8789492081693882211th_nat(A)
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% minus_1_div_2_eq
tff(fact_1159_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% bits_minus_1_mod_2_eq
tff(fact_1160_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( euclid8789492081693882211th_nat(A)
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% minus_1_mod_2_eq
tff(fact_1161_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).
% diff_numeral_special(3)
tff(fact_1162_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).
% diff_numeral_special(4)
tff(fact_1163_add__self__mod__2,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).
% add_self_mod_2
tff(fact_1164_power__minus1__even,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = one_one(A) ) ).
% power_minus1_even
tff(fact_1165_mod2__gr__0,axiom,
! [M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).
% mod2_gr_0
tff(fact_1166_lex__prod__def,axiom,
! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_cb(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Ra),Rb)))) ).
% lex_prod_def
tff(fact_1167_same__fst__def,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),R4: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R4) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_cd(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P),R4)))) ).
% same_fst_def
tff(fact_1168_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: fun(B,fun(C,set(A))),P3: product_prod(B,C)] :
( pp(member(A,Z2,aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C2),P3)))
=> ~ ! [X3: B,Y4: C] :
( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y4) )
=> ~ pp(member(A,Z2,aa(C,set(A),aa(B,fun(C,set(A)),C2,X3),Y4))) ) ) ).
% mem_case_prodE
tff(fact_1169_case__prodD,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,bool)),A3: A,B2: B] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A3),B2)) ) ).
% case_prodD
tff(fact_1170_case__prodE,axiom,
! [A: $tType,B: $tType,C2: fun(A,fun(B,bool)),P3: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C2),P3))
=> ~ ! [X3: A,Y4: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X3),Y4)) ) ) ).
% case_prodE
tff(fact_1171_add__One__commute,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).
% add_One_commute
tff(fact_1172_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R4: fun(A,fun(B,fun(C,bool))),A3: A,B2: B,C2: C] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),R4),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C2))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R4,A3),B2),C2)) ) ).
% case_prodD'
tff(fact_1173_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,bool))),P3: product_prod(A,B),Z2: C] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C2),P3),Z2))
=> ~ ! [X3: A,Y4: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X3),Y4),Z2)) ) ) ).
% case_prodE'
tff(fact_1174_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.atLeast_int_atMost_int_shift
tff(fact_1175_less__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X2: product_prod(nat,nat)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X2)))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X2)) ) ).
% less_int.abs_eq
tff(fact_1176_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X2: product_prod(nat,nat)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X2)))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X2)) ) ).
% less_eq_int.abs_eq
tff(fact_1177_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A4: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A4),B5))
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A4))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),B5)))) ) ).
% Collect_case_prod_mono
tff(fact_1178_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),Z2) ).
% zadd_int_left
tff(fact_1179_int__plus,axiom,
! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)) ).
% int_plus
tff(fact_1180_int__ops_I5_J,axiom,
! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% int_ops(5)
tff(fact_1181_int__ops_I2_J,axiom,
aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
% int_ops(2)
tff(fact_1182_zle__iff__zadd,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z2))
<=> ? [N7: nat] : Z2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N7)) ) ).
% zle_iff_zadd
tff(fact_1183_numeral__One,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).
% numeral_One
tff(fact_1184_numerals_I1_J,axiom,
aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).
% numerals(1)
tff(fact_1185_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_bind(B,A,F,G)),H) = aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),case_option(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat))),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),product_case_prod(B,product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cj(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),G))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),F),H)) ).
% execute_bind_case
tff(fact_1186_Sum__Icc__int,axiom,
! [M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N))
=> ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_ck(int,int)),set_or1337092689740270186AtMost(int,M,N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).
% Sum_Icc_int
tff(fact_1187_Heap__Time__Monad_Obind__def,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(A),G: fun(A,heap_Time_Heap(B))] : heap_Time_bind(A,B,F,G) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),aa(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cn(heap_Time_Heap(A),fun(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),F),G)) ).
% Heap_Time_Monad.bind_def
tff(fact_1188_zless__iff__Suc__zadd,axiom,
! [W: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z2))
<=> ? [N7: nat] : Z2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N7))) ) ).
% zless_iff_Suc_zadd
tff(fact_1189_int__ops_I4_J,axiom,
! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).
% int_ops(4)
tff(fact_1190_int__Suc,axiom,
! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).
% int_Suc
tff(fact_1191_int__bit__induct,axiom,
! [P: fun(int,bool),K: int] :
( pp(aa(int,bool,P,zero_zero(int)))
=> ( pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),one_one(int))))
=> ( ! [K3: int] :
( pp(aa(int,bool,P,K3))
=> ( ( K3 != zero_zero(int) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
=> ( ! [K3: int] :
( pp(aa(int,bool,P,K3))
=> ( ( K3 != aa(int,int,uminus_uminus(int),one_one(int)) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) )
=> pp(aa(int,bool,P,K)) ) ) ) ) ).
% int_bit_induct
tff(fact_1192_uminus__numeral__One,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% uminus_numeral_One
tff(fact_1193_int__def,axiom,
! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),zero_zero(nat))) ).
% int_def
tff(fact_1194_num__of__nat__One,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
=> ( aa(nat,num,num_of_nat,N) = one2 ) ) ).
% num_of_nat_One
tff(fact_1195_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A3) ) ) ).
% neg_zdiv_mult_2
tff(fact_1196_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A3) ) ) ).
% pos_zdiv_mult_2
tff(fact_1197_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),A3))) ) ) ).
% pos_zmod_mult_2
tff(fact_1198_mod__mult2__eq_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,semiring_1_of_nat(A),M))) ) ).
% mod_mult2_eq'
tff(fact_1199_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,N)) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).
% numeral_num_of_nat_unfold
tff(fact_1200_double__arith__series,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,D3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_co(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))) ) ).
% double_arith_series
tff(fact_1201_double__gauss__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).
% double_gauss_sum
tff(fact_1202_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A3))),one_one(int)) ) ) ).
% neg_zmod_mult_2
tff(fact_1203_mult__2,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2
tff(fact_1204_mult__2__right,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [Z2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2_right
tff(fact_1205_left__add__twice,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),B2) ) ).
% left_add_twice
tff(fact_1206_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
=> ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_1207_one__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) ) ) ).
% one_power2
tff(fact_1208_nat__1__add__1,axiom,
aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).
% nat_1_add_1
tff(fact_1209_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),N) ).
% Suc_nat_number_of_add
tff(fact_1210_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).
% double_gauss_sum_from_Suc_0
tff(fact_1211_arith__series,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,D3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cp(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).
% arith_series
tff(fact_1212_gauss__sum,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).
% gauss_sum
tff(fact_1213_nat__approx__posE,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [E2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
=> ~ ! [N3: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),E2)) ) ) ).
% nat_approx_posE
tff(fact_1214_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
=> ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R))
=> eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R)),one_one(int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_1215_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).
% gauss_sum_from_Suc_0
tff(fact_1216_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) ) ) ) ).
% exp_add_not_zero_imp_right
tff(fact_1217_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) != zero_zero(A) ) ) ) ).
% exp_add_not_zero_imp_left
tff(fact_1218_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(A) )
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_1219_div__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).
% div_exp_eq
tff(fact_1220_power2__eq__1__iff,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [A3: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
<=> ( ( A3 = one_one(A) )
| ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% power2_eq_1_iff
tff(fact_1221_plus__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X2: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X2)) ).
% plus_int.abs_eq
tff(fact_1222_abs__square__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
<=> ( aa(A,A,abs_abs(A),X2) = one_one(A) ) ) ) ).
% abs_square_eq_1
tff(fact_1223_minus__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X2: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ct(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X2)) ).
% minus_int.abs_eq
tff(fact_1224_nat__induct2,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( pp(aa(nat,bool,P,one_one(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
=> pp(aa(nat,bool,P,N)) ) ) ) ).
% nat_induct2
tff(fact_1225_times__int_Oabs__eq,axiom,
! [Xa: product_prod(nat,nat),X2: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X2)) ).
% times_int.abs_eq
tff(fact_1226_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)))
<=> ( ( X2 = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_1227_sum__power2__ge__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).
% sum_power2_ge_zero
tff(fact_1228_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ) ).
% bits_stable_imp_add_self
tff(fact_1229_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))) ) ).
% not_sum_power2_lt_zero
tff(fact_1230_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
<=> ( ( X2 != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_1231_power2__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X2: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X2)),Y)) ) ).
% power2_sum
tff(fact_1232_square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))) ) ) ) ).
% square_le_1
tff(fact_1233_abs__square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),one_one(A))) ) ) ).
% abs_square_le_1
tff(fact_1234_abs__square__less__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X2)),one_one(A))) ) ) ).
% abs_square_less_1
tff(fact_1235_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat,M: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).
% div_exp_mod_exp_eq
tff(fact_1236_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat))))) ) ) ) ).
% ex_power_ivl2
tff(fact_1237_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N3)),K))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat))))) ) ) ) ).
% ex_power_ivl1
tff(fact_1238_power2__diff,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X2: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X2)),Y)) ) ).
% power2_diff
tff(fact_1239_power__minus1__odd,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% power_minus1_odd
tff(fact_1240_mask__eq__sum__exp,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_cw(nat,fun(nat,bool),N))) ) ).
% mask_eq_sum_exp
tff(fact_1241_sum_Oin__pairs,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cx(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.in_pairs
tff(fact_1242_gauss__sum__nat,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bx(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).
% gauss_sum_nat
tff(fact_1243_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,R: A,Q3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Q3)),R) ) ) ) ) ).
% divmod_step_eq
tff(fact_1244_divmod__step__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_cy(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_1245_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_cz(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).
% divmod_step_nat_def
tff(fact_1246_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_da(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).
% divmod_step_int_def
tff(fact_1247_set__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% set_bit_0
tff(fact_1248_mlex__eq,axiom,
! [A: $tType,F: fun(A,nat),R4: set(product_prod(A,A))] : mlex_prod(A,F,R4) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_db(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F),R4))) ).
% mlex_eq
tff(fact_1249_unset__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% unset_bit_Suc
tff(fact_1250_flip__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% flip_bit_Suc
tff(fact_1251_set__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% set_bit_Suc
tff(fact_1252_split__part,axiom,
! [B: $tType,A: $tType,P: bool,Q: fun(A,fun(B,bool)),X: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_dc(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q)),X))
<=> ( pp(P)
& pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q),X)) ) ) ).
% split_part
tff(fact_1253_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_dd(A,fun(B,bool))),Prod)) ).
% prod.disc_eq_case
tff(fact_1254_divmod__step__integer__def,axiom,
! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_de(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).
% divmod_step_integer_def
tff(fact_1255_signed__take__bit__rec,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) )
& ( ( N != zero_zero(nat) )
=> ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ) ) ) ).
% signed_take_bit_rec
tff(fact_1256_round__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,Y: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),Y)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))
=> ( archimedean_round(A,X2) = Y ) ) ) ) ).
% round_unique
tff(fact_1257_rel__of__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),bool)] : rel_of(A,B,M,P) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_df(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),M),P))) ).
% rel_of_def
tff(fact_1258_dbl__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).
% dbl_simps(4)
tff(fact_1259_round__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X2)))
=> ( archimedean_round(A,X2) = archimedean_ceiling(A,X2) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X2)))
=> ( archimedean_round(A,X2) = archim6421214686448440834_floor(A,X2) ) ) ) ) ).
% round_altdef
tff(fact_1260_round__unique_H,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
=> ( archimedean_round(A,X2) = N ) ) ) ).
% round_unique'
tff(fact_1261_of__int__round__abs__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X2))),X2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% of_int_round_abs_le
tff(fact_1262_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_dg(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).
% divmod_algorithm_code(5)
tff(fact_1263_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% signed_take_bit_of_minus_1
tff(fact_1264_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).
% signed_take_bit_Suc_1
tff(fact_1265_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [K: num] : bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K),one_one(A)) = one_one(A) ) ).
% signed_take_bit_numeral_of_1
tff(fact_1266_round__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).
% round_1
tff(fact_1267_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).
% divmod_algorithm_code(2)
tff(fact_1268_dbl__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).
% dbl_simps(3)
tff(fact_1269_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(3)
tff(fact_1270_less__eq__integer__code_I1_J,axiom,
pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).
% less_eq_integer_code(1)
tff(fact_1271_sgn__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = zero_zero(code_integer) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,code_integer,sgn_sgn(code_integer),K) = one_one(code_integer) ) ) ) ) ) ).
% sgn_integer_code
tff(fact_1272_exhaustive__integer_H_Ocases,axiom,
! [X2: product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
~ ! [F3: fun(code_integer,option(product_prod(bool,list(code_term)))),D2: code_integer,I2: code_integer] : X2 != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(code_integer,option(product_prod(bool,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),F3),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D2),I2)) ).
% exhaustive_integer'.cases
tff(fact_1273_full__exhaustive__integer_H_Ocases,axiom,
! [X2: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
~ ! [F3: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: code_integer,I2: code_integer] : X2 != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer)),F3),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D2),I2)) ).
% full_exhaustive_integer'.cases
tff(fact_1274_times__integer__code_I1_J,axiom,
! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K),zero_zero(code_integer)) = zero_zero(code_integer) ).
% times_integer_code(1)
tff(fact_1275_times__integer__code_I2_J,axiom,
! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).
% times_integer_code(2)
tff(fact_1276_minus__integer__code_I1_J,axiom,
! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),K),zero_zero(code_integer)) = K ).
% minus_integer_code(1)
tff(fact_1277_plus__integer__code_I2_J,axiom,
! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),zero_zero(code_integer)),L) = L ).
% plus_integer_code(2)
tff(fact_1278_plus__integer__code_I1_J,axiom,
! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).
% plus_integer_code(1)
tff(fact_1279_minus__integer__code_I2_J,axiom,
! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).
% minus_integer_code(2)
tff(fact_1280_divmod__integer_H__def,axiom,
! [M: num,N: num] : unique8689654367752047608divmod(code_integer,M,N) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),N))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),N))) ).
% divmod_integer'_def
tff(fact_1281_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] : bit_ri4674362597316999326ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),bit_ri4674362597316999326ke_bit(int,N,K)),bit_ri4674362597316999326ke_bit(int,N,L))) = bit_ri4674362597316999326ke_bit(int,N,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).
% signed_take_bit_add
tff(fact_1282_dbl__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X2: A] : neg_numeral_dbl(A,X2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),X2) ) ).
% dbl_def
tff(fact_1283_divmod__int__def,axiom,
! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))) ).
% divmod_int_def
tff(fact_1284_divmod__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).
% divmod_def
tff(fact_1285_divmod_H__nat__def,axiom,
! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ).
% divmod'_nat_def
tff(fact_1286_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))),bit_ri4674362597316999326ke_bit(int,N,K))) ) ).
% signed_take_bit_int_greater_eq
tff(fact_1287_divmod__divmod__step,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] :
( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
=> ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
=> ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N))) ) ) ) ) ).
% divmod_divmod_step
tff(fact_1288_round__def,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : archimedean_round(A,X2) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% round_def
tff(fact_1289_signed__take__bit__Suc,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_ri4674362597316999326ke_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% signed_take_bit_Suc
tff(fact_1290_of__int__round__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,X2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).
% of_int_round_le
tff(fact_1291_of__int__round__ge,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X2)))) ) ).
% of_int_round_ge
tff(fact_1292_of__int__round__gt,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X2)))) ) ).
% of_int_round_gt
tff(fact_1293_integer__of__int__code,axiom,
! [K: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( aa(int,code_integer,code_integer_of_int,K) = aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( ( ( K = zero_zero(int) )
=> ( aa(int,code_integer,code_integer_of_int,K) = zero_zero(code_integer) ) )
& ( ( K != zero_zero(int) )
=> ( aa(int,code_integer,code_integer_of_int,K) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),one_one(code_integer))) ) ) ) ) ) ).
% integer_of_int_code
tff(fact_1294_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_dh(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).
% divmod_algorithm_code(6)
tff(fact_1295_of__nat__code,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_di(A,A),N,zero_zero(A)) ) ).
% of_nat_code
tff(fact_1296_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).
% signed_take_bit_Suc_minus_bit1
tff(fact_1297_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,N),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),bit_concat_bit(N,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),L))) ).
% concat_bit_Suc
tff(fact_1298_even__succ__mod__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))) ) ) ) ) ).
% even_succ_mod_exp
tff(fact_1299_even__succ__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ) ) ) ).
% even_succ_div_exp
tff(fact_1300_take__bit__rec,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = zero_zero(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).
% take_bit_rec
tff(fact_1301_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( pp(dvd_dvd(nat,M,one_one(nat)))
<=> ( M = one_one(nat) ) ) ).
% nat_dvd_1_iff_1
tff(fact_1302_integer__of__int__eq__of__int,axiom,
code_integer_of_int = ring_1_of_int(code_integer) ).
% integer_of_int_eq_of_int
tff(fact_1303_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_triv_right_iff
tff(fact_1304_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_triv_left_iff
tff(fact_1305_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_times_triv_right_iff
tff(fact_1306_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ).
% dvd_add_times_triv_left_iff
tff(fact_1307_unit__prod,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A))) ) ) ) ).
% unit_prod
tff(fact_1308_div__add,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C2,A3))
=> ( pp(dvd_dvd(A,C2,B2))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).
% div_add
tff(fact_1309_unit__div__1__div__1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = A3 ) ) ) ).
% unit_div_1_div_1
tff(fact_1310_unit__div__1__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3),one_one(A))) ) ) ).
% unit_div_1_unit
tff(fact_1311_unit__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),one_one(A))) ) ) ) ).
% unit_div
tff(fact_1312_take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).
% take_bit_Suc_1
tff(fact_1313_take__bit__numeral__1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [L: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).
% take_bit_numeral_1
tff(fact_1314_semiring__norm_I9_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% semiring_norm(9)
tff(fact_1315_semiring__norm_I7_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% semiring_norm(7)
tff(fact_1316_semiring__norm_I16_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).
% semiring_norm(16)
tff(fact_1317_unit__mult__div__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) ) ) ) ).
% unit_mult_div_div
tff(fact_1318_unit__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),A3) = B2 ) ) ) ).
% unit_div_mult_self
tff(fact_1319_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] :
( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),one_one(A)) = zero_zero(A) )
<=> ( N = zero_zero(nat) ) ) ) ).
% take_bit_of_1_eq_0_iff
tff(fact_1320_semiring__norm_I10_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),one2)) ).
% semiring_norm(10)
tff(fact_1321_semiring__norm_I8_J,axiom,
! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).
% semiring_norm(8)
tff(fact_1322_semiring__norm_I5_J,axiom,
! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),one2) = aa(num,num,bit1,M) ).
% semiring_norm(5)
tff(fact_1323_semiring__norm_I4_J,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).
% semiring_norm(4)
tff(fact_1324_semiring__norm_I3_J,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit0,N)) = aa(num,num,bit1,N) ).
% semiring_norm(3)
tff(fact_1325_odd__add,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,B2: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> ~ ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
<=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).
% odd_add
tff(fact_1326_even__add,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
<=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).
% even_add
tff(fact_1327_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2)) ) ) ).
% dbl_inc_simps(3)
tff(fact_1328_even__plus__one__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))))
<=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)) ) ) ).
% even_plus_one_iff
tff(fact_1329_even__diff,axiom,
! [A: $tType] :
( ring_parity(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
<=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ).
% even_diff
tff(fact_1330_neg__one__even__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).
% neg_one_even_power
tff(fact_1331_neg__one__odd__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).
% neg_one_odd_power
tff(fact_1332_even__diff__nat,axiom,
! [M: nat,N: nat] :
( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).
% even_diff_nat
tff(fact_1333_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(4)
tff(fact_1334_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).
% mod_Suc_eq_mod_add3
tff(fact_1335_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).
% Suc_mod_eq_add3_mod_numeral
tff(fact_1336_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).
% div_Suc_eq_div_add3
tff(fact_1337_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).
% Suc_div_eq_add3_div_numeral
tff(fact_1338_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))) ) ) ).
% dbl_dec_simps(4)
tff(fact_1339_even__succ__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).
% even_succ_div_2
tff(fact_1340_odd__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).
% odd_succ_div_two
tff(fact_1341_even__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).
% even_succ_div_two
tff(fact_1342_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)) ) ) ).
% odd_two_times_div_two_nat
tff(fact_1343_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A3 ) ) ) ).
% odd_two_times_div_two_succ
tff(fact_1344_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A))))
<=> ( N = zero_zero(nat) ) ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_1345_zmod__numeral__Bit1,axiom,
! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).
% zmod_numeral_Bit1
tff(fact_1346_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] :
( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
=> ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
=> ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).
% divmod_algorithm_code(8)
tff(fact_1347_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] :
( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
=> ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
=> ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).
% divmod_algorithm_code(7)
tff(fact_1348_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,N,aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).
% signed_take_bit_Suc_bit1
tff(fact_1349_uminus__integer__code_I1_J,axiom,
aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).
% uminus_integer_code(1)
tff(fact_1350_take__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Xa),X2)) ).
% take_bit_integer.abs_eq
tff(fact_1351_abs__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,code_integer,abs_abs(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),K) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,code_integer,abs_abs(code_integer),K) = K ) ) ) ).
% abs_integer_code
tff(fact_1352_less__integer__code_I1_J,axiom,
~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).
% less_integer_code(1)
tff(fact_1353_unset__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Xa),X2)) ).
% unset_bit_integer.abs_eq
tff(fact_1354_flip__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Xa),X2)) ).
% flip_bit_integer.abs_eq
tff(fact_1355_set__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xa),X2)) ).
% set_bit_integer.abs_eq
tff(fact_1356_take__bit__add,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,A3: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),B2))) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ).
% take_bit_add
tff(fact_1357_zero__integer__def,axiom,
zero_zero(code_integer) = aa(int,code_integer,code_integer_of_int,zero_zero(int)) ).
% zero_integer_def
tff(fact_1358_less__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa),X2)) ) ).
% less_integer.abs_eq
tff(fact_1359_uminus__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),X2)) ).
% uminus_integer.abs_eq
tff(fact_1360_abs__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_integer,abs_abs(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,abs_abs(int),X2)) ).
% abs_integer.abs_eq
tff(fact_1361_dvd__add__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_add_right_iff
tff(fact_1362_dvd__add__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(dvd_dvd(A,A3,C2))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(dvd_dvd(A,A3,B2)) ) ) ) ).
% dvd_add_left_iff
tff(fact_1363_dvd__add,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,A3,C2))
=> pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ) ).
% dvd_add
tff(fact_1364_divide__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),X2)) ).
% divide_integer.abs_eq
tff(fact_1365_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,B2))
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ) ).
% dvd_unit_imp_unit
tff(fact_1366_unit__imp__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> pp(dvd_dvd(A,B2,A3)) ) ) ).
% unit_imp_dvd
tff(fact_1367_one__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : pp(dvd_dvd(A,one_one(A),A3)) ) ).
% one_dvd
tff(fact_1368_modulo__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),X2)) ).
% modulo_integer.abs_eq
tff(fact_1369_sgn__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_integer,sgn_sgn(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,sgn_sgn(int),X2)) ).
% sgn_integer.abs_eq
tff(fact_1370_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R: int] : bit_concat_bit(N,K,bit_concat_bit(M,L,R)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),bit_concat_bit(N,K,L),R) ).
% concat_bit_assoc
tff(fact_1371_not__is__unit__0,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ~ pp(dvd_dvd(A,zero_zero(A),one_one(A))) ) ).
% not_is_unit_0
tff(fact_1372_pinf_I9_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X))
=> ( pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S)))
<=> pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S))) ) ) ) ).
% pinf(9)
tff(fact_1373_pinf_I10_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X))
=> ( ~ pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S)))
<=> ~ pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S))) ) ) ) ).
% pinf(10)
tff(fact_1374_minf_I9_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X),Z3))
=> ( pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S)))
<=> pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S))) ) ) ) ).
% minf(9)
tff(fact_1375_minf_I10_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X),Z3))
=> ( ~ pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S)))
<=> ~ pp(dvd_dvd(B,D3,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),S))) ) ) ) ).
% minf(10)
tff(fact_1376_unit__mult__right__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
tff(fact_1377_unit__mult__left__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
tff(fact_1378_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2))
<=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).
% mult_unit_dvd_iff'
tff(fact_1379_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_mult_unit_iff'
tff(fact_1380_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% mult_unit_dvd_iff
tff(fact_1381_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_mult_unit_iff
tff(fact_1382_is__unit__mult__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),one_one(A)))
<=> ( pp(dvd_dvd(A,A3,one_one(A)))
& pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).
% is_unit_mult_iff
tff(fact_1383_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C2,A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).
% div_plus_div_distrib_dvd_left
tff(fact_1384_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C2,B2))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).
% div_plus_div_distrib_dvd_right
tff(fact_1385_unit__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) )
<=> ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
tff(fact_1386_div__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2),C2))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% div_unit_dvd_iff
tff(fact_1387_dvd__div__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)))
<=> pp(dvd_dvd(A,A3,C2)) ) ) ) ).
% dvd_div_unit_iff
tff(fact_1388_xor__num_Ocases,axiom,
! [X2: product_prod(num,num)] :
( ( X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
=> ( ! [N3: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3))
=> ( ! [N3: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))
=> ( ! [M4: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),one2)
=> ( ! [M4: num,N3: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit0,N3))
=> ( ! [M4: num,N3: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit1,N3))
=> ( ! [M4: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2)
=> ( ! [M4: num,N3: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit0,N3))
=> ~ ! [M4: num,N3: num] : X2 != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N3)) ) ) ) ) ) ) ) ) ).
% xor_num.cases
tff(fact_1389_plus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),X2)) ).
% plus_integer.abs_eq
tff(fact_1390_bezout__lemma__nat,axiom,
! [D3: nat,A3: nat,B2: nat,X2: nat,Y: nat] :
( pp(dvd_dvd(nat,D3,A3))
=> ( pp(dvd_dvd(nat,D3,B2))
=> ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D3) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y)),D3) ) )
=> ? [X3: nat,Y4: nat] :
( pp(dvd_dvd(nat,D3,A3))
& pp(dvd_dvd(nat,D3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)))
& ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),Y4)),D3) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),D3) ) ) ) ) ) ) ).
% bezout_lemma_nat
tff(fact_1391_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X3: nat,Y4: nat] :
( pp(dvd_dvd(nat,D2,A3))
& pp(dvd_dvd(nat,D2,B2))
& ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D2) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),D2) ) ) ) ).
% bezout_add_nat
tff(fact_1392_one__integer__def,axiom,
one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).
% one_integer_def
tff(fact_1393_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Xa),X2)) ) ).
% less_eq_integer.abs_eq
tff(fact_1394_times__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),times_times(int),Xa),X2)) ).
% times_integer.abs_eq
tff(fact_1395_zdvd__period,axiom,
! [A3: int,D3: int,X2: int,T2: int,C2: int] :
( pp(dvd_dvd(int,A3,D3))
=> ( pp(dvd_dvd(int,A3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),T2)))
<=> pp(dvd_dvd(int,A3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D3))),T2))) ) ) ).
% zdvd_period
tff(fact_1396_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( pp(dvd_dvd(int,K,aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K),M))))
<=> pp(dvd_dvd(int,K,N)) ) ).
% zdvd_reduce
tff(fact_1397_minus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),X2)) ).
% minus_integer.abs_eq
tff(fact_1398_unity__coeff__ex,axiom,
! [A: $tType] :
( ( dvd(A)
& semiring_0(A) )
=> ! [P: fun(A,bool),L: A] :
( ? [X4: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
<=> ? [X4: A] :
( pp(dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
& pp(aa(A,bool,P,X4)) ) ) ) ).
% unity_coeff_ex
tff(fact_1399_unit__dvdE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ).
% unit_dvdE
tff(fact_1400_inf__period_I4_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D4: A,T2: A] :
( pp(dvd_dvd(A,D3,D4))
=> ! [X: A,K7: A] :
( ~ pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),T2)))
<=> ~ pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K7),D4))),T2))) ) ) ) ).
% inf_period(4)
tff(fact_1401_inf__period_I3_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D3: A,D4: A,T2: A] :
( pp(dvd_dvd(A,D3,D4))
=> ! [X: A,K7: A] :
( pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),T2)))
<=> pp(dvd_dvd(A,D3,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),K7),D4))),T2))) ) ) ) ).
% inf_period(3)
tff(fact_1402_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ) ).
% unit_div_eq_0_iff
tff(fact_1403_numeral__Bit1,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).
% numeral_Bit1
tff(fact_1404_unit__eq__div1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = C2 )
<=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).
% unit_eq_div1
tff(fact_1405_unit__eq__div2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = C2 ) ) ) ) ).
% unit_eq_div2
tff(fact_1406_div__mult__unit2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(dvd_dvd(A,C2,one_one(A)))
=> ( pp(dvd_dvd(A,B2,A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).
% div_mult_unit2
tff(fact_1407_unit__div__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) ) ) ) ).
% unit_div_commute
tff(fact_1408_unit__div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) ) ) ) ).
% unit_div_mult_swap
tff(fact_1409_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( pp(dvd_dvd(A,C2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2) ) ) ) ) ).
% is_unit_div_mult2_eq
tff(fact_1410_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = zero_zero(A) ) ) ) ).
% unit_imp_mod_eq_0
tff(fact_1411_is__unit__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),one_one(A)))
<=> ( pp(dvd_dvd(A,A3,one_one(A)))
| ( N = zero_zero(nat) ) ) ) ) ).
% is_unit_power_iff
tff(fact_1412_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero(nat) )
=> ? [D2: nat,X3: nat,Y4: nat] :
( pp(dvd_dvd(nat,D2,A3))
& pp(dvd_dvd(nat,D2,B2))
& ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D2) ) ) ) ).
% bezout_add_strong_nat
tff(fact_1413_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).
% take_bit_Suc_bit1
tff(fact_1414_numeral__code_I3_J,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).
% numeral_code(3)
tff(fact_1415_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A3 )
=> ( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = zero_zero(A) ) )
& ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_1416_is__unitE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [B3: A] :
( ( B3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B3,one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B3 )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A3 )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = one_one(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_1417_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).
% is_unit_div_mult_cancel_left
tff(fact_1418_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).
% is_unit_div_mult_cancel_right
tff(fact_1419_odd__even__add,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,B2: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2))
=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).
% odd_even_add
tff(fact_1420_odd__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),one_one(A))) ) ).
% odd_one
tff(fact_1421_dvd__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X2: A,M: nat,N: nat] :
( ( X2 != zero_zero(A) )
=> ( pp(dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)))
<=> ( pp(dvd_dvd(A,X2,one_one(A)))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ) ).
% dvd_power_iff
tff(fact_1422_dvd__power,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X2: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
| ( X2 = one_one(A) ) )
=> pp(dvd_dvd(A,X2,aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N))) ) ) ).
% dvd_power
tff(fact_1423_Suc3__eq__add__3,axiom,
! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).
% Suc3_eq_add_3
tff(fact_1424_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M),M))
<=> ( N = one_one(nat) ) ) ) ).
% dvd_mult_cancel2
tff(fact_1425_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N),M))
<=> ( N = one_one(nat) ) ) ) ).
% dvd_mult_cancel1
tff(fact_1426_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q3),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M)))
=> ( pp(dvd_dvd(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q3)))
<=> pp(dvd_dvd(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M)),Q3)))) ) ) ) ).
% dvd_minus_add
tff(fact_1427_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
% power_dvd_imp_le
tff(fact_1428_aset_I10_J,axiom,
! [D3: int,D4: int,A4: set(int),T2: int] :
( pp(dvd_dvd(int,D3,D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
=> ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),T2))) ) ) ) ).
% aset(10)
tff(fact_1429_aset_I9_J,axiom,
! [D3: int,D4: int,A4: set(int),T2: int] :
( pp(dvd_dvd(int,D3,D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,A4))
=> ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa4) ) ) )
=> ( pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
=> pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D4)),T2))) ) ) ) ).
% aset(9)
tff(fact_1430_bset_I10_J,axiom,
! [D3: int,D4: int,B5: set(int),T2: int] :
( pp(dvd_dvd(int,D3,D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
=> ~ pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),T2))) ) ) ) ).
% bset(10)
tff(fact_1431_bset_I9_J,axiom,
! [D3: int,D4: int,B5: set(int),T2: int] :
( pp(dvd_dvd(int,D3,D4))
=> ! [X: int] :
( ! [Xa4: int] :
( pp(member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb3: int] :
( pp(member(int,Xb3,B5))
=> ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa4) ) ) )
=> ( pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
=> pp(dvd_dvd(int,D3,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D4)),T2))) ) ) ) ).
% bset(9)
tff(fact_1432_num_Osize__gen_I3_J,axiom,
! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).
% num.size_gen(3)
tff(fact_1433_num_Osize_I6_J,axiom,
! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).
% num.size(6)
tff(fact_1434_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_1435_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).
% Suc_mod_eq_add3_mod
tff(fact_1436_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).
% Suc_div_eq_add3_div
tff(fact_1437_even__diff__iff,axiom,
! [K: int,L: int] :
( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)))
<=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).
% even_diff_iff
tff(fact_1438_even__abs__add__iff,axiom,
! [K: int,L: int] :
( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L)))
<=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).
% even_abs_add_iff
tff(fact_1439_even__add__abs__iff,axiom,
! [K: int,L: int] :
( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L))))
<=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).
% even_add_abs_iff
tff(fact_1440_signed__take__bit__eq__take__bit__shift,axiom,
! [N: nat,K: int] : bit_ri4674362597316999326ke_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).
% signed_take_bit_eq_take_bit_shift
tff(fact_1441_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) )
=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ) ).
% take_bit_incr_eq
tff(fact_1442_oddE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ~ ! [B3: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B3)),one_one(A)) ) ) ).
% oddE
tff(fact_1443_parity__cases,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
=> ~ ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).
% parity_cases
tff(fact_1444_mod2__eq__if,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) )
& ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ) ).
% mod2_eq_if
tff(fact_1445_minus__one__power__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) )
& ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% minus_one_power_iff
tff(fact_1446_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) ) ).
% take_bit_int_greater_eq
tff(fact_1447_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).
% take_bit_Suc_minus_1_eq
tff(fact_1448_take__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).
% take_bit_Suc
tff(fact_1449_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).
% take_bit_numeral_minus_1_eq
tff(fact_1450_even__mask__div__iff_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
% even_mask_div_iff'
tff(fact_1451_even__mask__div__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
<=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
% even_mask_div_iff
tff(fact_1452_Code__Numeral_Onegative__def,axiom,
code_negative = aa(fun(num,code_integer),fun(num,code_integer),aa(fun(code_integer,code_integer),fun(fun(num,code_integer),fun(num,code_integer)),comp(code_integer,code_integer,num),uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).
% Code_Numeral.negative_def
tff(fact_1453_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),aTP_Lamp_dj(num,fun(num,int),M)),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).
% take_bit_numeral_minus_numeral_int
tff(fact_1454_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).
% signed_take_bit_numeral_minus_bit1
tff(fact_1455_execute__change,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R: ref(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),ref_change(A,F,R)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(A,A,F,ref_get(A,H,R))),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R,aa(A,A,F,ref_get(A,H,R)),H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))) ) ).
% execute_change
tff(fact_1456_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] : bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),bit_ri4674362597316999326ke_bit(int,pred_numeral(L),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).
% signed_take_bit_numeral_bit1
tff(fact_1457_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).
% take_bit_Suc_minus_bit1
tff(fact_1458_divmod__BitM__2__eq,axiom,
! [M: num] : unique8689654367752047608divmod(int,bitM(M),aa(num,num,bit0,one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).
% divmod_BitM_2_eq
tff(fact_1459_flip__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).
% flip_bit_0
tff(fact_1460_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).
% of_bool_eq(2)
tff(fact_1461_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: bool] :
( ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) )
<=> pp(P) ) ) ).
% of_bool_eq_1_iff
tff(fact_1462_of__bool__less__one__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
<=> ~ pp(P) ) ) ).
% of_bool_less_one_iff
tff(fact_1463_of__bool__not__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(bool,A,zero_neq_one_of_bool(A),P)) ) ).
% of_bool_not_iff
tff(fact_1464_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_dk(num,option(num))),bit_take_bit_num(N,M)) ).
% take_bit_num_simps(3)
tff(fact_1465_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(N,M))) ).
% take_bit_num_simps(4)
tff(fact_1466_take__bit__num__simps_I6_J,axiom,
! [R: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit0,M)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_dk(num,option(num))),bit_take_bit_num(pred_numeral(R),M)) ).
% take_bit_num_simps(6)
tff(fact_1467_take__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% take_bit_of_1
tff(fact_1468_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).
% add_neg_numeral_special(6)
tff(fact_1469_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).
% add_neg_numeral_special(5)
tff(fact_1470_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).
% diff_numeral_special(5)
tff(fact_1471_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).
% diff_numeral_special(6)
tff(fact_1472_take__bit__num__simps_I7_J,axiom,
! [R: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit1,M)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(pred_numeral(R),M))) ).
% take_bit_num_simps(7)
tff(fact_1473_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: num,N: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).
% take_bit_numeral_numeral
tff(fact_1474_bits__1__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).
% bits_1_div_exp
tff(fact_1475_one__div__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).
% one_div_2_pow_eq
tff(fact_1476_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% one_mod_2_pow_eq
tff(fact_1477_mult__inc,axiom,
! [X2: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X2),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X2),Y)),X2) ).
% mult_inc
tff(fact_1478_add__inc,axiom,
! [X2: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),Y)) ).
% add_inc
tff(fact_1479_of__bool__less__eq__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).
% of_bool_less_eq_one
tff(fact_1480_of__bool__def,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P3: bool] :
( ( pp(P3)
=> ( aa(bool,A,zero_neq_one_of_bool(A),P3) = one_one(A) ) )
& ( ~ pp(P3)
=> ( aa(bool,A,zero_neq_one_of_bool(A),P3) = zero_zero(A) ) ) ) ) ).
% of_bool_def
tff(fact_1481_split__of__bool,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,bool),P3: bool] :
( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P3)))
<=> ( ( pp(P3)
=> pp(aa(A,bool,P,one_one(A))) )
& ( ~ pp(P3)
=> pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).
% split_of_bool
tff(fact_1482_split__of__bool__asm,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: fun(A,bool),P3: bool] :
( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P3)))
<=> ~ ( ( pp(P3)
& ~ pp(aa(A,bool,P,one_one(A))) )
| ( ~ pp(P3)
& ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).
% split_of_bool_asm
tff(fact_1483_add__One,axiom,
! [X2: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),one2) = inc(X2) ).
% add_One
tff(fact_1484_effect__updateI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H3: heap_ext(product_unit),R: ref(A),V: A,H: heap_ext(product_unit),N: nat,X2: product_unit] :
( ( H3 = ref_set(A,R,V,H) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(product_unit,ref_update(A,R,V),H,H3,X2,N) ) ) ) ).
% effect_updateI
tff(fact_1485_effect__updateE,axiom,
! [A: $tType] :
( heap(A)
=> ! [R3: ref(A),V: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: product_unit,N: nat] :
( heap_Time_effect(product_unit,ref_update(A,R3,V),H,H3,R,N)
=> ~ ( ( H3 = ref_set(A,R3,V,H) )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_updateE
tff(fact_1486_pred__numeral__def,axiom,
! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).
% pred_numeral_def
tff(fact_1487_BitM__plus__one,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = aa(num,num,bit0,N) ).
% BitM_plus_one
tff(fact_1488_one__plus__BitM,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = aa(num,num,bit0,N) ).
% one_plus_BitM
tff(fact_1489_numeral__inc,axiom,
! [A: $tType] :
( numeral(A)
=> ! [X2: num] : aa(num,A,numeral_numeral(A),inc(X2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X2)),one_one(A)) ) ).
% numeral_inc
tff(fact_1490_numeral__BitM,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))),one_one(A)) ) ).
% numeral_BitM
tff(fact_1491_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).
% take_bit_numeral_minus_bit1
tff(fact_1492_bits__induct,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [P: fun(A,bool),A3: A] :
( ! [A5: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A5),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A5 )
=> pp(aa(A,bool,P,A5)) )
=> ( ! [A5: A,B3: bool] :
( pp(aa(A,bool,P,A5))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A5))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A5 )
=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A5)))) ) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% bits_induct
tff(fact_1493_effect__changeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R3: ref(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,ref_change(A,F,R3),H,H3,R,N)
=> ~ ( ( H3 = ref_set(A,R3,aa(A,A,F,ref_get(A,H,R3)),H) )
=> ( ( R = aa(A,A,F,ref_get(A,H,R3)) )
=> ( N != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ) ) ) ).
% effect_changeE
tff(fact_1494_effect__changeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H3: heap_ext(product_unit),R: ref(A),F: fun(A,A),H: heap_ext(product_unit),X2: A,N: nat] :
( ( H3 = ref_set(A,R,aa(A,A,F,ref_get(A,H,R)),H) )
=> ( ( X2 = aa(A,A,F,ref_get(A,H,R)) )
=> ( ( N = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
=> heap_Time_effect(A,ref_change(A,F,R),H,H3,X2,N) ) ) ) ) ).
% effect_changeI
tff(fact_1495_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
| ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
| ( N = zero_zero(nat) ) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
& ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
| ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
| ( N = zero_zero(nat) ) )
=> ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) )
& ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,N,M)))))) ) ) ) ) ) ).
% divide_int_unfold
tff(fact_1496_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).
% take_bit_numeral_bit1
tff(fact_1497_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))))) ).
% Divides.adjust_div_eq
tff(fact_1498_mask__numeral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: num] : aa(nat,A,bit_se2239418461657761734s_mask(A),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),pred_numeral(N)))) ) ).
% mask_numeral
tff(fact_1499_and__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = zero_zero(int) )
| ( L = zero_zero(int) ) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = zero_zero(int) ) )
& ( ~ ( ( K = zero_zero(int) )
| ( L = zero_zero(int) ) )
=> ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = L ) )
& ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = K ) )
& ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).
% and_int_unfold
tff(fact_1500_the__res__def,axiom,
! [A: $tType,M: heap_Time_Heap(A),H: heap_ext(product_unit)] : time_the_res(A,M,H) = aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),A,aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),A),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),A),aa(A,fun(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),A),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),A)),case_option(A,product_prod(A,product_prod(heap_ext(product_unit),nat))),undefined(A)),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),A)),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),A),product_case_prod(A,product_prod(heap_ext(product_unit),nat),A),aTP_Lamp_dm(A,fun(product_prod(heap_ext(product_unit),nat),A)))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),M),H)) ).
% the_res_def
tff(fact_1501_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = aa(nat,int,bit_se2239418461657761734s_mask(int),N) )
<=> pp(dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))) ) ).
% take_bit_eq_mask_iff_exp_dvd
tff(fact_1502_Code__Numeral_Opositive__def,axiom,
code_positive = numeral_numeral(code_integer) ).
% Code_Numeral.positive_def
tff(fact_1503_divide__int__def,axiom,
! [L: int,K: int] :
( ( ( L = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) )
& ( ( L != zero_zero(int) )
=> ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ) )
& ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(int,L,K)))))) ) ) ) ) ) ).
% divide_int_def
tff(fact_1504_sum__gp0,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X2: A,N: nat] :
( ( ( X2 = one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
& ( ( X2 != one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)) ) ) ) ) ).
% sum_gp0
tff(fact_1505_atMost__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [X2: A,Y: A] :
( ( aa(A,set(A),set_ord_atMost(A),X2) = aa(A,set(A),set_ord_atMost(A),Y) )
<=> ( X2 = Y ) ) ) ).
% atMost_eq_iff
tff(fact_1506_atMost__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,aa(A,set(A),set_ord_atMost(A),K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),K)) ) ) ).
% atMost_iff
tff(fact_1507_atMost__subset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),X2)),aa(A,set(A),set_ord_atMost(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y)) ) ) ).
% atMost_subset_iff
tff(fact_1508_bit_Oconj__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X2),aa(A,A,uminus_uminus(A),one_one(A))) = X2 ) ).
% bit.conj_one_right
tff(fact_1509_and_Oright__neutral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).
% and.right_neutral
tff(fact_1510_and_Oleft__neutral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).
% and.left_neutral
tff(fact_1511_and__numerals_I2_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = one_one(A) ) ).
% and_numerals(2)
tff(fact_1512_and__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),one_one(A)) = one_one(A) ) ).
% and_numerals(8)
tff(fact_1513_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A,H: A,H3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H3)))
<=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H3)) ) ) ) ).
% Icc_subset_Iic_iff
tff(fact_1514_mask__Suc__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ( aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).
% mask_Suc_0
tff(fact_1515_sum_OatMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% sum.atMost_Suc
tff(fact_1516_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(nat,A,bit_se2239418461657761734s_mask(A),N) ) ).
% take_bit_minus_one_eq_mask
tff(fact_1517_and__numerals_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = zero_zero(A) ) ).
% and_numerals(1)
tff(fact_1518_and__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X2))),one_one(A)) = zero_zero(A) ) ).
% and_numerals(5)
tff(fact_1519_nat__numeral__diff__1,axiom,
! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).
% nat_numeral_diff_1
tff(fact_1520_and__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X2)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% and_numerals(7)
tff(fact_1521_mask__integer_Oabs__eq,axiom,
! [X2: nat] : aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),X2) = aa(int,code_integer,code_integer_of_int,aa(nat,int,bit_se2239418461657761734s_mask(int),X2)) ).
% mask_integer.abs_eq
tff(fact_1522_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [H3: A,L: A,H: A] : aa(A,set(A),set_ord_atMost(A),H3) != set_or1337092689740270186AtMost(A,L,H) ) ).
% not_Iic_eq_Icc
tff(fact_1523_atMost__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_dn(A,fun(A,bool),U)) ) ).
% atMost_def
tff(fact_1524_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
<=> ( ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) )
& ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% and_eq_minus_1_iff
tff(fact_1525_atMost__atLeast0,axiom,
! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).
% atMost_atLeast0
tff(fact_1526_nat__one__as__int,axiom,
one_one(nat) = aa(int,nat,nat2,one_one(int)) ).
% nat_one_as_int
tff(fact_1527_not__Iic__le__Icc,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [H: A,L4: A,H3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L4,H3))) ) ).
% not_Iic_le_Icc
tff(fact_1528_nat__int__add,axiom,
! [A3: nat,B2: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2) ).
% nat_int_add
tff(fact_1529_option_Othe__def,axiom,
! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),undefined(A)),aTP_Lamp_ae(A,A)),Option) ).
% option.the_def
tff(fact_1530_nat__plus__as__int,axiom,
! [X: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).
% nat_plus_as_int
tff(fact_1531_nat__add__distrib,axiom,
! [Z2: int,Z6: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z6))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z6)) ) ) ) ).
% nat_add_distrib
tff(fact_1532_Suc__as__int,axiom,
! [X: nat] : aa(nat,nat,suc,X) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X)),one_one(int))) ).
% Suc_as_int
tff(fact_1533_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L))))) ).
% nat_abs_triangle_ineq
tff(fact_1534_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).
% sum.atMost_Suc_shift
tff(fact_1535_sum__telescope,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [F: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_do(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,aa(nat,nat,suc,I))) ) ).
% sum_telescope
tff(fact_1536_the__heap__def,axiom,
! [A: $tType,M: heap_Time_Heap(A),H: heap_ext(product_unit)] : time_the_heap(A,M,H) = aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),heap_ext(product_unit)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),aa(heap_ext(product_unit),fun(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),heap_ext(product_unit)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit))),case_option(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),undefined(heap_ext(product_unit))),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit))),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),heap_ext(product_unit)),product_case_prod(A,product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),aTP_Lamp_dq(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit))))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),M),H)) ).
% the_heap_def
tff(fact_1537_and__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).
% and_one_eq
tff(fact_1538_one__and__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).
% one_and_eq
tff(fact_1539_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = aa(nat,int,bit_se2239418461657761734s_mask(int),N) )
<=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).
% take_bit_eq_mask_iff
tff(fact_1540_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(nat,nat,suc,aa(int,nat,nat2,Z2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z2)) ) ) ).
% Suc_nat_eq_nat_zadd1
tff(fact_1541_Divides_Oadjust__div__def,axiom,
! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_dr(int,fun(int,int))),Qr) ).
% Divides.adjust_div_def
tff(fact_1542_sum__gp__basic,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(nat,nat,suc,N))) ) ).
% sum_gp_basic
tff(fact_1543_sum__power__shift,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M: nat,N: nat,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ).
% sum_power_shift
tff(fact_1544_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ds(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_du(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).
% sum.triangle_reindex_eq
tff(fact_1545_and__int__rec,axiom,
! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).
% and_int_rec
tff(fact_1546_mask__half__int,axiom,
! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,bit_se2239418461657761734s_mask(int),N)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(nat,int,bit_se2239418461657761734s_mask(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).
% mask_half_int
tff(fact_1547_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_cx(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).
% sum.in_pairs_0
tff(fact_1548_mask__nat__def,axiom,
! [N: nat] : aa(nat,nat,bit_se2239418461657761734s_mask(nat),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)) ).
% mask_nat_def
tff(fact_1549_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),N) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ).
% mask_eq_exp_minus_1
tff(fact_1550_sum_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dw(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% sum.zero_middle
tff(fact_1551_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,bitM(N))) ).
% and_minus_numerals(3)
tff(fact_1552_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,bitM(N))) ).
% and_minus_numerals(7)
tff(fact_1553_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,aa(num,num,bit0,N))) ).
% and_minus_numerals(4)
tff(fact_1554_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,aa(num,num,bit0,N))) ).
% and_minus_numerals(8)
tff(fact_1555_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dx(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).
% gbinomial_partial_row_sum
tff(fact_1556_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ) ).
% gbinomial_r_part_sum
tff(fact_1557_and__num_Oelims,axiom,
! [X2: num,Xa: num,Y: option(num)] :
( ( bit_un7362597486090784418nd_num(X2,Xa) = Y )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y != aa(num,option(num),some(num),one2) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N3: num] : Xa = aa(num,num,bit0,N3)
=> ( Y != none(num) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N3: num] : Xa = aa(num,num,bit1,N3)
=> ( Y != aa(num,option(num),some(num),one2) ) ) )
=> ( ( ? [M4: num] : X2 = aa(num,num,bit0,M4)
=> ( ( Xa = one2 )
=> ( Y != none(num) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M4,N3)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M4,N3)) ) ) )
=> ( ( ? [M4: num] : X2 = aa(num,num,bit1,M4)
=> ( ( Xa = one2 )
=> ( Y != aa(num,option(num),some(num),one2) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M4,N3)) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_dy(num,option(num))),bit_un7362597486090784418nd_num(M4,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
tff(fact_1558_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,N)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ).
% take_bit_Suc_from_most
tff(fact_1559_gbinomial__1,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).
% gbinomial_1
tff(fact_1560_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).
% gbinomial_0(1)
tff(fact_1561_and__nat__numerals_I2_J,axiom,
! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = one_one(nat) ).
% and_nat_numerals(2)
tff(fact_1562_and__nat__numerals_I4_J,axiom,
! [X2: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X2))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).
% and_nat_numerals(4)
tff(fact_1563_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,B2: A,N: nat] :
( ! [N3: nat] :
( ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N3))
| ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),B2),N3)) )
=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N))
<=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))
| pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),B2),N)) ) ) ) ) ).
% bit_disjunctive_add_iff
tff(fact_1564_bit__1__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),one_one(A)),N))
<=> ( N = zero_zero(nat) ) ) ) ).
% bit_1_iff
tff(fact_1565_not__bit__1__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(nat,nat,suc,N))) ) ).
% not_bit_1_Suc
tff(fact_1566_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: num] : ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(num,nat,numeral_numeral(nat),N))) ) ).
% bit_numeral_simps(1)
tff(fact_1567_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).
% gbinomial_Suc_Suc
tff(fact_1568_and__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),X2)) ).
% and_integer.abs_eq
tff(fact_1569_gbinomial__addition__formula,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).
% gbinomial_addition_formula
tff(fact_1570_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).
% gbinomial_absorb_comp
tff(fact_1571_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).
% gbinomial_mult_1'
tff(fact_1572_gbinomial__mult__1,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,A,gbinomial(A,A3),K))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)))) ) ).
% gbinomial_mult_1
tff(fact_1573_Suc__times__gbinomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).
% Suc_times_gbinomial
tff(fact_1574_gbinomial__absorption,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).
% gbinomial_absorption
tff(fact_1575_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dz(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).
% gbinomial_parallel_sum
tff(fact_1576_gbinomial__rec,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).
% gbinomial_rec
tff(fact_1577_gbinomial__factors,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).
% gbinomial_factors
tff(fact_1578_gbinomial__negated__upper,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).
% gbinomial_negated_upper
tff(fact_1579_gbinomial__index__swap,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),N)) ) ).
% gbinomial_index_swap
tff(fact_1580_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),M5))
=> ( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),M5))
<=> pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N3)) ) )
=> ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N3),one_one(nat))))
<=> ~ pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N3)) ) ) ) ).
% int_bit_bound
tff(fact_1581_gbinomial__minus,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).
% gbinomial_minus
tff(fact_1582_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ) ) ).
% gbinomial_reduce_nat
tff(fact_1583_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ea(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),M)) ) ).
% gbinomial_sum_lower_neg
tff(fact_1584_numeral__and__num,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_un7362597486090784418nd_num(M,N)) ) ).
% numeral_and_num
tff(fact_1585_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eb(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).
% gbinomial_sum_up_index
tff(fact_1586_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,A3: A,X2: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ec(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X2),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ed(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X2),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).
% gbinomial_partial_sum_poly
tff(fact_1587_even__bit__succ__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N))
<=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))
| ( N = zero_zero(nat) ) ) ) ) ) ).
% even_bit_succ_iff
tff(fact_1588_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))
=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))
<=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N))
| ( N = zero_zero(nat) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
tff(fact_1589_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_dy(num,option(num))),bit_un7362597486090784418nd_num(M,N)) ).
% and_num.simps(9)
tff(fact_1590_gbinomial__absorption_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).
% gbinomial_absorption'
tff(fact_1591_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,A3: A,X2: A,Y: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ec(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X2),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ee(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X2),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).
% gbinomial_partial_sum_poly_xpos
tff(fact_1592_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ef(nat,fun(nat,A),M)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) ) ).
% gbinomial_sum_nat_pow2
tff(fact_1593_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A,N: nat] :
( ! [J3: nat] : ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,suc,J3)))
=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),N))
<=> ( ( ( N = zero_zero(nat) )
=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)) )
& ( ( N != zero_zero(nat) )
=> pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ) ) ).
% bit_sum_mult_2_cases
tff(fact_1594_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_dy(num,option(num))),bit_and_not_num(M,N)) ).
% and_not_num.simps(8)
tff(fact_1595_bit__rec,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))
<=> ( ( ( N = zero_zero(nat) )
=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)) )
& ( ( N != zero_zero(nat) )
=> pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ) ).
% bit_rec
tff(fact_1596_and__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( ( M = zero_zero(nat) )
| ( N = zero_zero(nat) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = zero_zero(nat) ) )
& ( ~ ( ( M = zero_zero(nat) )
| ( N = zero_zero(nat) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).
% and_nat_unfold
tff(fact_1597_and__nat__rec,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),M)),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).
% and_nat_rec
tff(fact_1598_set__bit__eq,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).
% set_bit_eq
tff(fact_1599_and__not__num_Oelims,axiom,
! [X2: num,Xa: num,Y: option(num)] :
( ( bit_and_not_num(X2,Xa) = Y )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y != none(num) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N3: num] : Xa = aa(num,num,bit0,N3)
=> ( Y != aa(num,option(num),some(num),one2) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N3: num] : Xa = aa(num,num,bit1,N3)
=> ( Y != none(num) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ( ( Xa = one2 )
=> ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M4)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M4,N3)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M4,N3)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ( ( Xa = one2 )
=> ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M4)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_dy(num,option(num))),bit_and_not_num(M4,N3)) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M4,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
tff(fact_1600_and__not__num_Opelims,axiom,
! [X2: num,Xa: num,Y: option(num)] :
( ( bit_and_not_num(X2,Xa) = Y )
=> ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X2),Xa)))
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = none(num) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3)))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = none(num) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_dy(num,option(num))),bit_and_not_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_and_not_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N3)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
tff(fact_1601_and__num_Opelims,axiom,
! [X2: num,Xa: num,Y: option(num)] :
( ( bit_un7362597486090784418nd_num(X2,Xa) = Y )
=> ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X2),Xa)))
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = none(num) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3)))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = none(num) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),one2) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un7362597486090784418nd_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_dy(num,option(num))),bit_un7362597486090784418nd_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N3)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
tff(fact_1602_gbinomial__code,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] :
( ( ( K = zero_zero(nat) )
=> ( aa(nat,A,gbinomial(A,A3),K) = one_one(A) ) )
& ( ( K != zero_zero(nat) )
=> ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_eg(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K)) ) ) ) ) ).
% gbinomial_code
tff(fact_1603_xor__num_Oelims,axiom,
! [X2: num,Xa: num,Y: option(num)] :
( ( bit_un2480387367778600638or_num(X2,Xa) = Y )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y != none(num) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(num,option(num),some(num),aa(num,num,bit1,N3)) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(num,option(num),some(num),aa(num,num,bit0,N3)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ( ( Xa = one2 )
=> ( Y != aa(num,option(num),some(num),aa(num,num,bit1,M4)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M4,N3)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_un2480387367778600638or_num(M4,N3))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ( ( Xa = one2 )
=> ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M4)) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( Y != aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_un2480387367778600638or_num(M4,N3))) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( Y != aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M4,N3)) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
tff(fact_1604_choose__alternating__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eh(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).
% choose_alternating_sum
tff(fact_1605_choose__linear__sum,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ei(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).
% choose_linear_sum
tff(fact_1606_binomial__r__part__sum,axiom,
! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ).
% binomial_r_part_sum
tff(fact_1607_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ej(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_du(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% sum.triangle_reindex
tff(fact_1608_lessThan__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y: A] :
( ( aa(A,set(A),set_ord_lessThan(A),X2) = aa(A,set(A),set_ord_lessThan(A),Y) )
<=> ( X2 = Y ) ) ) ).
% lessThan_eq_iff
tff(fact_1609_lessThan__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,aa(A,set(A),set_ord_lessThan(A),K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),K)) ) ) ).
% lessThan_iff
tff(fact_1610_binomial__n__n,axiom,
! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).
% binomial_n_n
tff(fact_1611_lessThan__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),X2)),aa(A,set(A),set_ord_lessThan(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y)) ) ) ).
% lessThan_subset_iff
tff(fact_1612_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ).
% binomial_Suc_Suc
tff(fact_1613_binomial__n__0,axiom,
! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).
% binomial_n_0
tff(fact_1614_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ).
% sum.lessThan_Suc
tff(fact_1615_choose__one,axiom,
! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).
% choose_one
tff(fact_1616_lessThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ek(A,fun(A,bool),U)) ) ).
% lessThan_def
tff(fact_1617_choose__mult__lemma,axiom,
! [M: nat,R: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),M)) ).
% choose_mult_lemma
tff(fact_1618_lessThan__Suc__atMost,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_atMost(nat),K) ).
% lessThan_Suc_atMost
tff(fact_1619_bit__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,fun(nat,bool),bit_se5641148757651400278ts_bit(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),X2) ).
% bit_integer.abs_eq
tff(fact_1620_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N)) ) ) ).
% lessThan_strict_subset_iff
tff(fact_1621_Suc__times__binomial__add,axiom,
! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A3)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),aa(nat,nat,suc,A3))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),A3)) ).
% Suc_times_binomial_add
tff(fact_1622_binomial__absorb__comp,axiom,
! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).
% binomial_absorb_comp
tff(fact_1623_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] : pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))) ) ).
% fact_fact_dvd_fact
tff(fact_1624_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% sum.nat_diff_reindex
tff(fact_1625_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% Iic_subset_Iio_iff
tff(fact_1626_binomial__absorption,axiom,
! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).
% binomial_absorption
tff(fact_1627_sum__diff__distrib,axiom,
! [A: $tType] :
( ord(A)
=> ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
( ! [X3: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P,X3)))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),N))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),N))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_em(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),N)) ) ) ) ).
% sum_diff_distrib
tff(fact_1628_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% sum.lessThan_Suc_shift
tff(fact_1629_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_br(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,M)),aa(nat,A,F,zero_zero(nat))) ) ).
% sum_lessThan_telescope
tff(fact_1630_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_do(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,M)) ) ).
% sum_lessThan_telescope'
tff(fact_1631_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% sum.atLeast1_atMost_eq
tff(fact_1632_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).
% sum_bounds_lt_plus1
tff(fact_1633_sum_Onested__swap_H,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_en(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ep(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% sum.nested_swap'
tff(fact_1634_sum__choose__lower,axiom,
! [R: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_eq(nat,fun(nat,nat),R)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),N))),N) ).
% sum_choose_lower
tff(fact_1635_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_er(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),M) ).
% choose_rising_sum(2)
tff(fact_1636_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_er(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).
% choose_rising_sum(1)
tff(fact_1637_binomial__code,axiom,
! [N: nat,K: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
=> ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).
% binomial_code
tff(fact_1638_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ) ).
% choose_reduce_nat
tff(fact_1639_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).
% times_binomial_minus1_eq
tff(fact_1640_one__diff__power__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% one_diff_power_eq
tff(fact_1641_power__diff__1__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% power_diff_1_eq
tff(fact_1642_geometric__sum,axiom,
! [A: $tType] :
( field(A)
=> ! [X2: A,N: nat] :
( ( X2 != one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),one_one(A))) ) ) ) ).
% geometric_sum
tff(fact_1643_sum_OatMost__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% sum.atMost_shift
tff(fact_1644_vandermonde,axiom,
! [M: nat,N: nat,R: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_es(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R)),aa(nat,set(nat),set_ord_atMost(nat),R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R) ).
% vandermonde
tff(fact_1645_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ).
% binomial_addition_formula
tff(fact_1646_power__diff__sumr2,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_et(A,fun(nat,fun(A,fun(nat,A))),X2),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% power_diff_sumr2
tff(fact_1647_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,N: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_eu(A,fun(nat,fun(A,fun(nat,A))),X2),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ).
% diff_power_eq_sum
tff(fact_1648_sum__gp__strict,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X2: A,N: nat] :
( ( ( X2 = one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
& ( ( X2 != one_one(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X2)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)) ) ) ) ) ).
% sum_gp_strict
tff(fact_1649_binomial,axiom,
! [A3: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),N) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ev(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ).
% binomial
tff(fact_1650_choose__two,axiom,
! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).
% choose_two
tff(fact_1651_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X2: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ew(A,fun(nat,fun(nat,A)),X2),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% one_diff_power_eq'
tff(fact_1652_binomial__ring,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ex(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).
% binomial_ring
tff(fact_1653_xor__num_Osimps_I6_J,axiom,
! [M: num,N: num] : bit_un2480387367778600638or_num(aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_un2480387367778600638or_num(M,N))) ).
% xor_num.simps(6)
tff(fact_1654_xor__num_Osimps_I8_J,axiom,
! [M: num,N: num] : bit_un2480387367778600638or_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_un2480387367778600638or_num(M,N))) ).
% xor_num.simps(8)
tff(fact_1655_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [N: nat] :
( ( N != one_one(nat) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ey(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).
% choose_alternating_linear_sum
tff(fact_1656_fact__Suc__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).
% fact_Suc_0
tff(fact_1657_fact__1,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).
% fact_1
tff(fact_1658_fact__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).
% fact_0
tff(fact_1659_fact__code,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N,one_one(nat))) ) ).
% fact_code
tff(fact_1660_fact__reduce,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).
% fact_reduce
tff(fact_1661_fact__num__eq__if,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M: nat] :
( ( ( M = zero_zero(nat) )
=> ( semiring_char_0_fact(A,M) = one_one(A) ) )
& ( ( M != zero_zero(nat) )
=> ( semiring_char_0_fact(A,M) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).
% fact_num_eq_if
tff(fact_1662_xor__num_Opelims,axiom,
! [X2: num,Xa: num,Y: option(num)] :
( ( bit_un2480387367778600638or_num(X2,Xa) = Y )
=> ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X2),Xa)))
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = none(num) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3)))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_un2480387367778600638or_num(M4,N3))) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_un2480387367778600638or_num(M4,N3))) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(option(num),option(num),aa(fun(num,num),fun(option(num),option(num)),map_option(num,num),bit0),bit_un2480387367778600638or_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N3)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
tff(fact_1663_dvd__fact,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> pp(dvd_dvd(nat,M,semiring_char_0_fact(nat,N))) ) ) ).
% dvd_fact
tff(fact_1664_fact__ge__1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).
% fact_ge_1
tff(fact_1665_pochhammer__double,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Z2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),comm_s3205402744901411588hammer(A,Z2,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N)) ) ).
% pochhammer_double
tff(fact_1666_fact__double,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))),semiring_char_0_fact(A,N)) ) ).
% fact_double
tff(fact_1667_pochhammer__code,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,A3,N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,A3,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ez(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A)) ) ) ) ) ).
% pochhammer_code
tff(fact_1668_xor__Suc__0__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))) ).
% xor_Suc_0_eq
tff(fact_1669_Suc__0__xor__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))) ).
% Suc_0_xor_eq
tff(fact_1670_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).
% push_bit_numeral_minus_1
tff(fact_1671_num__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
=> ( aa(code_integer,num,code_num_of_integer,K) = one2 ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
=> ( aa(code_integer,num,code_num_of_integer,K) = aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_fa(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).
% num_of_integer_code
tff(fact_1672_Sum__Ico__nat,axiom,
! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bx(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).
% Sum_Ico_nat
tff(fact_1673_ivl__diff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [I: A,N: A,M: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),N))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).
% ivl_diff
tff(fact_1674_push__bit__push__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3) ) ).
% push_bit_push_bit
tff(fact_1675_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [N: A,M: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(A,set(A),set_ord_lessThan(A),N)),aa(A,set(A),set_ord_lessThan(A),M)) = set_or7035219750837199246ssThan(A,M,N) ) ).
% lessThan_minus_lessThan
tff(fact_1676_pochhammer__1,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).
% pochhammer_1
tff(fact_1677_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,L: A,U: A] :
( pp(member(A,I,set_or7035219750837199246ssThan(A,L,U)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),U)) ) ) ) ).
% atLeastLessThan_iff
tff(fact_1678_ivl__subset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [I: A,J: A,M: A,N: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,N)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),I))
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),I))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),N)) ) ) ) ) ).
% ivl_subset
tff(fact_1679_pochhammer__0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).
% pochhammer_0
tff(fact_1680_xor__numerals_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% xor_numerals(1)
tff(fact_1681_xor__numerals_I2_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y)) ) ).
% xor_numerals(2)
tff(fact_1682_xor__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X2))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2)) ) ).
% xor_numerals(5)
tff(fact_1683_xor__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,X2)) ) ).
% xor_numerals(8)
tff(fact_1684_push__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ).
% push_bit_of_1
tff(fact_1685_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [N: nat,M: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).
% sum.op_ivl_Suc
tff(fact_1686_xor__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X2)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% xor_numerals(4)
tff(fact_1687_xor__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X2)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% xor_numerals(6)
tff(fact_1688_flip__bit__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).
% flip_bit_nat_def
tff(fact_1689_push__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Xa),X2)) ).
% push_bit_integer.abs_eq
tff(fact_1690_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
=> ( B2 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
tff(fact_1691_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
tff(fact_1692_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D3))
=> ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D3) )
<=> ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
tff(fact_1693_flip__bit__eq__xor,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) ) ).
% flip_bit_eq_xor
tff(fact_1694_push__bit__add,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),B2)) ) ).
% push_bit_add
tff(fact_1695_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% atLeastLessThan_subset_iff
tff(fact_1696_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).
% atLeastLessThanSuc_atLeastAtMost
tff(fact_1697_lessThan__atLeast0,axiom,
! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).
% lessThan_atLeast0
tff(fact_1698_push__bit__take__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A3)) ) ).
% push_bit_take_bit
tff(fact_1699_pochhammer__fact,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& comm_semiring_1(A) )
=> ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).
% pochhammer_fact
tff(fact_1700_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bn(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.shift_bounds_Suc_ivl
tff(fact_1701_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.shift_bounds_nat_ivl
tff(fact_1702_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& comm_monoid_add(A) )
=> ! [A3: B,C2: B,B2: B,D3: B,G: fun(B,A),H: fun(B,A)] :
( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ).
% sum.ivl_cong
tff(fact_1703_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,P3: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).
% sum.atLeastLessThan_concat
tff(fact_1704_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M: nat,N: nat,P3: nat,F: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,M,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,M,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,N,P3)) ) ) ) ) ).
% sum_diff_nat_ivl
tff(fact_1705_pochhammer__0__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).
% pochhammer_0_left
tff(fact_1706_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_1707_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.atLeastLessThan_shift_bounds
tff(fact_1708_concat__bit__eq,axiom,
! [N: nat,K: int,L: int] : bit_concat_bit(N,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L)) ).
% concat_bit_eq
tff(fact_1709_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_1710_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_1711_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(nat,A),K: nat] :
( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
tff(fact_1712_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).
% sum.atLeast0_lessThan_Suc
tff(fact_1713_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).
% sum.atLeast_Suc_lessThan
tff(fact_1714_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: nat,B2: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).
% sum.atLeastLessThan_Suc
tff(fact_1715_sum_Olast__plus,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).
% sum.last_plus
tff(fact_1716_pochhammer__rec,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),N)) ) ).
% pochhammer_rec
tff(fact_1717_pochhammer__Suc,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))) ) ).
% pochhammer_Suc
tff(fact_1718_pochhammer__rec_H,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z2: A,N: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N))),comm_s3205402744901411588hammer(A,Z2,N)) ) ).
% pochhammer_rec'
tff(fact_1719_pochhammer__product_H,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z2: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),N)),M)) ) ).
% pochhammer_product'
tff(fact_1720_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))
<=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) != zero_zero(A) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
tff(fact_1721_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [M: nat,N: nat,F: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_br(fun(nat,A),fun(nat,A),F)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,N)),aa(nat,A,F,M)) ) ) ) ).
% sum_Suc_diff'
tff(fact_1722_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fb(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).
% sum.atLeastLessThan_rev
tff(fact_1723_sum_Onested__swap,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fc(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ep(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% sum.nested_swap
tff(fact_1724_sum_Onat__group,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).
% sum.nat_group
tff(fact_1725_numeral__xor__num,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_un2480387367778600638or_num(M,N)) ) ).
% numeral_xor_num
tff(fact_1726_sum_Ohead__if,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [N: nat,M: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).
% sum.head_if
tff(fact_1727_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% sum.atLeast0_lessThan_Suc_shift
tff(fact_1728_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).
% sum.atLeastLessThan_shift_0
tff(fact_1729_pochhammer__product,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: nat,N: nat,Z2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( comm_s3205402744901411588hammer(A,Z2,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).
% pochhammer_product
tff(fact_1730_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_1731_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_bt(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.atLeast_lessThan_pred_shift
tff(fact_1732_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [R: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K)) ) ).
% pochhammer_absorb_comp
tff(fact_1733_pochhammer__same,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& comm_ring_1(A)
& semiri3467727345109120633visors(A) )
=> ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).
% pochhammer_same
tff(fact_1734_pochhammer__minus,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).
% pochhammer_minus
tff(fact_1735_pochhammer__minus_H,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),K)) ) ).
% pochhammer_minus'
tff(fact_1736_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).
% gbinomial_pochhammer'
tff(fact_1737_gbinomial__pochhammer,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K))),semiring_char_0_fact(A,K)) ) ).
% gbinomial_pochhammer
tff(fact_1738_xor__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = N ) )
& ( ( M != zero_zero(nat) )
=> ( ( ( N = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = M ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).
% xor_nat_unfold
tff(fact_1739_num__of__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,num,code_num_of_integer,aa(int,code_integer,code_integer_of_int,X2)) = aa(nat,num,num_of_nat,aa(int,nat,nat2,X2)) ).
% num_of_integer.abs_eq
tff(fact_1740_signed__take__bit__code,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = if(A,aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),A3)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),A3)) ) ).
% signed_take_bit_code
tff(fact_1741_xor__nat__rec,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),M))),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).
% xor_nat_rec
tff(fact_1742_xor__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))) ) ).
% xor_one_eq
tff(fact_1743_one__xor__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3)))) ) ).
% one_xor_eq
tff(fact_1744_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [A3: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fe(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).
% pochhammer_binomial_sum
tff(fact_1745_divmod__integer__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L)) ).
% divmod_integer_def
tff(fact_1746_sum__power2,axiom,
! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).
% sum_power2
tff(fact_1747_nat__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,nat,code_nat_of_integer,K) = aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_ff(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).
% nat_of_integer_code
tff(fact_1748_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Z2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z2,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fg(A,fun(nat,A),Z2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)))) ) ).
% pochhammer_times_pochhammer_half
tff(fact_1749_int__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,int,code_int_of_integer,K) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( ( ( K = zero_zero(code_integer) )
=> ( aa(code_integer,int,code_int_of_integer,K) = zero_zero(int) ) )
& ( ( K != zero_zero(code_integer) )
=> ( aa(code_integer,int,code_int_of_integer,K) = aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_fh(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ) ) ).
% int_of_integer_code
tff(fact_1750_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( ( L = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
& ( ( L != zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),aa(fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer),aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),aa(fun(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer),product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_fi(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ).
% divmod_integer_eq_cases
tff(fact_1751_time__def,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit)] : time_time(A,C2,H) = aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),nat,aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),nat),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),nat),aa(nat,fun(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),nat),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),nat)),case_option(nat,product_prod(A,product_prod(heap_ext(product_unit),nat))),zero_zero(nat)),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),nat)),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),nat),product_case_prod(A,product_prod(heap_ext(product_unit),nat),nat),aTP_Lamp_fk(A,fun(product_prod(heap_ext(product_unit),nat),nat)))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H)) ).
% time_def
tff(fact_1752_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,bool,fNot,dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)),K))) ).
% bit_cut_integer_def
tff(fact_1753_integer__of__num_I3_J,axiom,
! [N: num] : aa(num,code_integer,code_integer_of_num,aa(num,num,bit1,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(num,code_integer,code_integer_of_num,N)),aa(num,code_integer,code_integer_of_num,N))),one_one(code_integer)) ).
% integer_of_num(3)
tff(fact_1754_less__eq__int_Orep__eq,axiom,
! [X2: int,Xa: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),Xa))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X2)),aa(int,product_prod(nat,nat),rep_Integ,Xa))) ) ).
% less_eq_int.rep_eq
tff(fact_1755_nat__of__integer__of__nat,axiom,
! [N: nat] : aa(code_integer,nat,code_nat_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),N)) = N ).
% nat_of_integer_of_nat
tff(fact_1756_int__of__integer__of__nat,axiom,
! [N: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),N)) = aa(nat,int,semiring_1_of_nat(int),N) ).
% int_of_integer_of_nat
tff(fact_1757_integer__of__int__int__of__integer,axiom,
! [K: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,K)) = K ).
% integer_of_int_int_of_integer
tff(fact_1758_int__of__integer__integer__of__int,axiom,
! [K: int] : aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,K)) = K ).
% int_of_integer_integer_of_int
tff(fact_1759_int__of__integer__inverse,axiom,
! [X2: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,X2)) = X2 ).
% int_of_integer_inverse
tff(fact_1760_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X2: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X2),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),aa(C,B,F,Y)) ).
% apsnd_conv
tff(fact_1761_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fl(B,A)),A4) = one_one(A) ) ).
% prod.neutral_const
tff(fact_1762_of__int__integer__of,axiom,
! [K: code_integer] : aa(int,code_integer,ring_1_of_int(code_integer),aa(code_integer,int,code_int_of_integer,K)) = K ).
% of_int_integer_of
tff(fact_1763_int__of__integer__of__int,axiom,
! [K: int] : aa(code_integer,int,code_int_of_integer,aa(int,code_integer,ring_1_of_int(code_integer),K)) = K ).
% int_of_integer_of_int
tff(fact_1764_zero__integer_Orep__eq,axiom,
aa(code_integer,int,code_int_of_integer,zero_zero(code_integer)) = zero_zero(int) ).
% zero_integer.rep_eq
tff(fact_1765_int__of__integer__numeral,axiom,
! [K: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).
% int_of_integer_numeral
tff(fact_1766_plus__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% plus_integer.rep_eq
tff(fact_1767_one__integer_Orep__eq,axiom,
aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).
% one_integer.rep_eq
tff(fact_1768_uminus__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),X2)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% uminus_integer.rep_eq
tff(fact_1769_times__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% times_integer.rep_eq
tff(fact_1770_minus__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% minus_integer.rep_eq
tff(fact_1771_abs__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,abs_abs(code_integer),X2)) = aa(int,int,abs_abs(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% abs_integer.rep_eq
tff(fact_1772_divide__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% divide_integer.rep_eq
tff(fact_1773_modulo__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% modulo_integer.rep_eq
tff(fact_1774_sgn__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,sgn_sgn(code_integer),X2)) = aa(int,int,sgn_sgn(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% sgn_integer.rep_eq
tff(fact_1775_prod_OatMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% prod.atMost_Suc
tff(fact_1776_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ).
% prod.lessThan_Suc
tff(fact_1777_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) ) ).
% nat_of_integer_non_positive
tff(fact_1778_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,M: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).
% prod.cl_ivl_Suc
tff(fact_1779_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,M: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).
% prod.op_ivl_Suc
tff(fact_1780_xor__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% xor_integer.rep_eq
tff(fact_1781_push__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),X2),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),X2),aa(code_integer,int,code_int_of_integer,Xa)) ).
% push_bit_integer.rep_eq
tff(fact_1782_xor__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),X2)) ).
% xor_integer.abs_eq
tff(fact_1783_nat__of__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,nat,code_nat_of_integer,X2) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X2)) ).
% nat_of_integer.rep_eq
tff(fact_1784_integer__eq__iff,axiom,
! [K: code_integer,L: code_integer] :
( ( K = L )
<=> ( aa(code_integer,int,code_int_of_integer,K) = aa(code_integer,int,code_int_of_integer,L) ) ) ).
% integer_eq_iff
tff(fact_1785_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( aa(code_integer,int,code_int_of_integer,K) = aa(code_integer,int,code_int_of_integer,L) )
=> ( K = L ) ) ).
% integer_eqI
tff(fact_1786_int__of__integer__inject,axiom,
! [X2: code_integer,Y: code_integer] :
( ( aa(code_integer,int,code_int_of_integer,X2) = aa(code_integer,int,code_int_of_integer,Y) )
<=> ( X2 = Y ) ) ).
% int_of_integer_inject
tff(fact_1787_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A4: set(B)] :
( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) != one_one(A) )
=> ~ ! [A5: B] :
( pp(member(B,A5,A4))
=> ( aa(B,A,G,A5) = one_one(A) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_1788_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( ! [X3: B] :
( pp(member(B,X3,A4))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = one_one(A) ) ) ) ).
% prod.neutral
tff(fact_1789_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(C,B),G: fun(D,C),X2: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G),X2)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),F),G)),X2) ).
% apsnd_compose
tff(fact_1790_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: set(B),F: fun(B,A)] :
( ! [X3: B] :
( pp(member(B,X3,A4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F,X3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4))) ) ) ).
% prod_ge_1
tff(fact_1791_int__of__integer__less__iff,axiom,
! [X2: code_integer,Y: code_integer] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Y)))
<=> pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X2),Y)) ) ).
% int_of_integer_less_iff
tff(fact_1792_integer__less__iff,axiom,
! [K: code_integer,L: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),L))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L))) ) ).
% integer_less_iff
tff(fact_1793_less__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X2),Xa))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa))) ) ).
% less_integer.rep_eq
tff(fact_1794_less__eq__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),X2),Xa))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa))) ) ).
% less_eq_integer.rep_eq
tff(fact_1795_integer__less__eq__iff,axiom,
! [K: code_integer,L: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),L))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L))) ) ).
% integer_less_eq_iff
tff(fact_1796_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.shift_bounds_cl_Suc_ivl
tff(fact_1797_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.shift_bounds_Suc_ivl
tff(fact_1798_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.shift_bounds_cl_nat_ivl
tff(fact_1799_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.shift_bounds_nat_ivl
tff(fact_1800_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_1801_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( linord181362715937106298miring(A)
=> ! [A4: set(B),F: fun(B,A)] :
( ! [X3: B] :
( pp(member(B,X3,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X3)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X3)),one_one(A))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),one_one(A))) ) ) ).
% prod_le_1
tff(fact_1802_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& comm_monoid_mult(A) )
=> ! [A3: B,C2: B,B2: B,D3: B,G: fun(B,A),H: fun(B,A)] :
( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ).
% prod.ivl_cong
tff(fact_1803_take__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),X2),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),X2),aa(code_integer,int,code_int_of_integer,Xa)) ).
% take_bit_integer.rep_eq
tff(fact_1804_and__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% and_integer.rep_eq
tff(fact_1805_bit__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,fun(nat,bool),bit_se5641148757651400278ts_bit(code_integer),X2) = aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% bit_integer.rep_eq
tff(fact_1806_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,P3: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).
% prod.atLeastLessThan_concat
tff(fact_1807_mask__integer_Orep__eq,axiom,
! [X2: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),X2)) = aa(nat,int,bit_se2239418461657761734s_mask(int),X2) ).
% mask_integer.rep_eq
tff(fact_1808_integer__of__num__def,axiom,
code_integer_of_num = numeral_numeral(code_integer) ).
% integer_of_num_def
tff(fact_1809_unset__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),X2),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),X2),aa(code_integer,int,code_int_of_integer,Xa)) ).
% unset_bit_integer.rep_eq
tff(fact_1810_set__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),X2),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),X2),aa(code_integer,int,code_int_of_integer,Xa)) ).
% set_bit_integer.rep_eq
tff(fact_1811_flip__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),X2),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),X2),aa(code_integer,int,code_int_of_integer,Xa)) ).
% flip_bit_integer.rep_eq
tff(fact_1812_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_1813_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_1814_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_1815_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.atLeastAtMost_shift_bounds
tff(fact_1816_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.atLeastLessThan_shift_bounds
tff(fact_1817_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fo(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).
% prod.atLeastAtMost_rev
tff(fact_1818_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fp(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% prod.nat_diff_reindex
tff(fact_1819_nat__of__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,nat,code_nat_of_integer,aa(int,code_integer,code_integer_of_int,X2)) = aa(int,nat,nat2,X2) ).
% nat_of_integer.abs_eq
tff(fact_1820_num__of__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,num,code_num_of_integer,X2) = aa(nat,num,num_of_nat,aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X2))) ).
% num_of_integer.rep_eq
tff(fact_1821_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% prod.atLeast0_atMost_Suc
tff(fact_1822_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).
% prod.atLeast0_lessThan_Suc
tff(fact_1823_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).
% prod.atLeast_Suc_lessThan
tff(fact_1824_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% prod.nat_ivl_Suc'
tff(fact_1825_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).
% prod.atLeast_Suc_atMost
tff(fact_1826_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: nat,B2: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).
% prod.atLeastLessThan_Suc
tff(fact_1827_divmod__abs__code_I6_J,axiom,
! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).
% divmod_abs_code(6)
tff(fact_1828_prod_Olast__plus,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).
% prod.last_plus
tff(fact_1829_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_1830_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).
% prod.atMost_Suc_shift
tff(fact_1831_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% prod.lessThan_Suc_shift
tff(fact_1832_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).
% prod.atLeastLessThan_rev
tff(fact_1833_prod_Onested__swap,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fr(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% prod.nested_swap
tff(fact_1834_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% prod.atLeast1_atMost_eq
tff(fact_1835_prod_Onat__group,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).
% prod.nat_group
tff(fact_1836_fact__prod,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bx(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).
% fact_prod
tff(fact_1837_prod_Onested__swap_H,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fv(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% prod.nested_swap'
tff(fact_1838_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fw(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,one_one(A)) ) ).
% prod_atLeastAtMost_code
tff(fact_1839_prod_Ohead__if,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,M: nat,G: fun(nat,A)] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).
% prod.head_if
tff(fact_1840_prod_Oub__add__nat,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A),P3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).
% prod.ub_add_nat
tff(fact_1841_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).
% prod.atLeast0_atMost_Suc_shift
tff(fact_1842_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% prod.atLeast0_lessThan_Suc_shift
tff(fact_1843_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).
% prod.atLeastLessThan_shift_0
tff(fact_1844_integer__of__num__triv_I1_J,axiom,
aa(num,code_integer,code_integer_of_num,one2) = one_one(code_integer) ).
% integer_of_num_triv(1)
tff(fact_1845_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fo(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_1846_prod_OatMost__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).
% prod.atMost_shift
tff(fact_1847_pochhammer__prod,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fx(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% pochhammer_prod
tff(fact_1848_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_bt(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.atLeast_atMost_pred_shift
tff(fact_1849_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_bt(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.atLeast_lessThan_pred_shift
tff(fact_1850_integer__of__num_I2_J,axiom,
! [N: num] : aa(num,code_integer,code_integer_of_num,aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(num,code_integer,code_integer_of_num,N)),aa(num,code_integer,code_integer_of_num,N)) ).
% integer_of_num(2)
tff(fact_1851_divmod__abs__code_I5_J,axiom,
! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).
% divmod_abs_code(5)
tff(fact_1852_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).
% prod.atLeastAtMost_shift_0
tff(fact_1853_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.atLeast_int_lessThan_int_shift
tff(fact_1854_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fx(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).
% pochhammer_Suc_prod
tff(fact_1855_pochhammer__prod__rev,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fy(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).
% pochhammer_prod_rev
tff(fact_1856_fact__div__fact,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bx(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).
% fact_div_fact
tff(fact_1857_integer__of__num__triv_I2_J,axiom,
aa(num,code_integer,code_integer_of_num,aa(num,num,bit0,one2)) = aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)) ).
% integer_of_num_triv(2)
tff(fact_1858_prod_Oin__pairs,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fz(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.in_pairs
tff(fact_1859_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fz(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).
% prod.in_pairs_0
tff(fact_1860_nat__of__integer__less__iff,axiom,
! [X2: code_integer,Y: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),X2))
=> ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),Y))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(code_integer,nat,code_nat_of_integer,X2)),aa(code_integer,nat,code_nat_of_integer,Y)))
<=> pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X2),Y)) ) ) ) ).
% nat_of_integer_less_iff
tff(fact_1861_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fy(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).
% pochhammer_Suc_prod_rev
tff(fact_1862_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
=> ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ga(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
=> ( ( ( L = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
& ( ( L != zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_gb(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).
% divmod_integer_code
tff(fact_1863_bit__cut__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),zero_zero(code_integer)),fFalse) ) )
& ( ( K != zero_zero(code_integer) )
=> ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,bool)),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool)),aTP_Lamp_gc(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).
% bit_cut_integer_code
tff(fact_1864_prod_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ge(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% prod.zero_middle
tff(fact_1865_xor__int__rec,axiom,
! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K))),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).
% xor_int_rec
tff(fact_1866_divmod__abs__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).
% divmod_abs_def
tff(fact_1867_less__int_Orep__eq,axiom,
! [X2: int,Xa: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Xa))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X2)),aa(int,product_prod(nat,nat),rep_Integ,Xa))) ) ).
% less_int.rep_eq
tff(fact_1868_make_H__def,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& heap(A) )
=> ! [I: code_integer,F: fun(B,A)] : array_make2(B,A,I,F) = array_make(A,aa(code_integer,nat,code_nat_of_integer,I),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),F),semiring_1_of_nat(B))) ) ).
% make'_def
tff(fact_1869_new_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ( array_new2(A) = aa(fun(code_integer,nat),fun(code_integer,fun(A,heap_Time_Heap(array(A)))),aa(fun(nat,fun(A,heap_Time_Heap(array(A)))),fun(fun(code_integer,nat),fun(code_integer,fun(A,heap_Time_Heap(array(A))))),comp(nat,fun(A,heap_Time_Heap(array(A))),code_integer),array_new(A)),code_nat_of_integer) ) ) ).
% new'_def
tff(fact_1870_xor__int__unfold,axiom,
! [K: int,L: int] :
( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),L) ) )
& ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),K) ) )
& ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
=> ( ( ( K = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = L ) )
& ( ( K != zero_zero(int) )
=> ( ( ( L = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = K ) )
& ( ( L != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
tff(fact_1871_take__bit__num__code,axiom,
! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_gi(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).
% take_bit_num_code
tff(fact_1872_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ej(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).
% prod.triangle_reindex
tff(fact_1873_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: fun(nat,A),V: num,N: nat] : case_nat(A,A3,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).
% case_nat_add_eq_if
tff(fact_1874_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num,N: nat] : rec_nat(A,A3,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(A,A,aa(nat,fun(A,A),F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)),rec_nat(A,A3,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N))) ).
% rec_nat_add_eq_if
tff(fact_1875_bit_Ocompl__zero,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% bit.compl_zero
tff(fact_1876_bit_Ocompl__one,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).
% bit.compl_one
tff(fact_1877_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X2),aa(A,A,bit_ri4277139882892585799ns_not(A),X2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.xor_cancel_right
tff(fact_1878_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X2)),X2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.xor_cancel_left
tff(fact_1879_bit_Oxor__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X2) ) ).
% bit.xor_one_right
tff(fact_1880_bit_Oxor__one__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X2) = aa(A,A,bit_ri4277139882892585799ns_not(A),X2) ) ).
% bit.xor_one_left
tff(fact_1881_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),N)) ) ).
% push_bit_minus_one_eq_not_mask
tff(fact_1882_not__one__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).
% not_one_eq
tff(fact_1883_not__add__distrib,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B2) ) ).
% not_add_distrib
tff(fact_1884_not__diff__distrib,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B2) ) ).
% not_diff_distrib
tff(fact_1885_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).
% minus_eq_not_plus_1
tff(fact_1886_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).
% minus_eq_not_minus_1
tff(fact_1887_not__eq__complement,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).
% not_eq_complement
tff(fact_1888_prod__int__eq,axiom,
! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_ck(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).
% prod_int_eq
tff(fact_1889_prod__int__plus__eq,axiom,
! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_ck(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).
% prod_int_plus_eq
tff(fact_1890_push__bit__mask__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [M: nat,N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(nat,A,bit_se2239418461657761734s_mask(A),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),M))) ) ).
% push_bit_mask_eq
tff(fact_1891_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.atLeast_int_lessThan_int_shift
tff(fact_1892_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A)))) ) ).
% unset_bit_eq_and_not
tff(fact_1893_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.atLeast_int_atMost_int_shift
tff(fact_1894_int__numeral__not__and__num,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(N,M)) ).
% int_numeral_not_and_num
tff(fact_1895_int__numeral__and__not__num,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,N)) ).
% int_numeral_and_not_num
tff(fact_1896_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ds(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).
% prod.triangle_reindex_eq
tff(fact_1897_and__not__numerals_I8_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% and_not_numerals(8)
tff(fact_1898_not__int__rec,axiom,
! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).
% not_int_rec
tff(fact_1899_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType,N: nat,X2: A,F: fun(nat,A)] :
( ( ( N = zero_zero(nat) )
=> ( case_nat(A,X2,F,N) = X2 ) )
& ( ( N != zero_zero(nat) )
=> ( case_nat(A,X2,F,N) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).
% Nitpick.case_nat_unfold
tff(fact_1900_or__int__rec,axiom,
! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fdisj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).
% or_int_rec
tff(fact_1901_or__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X2)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(4)
tff(fact_1902_or__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X2)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(6)
tff(fact_1903_or__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X2)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(7)
tff(fact_1904_or__nat__rec,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),M)),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).
% or_nat_rec
tff(fact_1905_or__not__numerals_I9_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% or_not_numerals(9)
tff(fact_1906_bit_Odisj__one__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_one_left
tff(fact_1907_bit_Odisj__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_one_right
tff(fact_1908_or__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2)) ) ).
% or_numerals(8)
tff(fact_1909_or__numerals_I2_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% or_numerals(2)
tff(fact_1910_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X2)),X2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_cancel_left
tff(fact_1911_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X2),aa(A,A,bit_ri4277139882892585799ns_not(A),X2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_cancel_right
tff(fact_1912_or__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X2: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X2))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X2)) ) ).
% or_numerals(5)
tff(fact_1913_or__numerals_I1_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% or_numerals(1)
tff(fact_1914_or__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),X2)) ).
% or_integer.abs_eq
tff(fact_1915_or__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),X2),Xa)) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,Xa)) ).
% or_integer.rep_eq
tff(fact_1916_disjunctive__add,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A] :
( ! [N3: nat] :
( ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N3))
| ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),B2),N3)) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) ) ) ) ).
% disjunctive_add
tff(fact_1917_plus__and__or,axiom,
! [X2: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X2),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X2),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),Y) ).
% plus_and_or
tff(fact_1918_not__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),X2)) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% not_integer.rep_eq
tff(fact_1919_set__bit__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).
% set_bit_nat_def
tff(fact_1920_not__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,bit_ri4277139882892585799ns_not(int),X2)) ).
% not_integer.abs_eq
tff(fact_1921_set__bit__eq__or,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) ) ).
% set_bit_eq_or
tff(fact_1922_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,X2: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X2) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X2) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( X2 = Y ) ) ) ) ) ) ).
% bit.complement_unique
tff(fact_1923_bit_Ocompl__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X2: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X2),Y) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),X2) = Y ) ) ) ) ).
% bit.compl_unique
tff(fact_1924_or__one__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).
% or_one_eq
tff(fact_1925_one__or__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A3))) ) ).
% one_or_eq
tff(fact_1926_mask__Suc__double,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),N))) ) ).
% mask_Suc_double
tff(fact_1927_or__not__numerals_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% or_not_numerals(5)
tff(fact_1928_Suc__0__or__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))) ).
% Suc_0_or_eq
tff(fact_1929_or__Suc__0__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))) ).
% or_Suc_0_eq
tff(fact_1930_or__not__numerals_I8_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).
% or_not_numerals(8)
tff(fact_1931_or__int__unfold,axiom,
! [K: int,L: int] :
( ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
| ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
& ( ~ ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
| ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( ( ( K = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = L ) )
& ( ( K != zero_zero(int) )
=> ( ( ( L = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = K ) )
& ( ( L != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,ord_max(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).
% or_int_unfold
tff(fact_1932_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> boolea3799213064322606851m_diff(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A)),bit_se5824344971392196577ns_xor(A)) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_1933_or__nat__unfold,axiom,
! [M: nat,N: nat] :
( ( ( M = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = N ) )
& ( ( M != zero_zero(nat) )
=> ( ( ( N = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = M ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).
% or_nat_unfold
tff(fact_1934_drop__bit__rec,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = A3 ) )
& ( ( N != zero_zero(nat) )
=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).
% drop_bit_rec
tff(fact_1935_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),U))
=> ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image2(nat,code_integer,semiring_1_of_nat(code_integer)),aa(nat,set(nat),set_ord_lessThan(nat),aa(code_integer,nat,code_nat_of_integer,U))) ) ) ).
% image_atLeastZeroLessThan_integer
tff(fact_1936_uminus__int__def,axiom,
uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat))))) ).
% uminus_int_def
tff(fact_1937_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> boolea2506097494486148201lgebra(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% bit.abstract_boolean_algebra_axioms
tff(fact_1938_map__fun__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,F: fun(B,C),G: fun(D,A),H: fun(C,D),X2: B] : aa(B,A,aa(fun(C,D),fun(B,A),map_fun(B,C,D,A,F,G),H),X2) = aa(D,A,G,aa(C,D,H,aa(B,C,F,X2))) ).
% map_fun_apply
tff(fact_1939_drop__bit__drop__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3) ) ).
% drop_bit_drop_bit
tff(fact_1940_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).
% image_add_atLeastAtMost
tff(fact_1941_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [D3: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D3),A3)) ) ).
% image_diff_atLeastAtMost
tff(fact_1942_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).
% image_add_atLeastLessThan
tff(fact_1943_max__0__1_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,ord_max(A,zero_zero(A)),one_one(A)) = one_one(A) ) ) ).
% max_0_1(1)
tff(fact_1944_max__0__1_I2_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,ord_max(A,one_one(A)),zero_zero(A)) = one_one(A) ) ) ).
% max_0_1(2)
tff(fact_1945_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or1337092689740270186AtMost(A,X2,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atLeastAtMost
tff(fact_1946_max__0__1_I6_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X2: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),X2)),one_one(A)) = aa(num,A,numeral_numeral(A),X2) ) ).
% max_0_1(6)
tff(fact_1947_max__0__1_I5_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X2: num] : aa(A,A,ord_max(A,one_one(A)),aa(num,A,numeral_numeral(A),X2)) = aa(num,A,numeral_numeral(A),X2) ) ).
% max_0_1(5)
tff(fact_1948_image__add__atMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [C2: A,A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C2)),aa(A,set(A),set_ord_atMost(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)) ) ).
% image_add_atMost
tff(fact_1949_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_gl(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).
% image_add_atLeastAtMost'
tff(fact_1950_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [D3: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_gm(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D3)) ) ).
% image_minus_const_atLeastAtMost'
tff(fact_1951_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_gl(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).
% image_add_atLeastLessThan'
tff(fact_1952_drop__bit__of__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).
% drop_bit_of_1
tff(fact_1953_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D3),B2)) ) ) ) ).
% image_mult_atLeastAtMost
tff(fact_1954_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D3))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_gn(A,fun(A,A),D3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D3)) ) ) ) ).
% image_divide_atLeastAtMost
tff(fact_1955_drop__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),X2),Xa)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),X2),aa(code_integer,int,code_int_of_integer,Xa)) ).
% drop_bit_integer.rep_eq
tff(fact_1956_drop__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa),X2)) ).
% drop_bit_integer.abs_eq
tff(fact_1957_max__add__distrib__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X2: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,ord_max(A,X2),Y)),Z2) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).
% max_add_distrib_left
tff(fact_1958_max__add__distrib__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X2: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,ord_max(A,Y),Z2)) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Z2)) ) ).
% max_add_distrib_right
tff(fact_1959_max__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,ord_max(A,X2),Y)),Z2) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).
% max_diff_distrib_left
tff(fact_1960_translation__Compl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,T2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).
% translation_Compl
tff(fact_1961_nat__add__max__right,axiom,
! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,ord_max(nat,N),Q3)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ).
% nat_add_max_right
tff(fact_1962_nat__add__max__left,axiom,
! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,M),N)),Q3) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q3)) ).
% nat_add_max_left
tff(fact_1963_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(B,A),A4: set(B),G: fun(C,A),B5: set(C),H: fun(A,D)] :
( ( aa(set(B),set(A),image2(B,A,F),A4) = aa(set(C),set(A),image2(C,A,G),B5) )
=> ( aa(set(B),set(D),image2(B,D,aa(fun(B,A),fun(B,D),aa(fun(A,D),fun(fun(B,A),fun(B,D)),comp(A,D,B),H),F)),A4) = aa(set(C),set(D),image2(C,D,aa(fun(C,A),fun(C,D),aa(fun(A,D),fun(fun(C,A),fun(C,D)),comp(A,D,C),H),G)),B5) ) ) ).
% image_eq_imp_comp
tff(fact_1964_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),R: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(C),set(B),image2(C,B,G),R)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G)),R) ).
% image_comp
tff(fact_1965_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(B,D),H: fun(A,B)] : aa(fun(A,B),fun(C,D),map_fun(C,A,B,D,F,G),H) = aa(fun(C,A),fun(C,D),aa(fun(A,D),fun(fun(C,A),fun(C,D)),comp(A,D,C),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),G),H)),F) ).
% map_fun_def
tff(fact_1966_map__fun_Ocompositionality,axiom,
! [D: $tType,F4: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: fun(E,C),G: fun(D,F4),H: fun(C,A),I: fun(B,D),Fun: fun(A,B)] : aa(fun(C,D),fun(E,F4),map_fun(E,C,D,F4,F,G),aa(fun(A,B),fun(C,D),map_fun(C,A,B,D,H,I),Fun)) = aa(fun(A,B),fun(E,F4),map_fun(E,A,B,F4,aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),H),F),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G),I)),Fun) ).
% map_fun.compositionality
tff(fact_1967_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F4: $tType,D: $tType,B: $tType,F: fun(E,C),G: fun(D,F4),H: fun(C,A),I: fun(B,D)] : aa(fun(fun(A,B),fun(C,D)),fun(fun(A,B),fun(E,F4)),aa(fun(fun(C,D),fun(E,F4)),fun(fun(fun(A,B),fun(C,D)),fun(fun(A,B),fun(E,F4))),comp(fun(C,D),fun(E,F4),fun(A,B)),map_fun(E,C,D,F4,F,G)),map_fun(C,A,B,D,H,I)) = map_fun(E,A,B,F4,aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),H),F),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G),I)) ).
% map_fun.comp
tff(fact_1968_translation__subtract__Compl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,T2: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),T2)) ) ).
% translation_subtract_Compl
tff(fact_1969_nat__minus__add__max,axiom,
! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),M) = aa(nat,nat,ord_max(nat,N),M) ).
% nat_minus_add_max
tff(fact_1970_translation__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).
% translation_diff
tff(fact_1971_translation__subtract__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),S)),aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),T2)) ) ).
% translation_subtract_diff
tff(fact_1972_take__bit__drop__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3)) ) ).
% take_bit_drop_bit
tff(fact_1973_bit__drop__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) = aa(fun(nat,nat),fun(nat,bool),aa(fun(nat,bool),fun(fun(nat,nat),fun(nat,bool)),comp(nat,bool,nat),aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3)),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).
% bit_drop_bit_eq
tff(fact_1974_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) ) ).
% div_push_bit_of_1_eq_drop_bit
tff(fact_1975_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))
<=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)),one_one(A)) = one_one(A) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
tff(fact_1976_bits__ident,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)) = A3 ) ).
% bits_ident
tff(fact_1977_slice__eq__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,M: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),N)))) ) ).
% slice_eq_mask
tff(fact_1978_image__add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [S2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).
% image_add_0
tff(fact_1979_times__int__def,axiom,
times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int_def
tff(fact_1980_minus__int__def,axiom,
minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ct(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% minus_int_def
tff(fact_1981_plus__int__def,axiom,
plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% plus_int_def
tff(fact_1982_mlex__prod__def,axiom,
! [A: $tType,F: fun(A,nat),R4: set(product_prod(A,A))] : mlex_prod(A,F,R4) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R4),aTP_Lamp_gp(fun(A,nat),fun(A,product_prod(nat,A)),F)) ).
% mlex_prod_def
tff(fact_1983_rat__minus__code,axiom,
! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_gr(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).
% rat_minus_code
tff(fact_1984_rat__plus__code,axiom,
! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_gt(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).
% rat_plus_code
tff(fact_1985_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).
% image_Suc_atLeastAtMost
tff(fact_1986_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).
% image_Suc_atLeastLessThan
tff(fact_1987_in__inv__image,axiom,
! [A: $tType,B: $tType,X2: A,Y: A,R: set(product_prod(B,B)),F: fun(A,B)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),inv_image(B,A,R,F)))
<=> pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X2)),aa(A,B,F,Y)),R)) ) ).
% in_inv_image
tff(fact_1988_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set(product_prod(A,B)),F: fun(A,fun(B,C))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),A4))
=> pp(member(C,aa(B,C,aa(A,fun(B,C),F,A3),B2),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),A4))) ) ).
% pair_imageI
tff(fact_1989_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,ord_max(code_integer,K),L)) = aa(int,int,ord_max(int,aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ).
% int_of_integer_max
tff(fact_1990_rat__one__code,axiom,
quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).
% rat_one_code
tff(fact_1991_of__nat__of__integer,axiom,
! [K: code_integer] : aa(nat,code_integer,semiring_1_of_nat(code_integer),aa(code_integer,nat,code_nat_of_integer,K)) = aa(code_integer,code_integer,ord_max(code_integer,zero_zero(code_integer)),K) ).
% of_nat_of_integer
tff(fact_1992_rat__zero__code,axiom,
quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).
% rat_zero_code
tff(fact_1993_quotient__of__number_I3_J,axiom,
! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).
% quotient_of_number(3)
tff(fact_1994_quotient__of__number_I4_J,axiom,
quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).
% quotient_of_number(4)
tff(fact_1995_quotient__of__number_I5_J,axiom,
! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).
% quotient_of_number(5)
tff(fact_1996_diff__rat__def,axiom,
! [Q3: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q3),R) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q3),aa(rat,rat,uminus_uminus(rat),R)) ).
% diff_rat_def
tff(fact_1997_quotient__of__div,axiom,
! [R: rat,N: int,D3: int] :
( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),D3) )
=> ( R = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),N)),aa(int,rat,ring_1_of_int(rat),D3)) ) ) ).
% quotient_of_div
tff(fact_1998_zero__notin__Suc__image,axiom,
! [A4: set(nat)] : ~ pp(member(nat,zero_zero(nat),aa(set(nat),set(nat),image2(nat,nat,suc),A4))) ).
% zero_notin_Suc_image
tff(fact_1999_None__notin__image__Some,axiom,
! [A: $tType,A4: set(A)] : ~ pp(member(option(A),none(A),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A4))) ).
% None_notin_image_Some
tff(fact_2000_plus__integer__def,axiom,
plus_plus(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),plus_plus(int)) ).
% plus_integer_def
tff(fact_2001_times__integer__def,axiom,
times_times(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),times_times(int)) ).
% times_integer_def
tff(fact_2002_minus__integer__def,axiom,
minus_minus(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),minus_minus(int)) ).
% minus_integer_def
tff(fact_2003_divide__integer__def,axiom,
divide_divide(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),divide_divide(int)) ).
% divide_integer_def
tff(fact_2004_modulo__integer__def,axiom,
modulo_modulo(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),modulo_modulo(int)) ).
% modulo_integer_def
tff(fact_2005_and__integer__def,axiom,
bit_se5824344872417868541ns_and(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se5824344872417868541ns_and(int)) ).
% and_integer_def
tff(fact_2006_or__integer__def,axiom,
bit_se1065995026697491101ons_or(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se1065995026697491101ons_or(int)) ).
% or_integer_def
tff(fact_2007_xor__integer__def,axiom,
bit_se5824344971392196577ns_xor(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se5824344971392196577ns_xor(int)) ).
% xor_integer_def
tff(fact_2008_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] : aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A3,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% image_int_atLeastAtMost
tff(fact_2009_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] : aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A3,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% image_int_atLeastLessThan
tff(fact_2010_quotient__of__denom__pos,axiom,
! [R: rat,P3: int,Q3: int] :
( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3)) ) ).
% quotient_of_denom_pos
tff(fact_2011_image__Suc__atMost,axiom,
! [N: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).
% image_Suc_atMost
tff(fact_2012_image__Suc__lessThan,axiom,
! [N: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).
% image_Suc_lessThan
tff(fact_2013_inv__image__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(B,B)),F: fun(A,B)] : inv_image(B,A,R,F) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_gu(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),R),F))) ).
% inv_image_def
tff(fact_2014_rat__uminus__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_gv(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).
% rat_uminus_code
tff(fact_2015_rat__abs__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_gw(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).
% rat_abs_code
tff(fact_2016_uminus__integer__def,axiom,
uminus_uminus(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),uminus_uminus(int)) ).
% uminus_integer_def
tff(fact_2017_abs__integer__def,axiom,
abs_abs(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),abs_abs(int)) ).
% abs_integer_def
tff(fact_2018_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_gx(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).
% image_add_int_atLeastLessThan
tff(fact_2019_sgn__integer__def,axiom,
sgn_sgn(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),sgn_sgn(int)) ).
% sgn_integer_def
tff(fact_2020_not__integer__def,axiom,
bit_ri4277139882892585799ns_not(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),bit_ri4277139882892585799ns_not(int)) ).
% not_integer_def
tff(fact_2021_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image2(code_integer,code_integer,aTP_Lamp_gy(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).
% image_add_integer_atLeastLessThan
tff(fact_2022_rat__times__code,axiom,
! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ha(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).
% rat_times_code
tff(fact_2023_rat__divide__code,axiom,
! [P3: rat,Q3: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_hc(rat,fun(int,fun(int,product_prod(int,int))),Q3)),quotient_of(P3)) ).
% rat_divide_code
tff(fact_2024_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
=> ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),aa(int,nat,nat2,U))) ) ) ).
% image_atLeastZeroLessThan_int
tff(fact_2025_quotient__of__int,axiom,
! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),one_one(int)) ).
% quotient_of_int
tff(fact_2026_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] : fun_rp_inv_image(A,B) = aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_hd(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))))) ).
% rp_inv_image_def
tff(fact_2027_rat__inverse__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_he(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).
% rat_inverse_code
tff(fact_2028_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M) ) ).
% diff_numeral_special(8)
tff(fact_2029_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).
% diff_numeral_special(7)
tff(fact_2030_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) ) ).
% minus_sub_one_diff_one
tff(fact_2031_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,one2) ) ).
% add_neg_numeral_special(4)
tff(fact_2032_int__of__integer__sub,axiom,
! [K: num,L: num] : aa(code_integer,int,code_int_of_integer,neg_numeral_sub(code_integer,K,L)) = neg_numeral_sub(int,K,L) ).
% int_of_integer_sub
tff(fact_2033_inverse__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).
% inverse_1
tff(fact_2034_inverse__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [X2: A] :
( ( aa(A,A,inverse_inverse(A),X2) = one_one(A) )
<=> ( X2 = one_one(A) ) ) ) ).
% inverse_eq_1_iff
tff(fact_2035_right__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).
% right_inverse
tff(fact_2036_left__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).
% left_inverse
tff(fact_2037_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W)) ) ).
% inverse_eq_divide_numeral
tff(fact_2038_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M) ) ).
% add_neg_numeral_simps(2)
tff(fact_2039_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M,N) ) ).
% add_neg_numeral_simps(1)
tff(fact_2040_semiring__norm_I166_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W)),Y) ) ).
% semiring_norm(166)
tff(fact_2041_semiring__norm_I167_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V)),Y) ) ).
% semiring_norm(167)
tff(fact_2042_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).
% inverse_eq_divide_neg_numeral
tff(fact_2043_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),one_one(A)) = neg_numeral_sub(A,M,one2) ) ).
% diff_numeral_special(2)
tff(fact_2044_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,one2,N) ) ).
% diff_numeral_special(1)
tff(fact_2045_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) = neg_numeral_sub(A,one2,M) ) ).
% add_neg_numeral_special(1)
tff(fact_2046_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one2,M) ) ).
% add_neg_numeral_special(2)
tff(fact_2047_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one2) ) ).
% add_neg_numeral_special(3)
tff(fact_2048_inverse__unique,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = one_one(A) )
=> ( aa(A,A,inverse_inverse(A),A3) = B2 ) ) ) ).
% inverse_unique
tff(fact_2049_inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ).
% inverse_eq_divide
tff(fact_2050_prod__inversef,axiom,
! [A: $tType,B: $tType] :
( field(A)
=> ! [F: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),inverse_inverse(A)),F)),A4) = aa(A,A,inverse_inverse(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)) ) ).
% prod_inversef
tff(fact_2051_inverse__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X2)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X2)) ) ) ) ).
% inverse_le_1_iff
tff(fact_2052_one__less__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),one_one(A))) ) ) ) ).
% one_less_inverse_iff
tff(fact_2053_one__less__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% one_less_inverse
tff(fact_2054_division__ring__inverse__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% division_ring_inverse_add
tff(fact_2055_inverse__add,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% inverse_add
tff(fact_2056_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).
% field_class.field_inverse
tff(fact_2057_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).
% nonzero_inverse_eq_divide
tff(fact_2058_one__le__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),one_one(A))) ) ) ) ).
% one_le_inverse_iff
tff(fact_2059_inverse__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X2)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X2)) ) ) ) ).
% inverse_less_1_iff
tff(fact_2060_one__le__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% one_le_inverse
tff(fact_2061_Frct__code__post_I5_J,axiom,
! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).
% Frct_code_post(5)
tff(fact_2062_Code__Numeral_Odup__def,axiom,
code_dup = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),aTP_Lamp_hf(int,int)) ).
% Code_Numeral.dup_def
tff(fact_2063_Frct__code__post_I6_J,axiom,
! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).
% Frct_code_post(6)
tff(fact_2064_Frct__code__post_I4_J,axiom,
! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).
% Frct_code_post(4)
tff(fact_2065_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Y: num] :
( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y))) ) ) ).
% eq_numeral_iff_iszero(8)
tff(fact_2066_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X2)) = one_one(A) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),one2))) ) ) ).
% eq_numeral_iff_iszero(7)
tff(fact_2067_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: fun(set(A),A),G: fun(B,A),F: fun(C,B),A4: set(C)] : aa(set(A),A,Inf,aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A4))) = aa(set(A),A,Inf,aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A4)) ).
% Inf.INF_image
tff(fact_2068_not__iszero__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,one_one(A)) ) ).
% not_iszero_1
tff(fact_2069_Code__Numeral_Odup__code_I1_J,axiom,
aa(code_integer,code_integer,code_dup,zero_zero(code_integer)) = zero_zero(code_integer) ).
% Code_Numeral.dup_code(1)
tff(fact_2070_not__iszero__neg__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% not_iszero_neg_1
tff(fact_2071_dup_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,code_dup,X2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X2)),aa(code_integer,int,code_int_of_integer,X2)) ).
% dup.rep_eq
tff(fact_2072_dup_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_integer,code_dup,aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),X2)) ).
% dup.abs_eq
tff(fact_2073_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: num,Y: num] :
( ( aa(num,A,numeral_numeral(A),X2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),Y))) ) ) ).
% eq_numeral_iff_iszero(2)
tff(fact_2074_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: num,Y: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X2)) = aa(num,A,numeral_numeral(A),Y) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),Y))) ) ) ).
% eq_numeral_iff_iszero(3)
tff(fact_2075_Frct__code__post_I2_J,axiom,
! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),zero_zero(int))) = zero_zero(rat) ).
% Frct_code_post(2)
tff(fact_2076_Frct__code__post_I1_J,axiom,
! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A3)) = zero_zero(rat) ).
% Frct_code_post(1)
tff(fact_2077_Frct__code__post_I3_J,axiom,
frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).
% Frct_code_post(3)
tff(fact_2078_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Y: num] :
( ( one_one(A) = aa(num,A,numeral_numeral(A),Y) )
<=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y)) ) ) ).
% eq_numeral_iff_iszero(6)
tff(fact_2079_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: num] :
( ( aa(num,A,numeral_numeral(A),X2) = one_one(A) )
<=> ring_1_iszero(A,neg_numeral_sub(A,X2,one2)) ) ) ).
% eq_numeral_iff_iszero(5)
tff(fact_2080_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),A3)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).
% Frct_code_post(7)
tff(fact_2081_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2))) ).
% Frct_code_post(8)
tff(fact_2082_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: fun(set(A),A),G: fun(B,A),F: fun(C,B),A4: set(C)] : aa(set(A),A,Sup,aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A4))) = aa(set(A),A,Sup,aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A4)) ).
% Sup.SUP_image
tff(fact_2083_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N))),one_one(code_integer)) ).
% Code_Numeral.sub_code(9)
tff(fact_2084_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N))),one_one(code_integer)) ).
% Code_Numeral.sub_code(8)
tff(fact_2085_less__eq__int__def,axiom,
ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).
% less_eq_int_def
tff(fact_2086_less__int__def,axiom,
ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).
% less_int_def
tff(fact_2087_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hg(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
tff(fact_2088_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hh(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
tff(fact_2089_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hi(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
tff(fact_2090_id__apply,axiom,
! [A: $tType,X2: A] : aa(A,A,id(A),X2) = X2 ).
% id_apply
tff(fact_2091_bijective__Empty,axiom,
! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).
% bijective_Empty
tff(fact_2092_image__id,axiom,
! [A: $tType] : image2(A,A,id(A)) = id(set(A)) ).
% image_id
tff(fact_2093_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: fun(D,A)] : aa(fun(D,A),fun(D,A),aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),id(A)),T2) = T2 ).
% fun.map_id
tff(fact_2094_id__comp,axiom,
! [B: $tType,A: $tType,G: fun(A,B)] : aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),id(B)),G) = G ).
% id_comp
tff(fact_2095_comp__id,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),id(A)) = F ).
% comp_id
tff(fact_2096_lessThan__0,axiom,
aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).
% lessThan_0
tff(fact_2097_apsnd__id,axiom,
! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).
% apsnd_id
tff(fact_2098_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),V: B] : map_mmupd(A,B,M,bot_bot(set(A)),V) = M ).
% map_mmupd_empty
tff(fact_2099_set__to__map__empty,axiom,
! [A: $tType,B: $tType,X: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X) = none(B) ).
% set_to_map_empty
tff(fact_2100_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B2) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% atLeastatMost_empty_iff2
tff(fact_2101_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% atLeastatMost_empty_iff
tff(fact_2102_atLeastatMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) ) ) ) ).
% atLeastatMost_empty
tff(fact_2103_atLeastLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) ) ) ) ).
% atLeastLessThan_empty
tff(fact_2104_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = bot_bot(set(A)) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% atLeastLessThan_empty_iff
tff(fact_2105_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B2) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% atLeastLessThan_empty_iff2
tff(fact_2106_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty
tff(fact_2107_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).
% group_add_class.minus_comp_minus
tff(fact_2108_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).
% boolean_algebra_class.minus_comp_minus
tff(fact_2109_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : rel_of(A,B,aTP_Lamp_hj(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).
% rel_of_empty
tff(fact_2110_comp__the__Some,axiom,
! [A: $tType] : aa(fun(A,option(A)),fun(A,A),aa(fun(option(A),A),fun(fun(A,option(A)),fun(A,A)),comp(option(A),A,A),the2(A)),some(A)) = id(A) ).
% comp_the_Some
tff(fact_2111_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] : aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),id(A)) = id(fun(D,A)) ).
% fun.map_id0
tff(fact_2112_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( linorder(A)
& order_bot(A) )
=> ! [N: A] :
( ( aa(A,set(A),set_ord_lessThan(A),N) = bot_bot(set(A)) )
<=> ( N = bot_bot(A) ) ) ) ).
% Iio_eq_empty_iff
tff(fact_2113_option_Omap__id0,axiom,
! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)) = id(option(A)) ).
% option.map_id0
tff(fact_2114_set__notEmptyE,axiom,
! [A: $tType,S2: set(A)] :
( ( S2 != bot_bot(set(A)) )
=> ~ ! [X3: A] : ~ pp(member(A,X3,S2)) ) ).
% set_notEmptyE
tff(fact_2115_memb__imp__not__empty,axiom,
! [A: $tType,X2: A,S2: set(A)] :
( pp(member(A,X2,S2))
=> ( S2 != bot_bot(set(A)) ) ) ).
% memb_imp_not_empty
tff(fact_2116_id__def,axiom,
! [A: $tType,X: A] : aa(A,A,id(A),X) = X ).
% id_def
tff(fact_2117_eq__id__iff,axiom,
! [A: $tType,F: fun(A,A)] :
( ! [X4: A] : aa(A,A,F,X4) = X4
<=> ( F = id(A) ) ) ).
% eq_id_iff
tff(fact_2118_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(A,B),V: A] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),id(B)),C2) )
=> ( aa(C,B,A3,aa(A,C,B2,V)) = aa(A,B,C2,V) ) ) ).
% comp_eq_id_dest
tff(fact_2119_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B),X2: A] :
( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),G) = id(A) )
=> ( aa(B,A,F,aa(A,B,G,X2)) = X2 ) ) ).
% pointfree_idE
tff(fact_2120_option_Omap__id,axiom,
! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)),T2) = T2 ).
% option.map_id
tff(fact_2121_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).
% not_empty_eq_Iic_eq_empty
tff(fact_2122_lessThan__non__empty,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X2: A] : aa(A,set(A),set_ord_lessThan(A),X2) != bot_bot(set(A)) ) ).
% lessThan_non_empty
tff(fact_2123_subset__minus__empty,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = bot_bot(set(A)) ) ) ).
% subset_minus_empty
tff(fact_2124_atLeastLessThan0,axiom,
! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).
% atLeastLessThan0
tff(fact_2125_lessThan__empty__iff,axiom,
! [N: nat] :
( ( aa(nat,set(nat),set_ord_lessThan(nat),N) = bot_bot(set(nat)) )
<=> ( N = zero_zero(nat) ) ) ).
% lessThan_empty_iff
tff(fact_2126_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S2: set(product_prod(A,B))] :
( ! [X4: A] : aa(A,option(B),set_to_map(A,B,S2),X4) = none(B)
<=> ( S2 = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(1)
tff(fact_2127_Code__Numeral_Osub__code_I1_J,axiom,
aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),one2) = zero_zero(code_integer) ).
% Code_Numeral.sub_code(1)
tff(fact_2128_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S2: set(product_prod(A,B))] :
( ( aTP_Lamp_hj(A,option(B)) = set_to_map(A,B,S2) )
<=> ( S2 = bot_bot(set(product_prod(A,B))) ) ) ).
% set_to_map_empty_iff(2)
tff(fact_2129_less__integer__def,axiom,
ord_less(code_integer) = aa(fun(int,fun(int,bool)),fun(code_integer,fun(code_integer,bool)),map_fun(code_integer,int,fun(int,bool),fun(code_integer,bool),code_int_of_integer,map_fun(code_integer,int,bool,bool,code_int_of_integer,id(bool))),ord_less(int)) ).
% less_integer_def
tff(fact_2130_less__eq__integer__def,axiom,
ord_less_eq(code_integer) = aa(fun(int,fun(int,bool)),fun(code_integer,fun(code_integer,bool)),map_fun(code_integer,int,fun(int,bool),fun(code_integer,bool),code_int_of_integer,map_fun(code_integer,int,bool,bool,code_int_of_integer,id(bool))),ord_less_eq(int)) ).
% less_eq_integer_def
tff(fact_2131_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N)) ).
% Code_Numeral.sub_code(6)
tff(fact_2132_Code__Numeral_Osub__code_I7_J,axiom,
! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(code_integer,code_integer,code_dup,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N)) ).
% Code_Numeral.sub_code(7)
tff(fact_2133_sub_Orep__eq,axiom,
! [X2: num,Xa: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,X2),Xa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),X2)),aa(num,int,numeral_numeral(int),Xa)) ).
% sub.rep_eq
tff(fact_2134_sub_Oabs__eq,axiom,
! [Xa: num,X2: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Xa),X2) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Xa)),aa(num,int,numeral_numeral(int),X2))) ).
% sub.abs_eq
tff(fact_2135_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X2: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X2,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X2,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X2,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_2136_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,Y: A,C2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_hk(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X2,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_hk(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X2,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X2),C2)) ) ) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> ( aa(set(A),set(A),image2(A,A,aTP_Lamp_hk(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X2,Y)) = bot_bot(set(A)) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
tff(fact_2137_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( set_or1337092689740270186AtMost(A,A3,B2) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hl(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hl(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_hl(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
tff(fact_2138_and__int_Osimps,axiom,
! [K: int,L: int] :
( ( ( pp(member(int,K,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,L,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))) ) )
& ( ~ ( pp(member(int,K,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,L,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).
% and_int.simps
tff(fact_2139_and__int_Oelims,axiom,
! [X2: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X2),Xa) = Y )
=> ( ( ( pp(member(int,X2,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,Xa,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X2)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))) ) )
& ( ~ ( pp(member(int,X2,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,Xa,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X2)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).
% and_int.elims
tff(fact_2140_and__int_Opelims,axiom,
! [X2: int,Xa: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X2),Xa) = Y )
=> ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X2),Xa)))
=> ~ ( ( ( ( pp(member(int,X2,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,Xa,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X2)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))) ) )
& ( ~ ( pp(member(int,X2,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,Xa,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X2)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) )
=> ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X2),Xa))) ) ) ) ).
% and_int.pelims
tff(fact_2141_and__int_Opsimps,axiom,
! [K: int,L: int] :
( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L)))
=> ( ( ( pp(member(int,K,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,L,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))) ) )
& ( ~ ( pp(member(int,K,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,L,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).
% and_int.psimps
tff(fact_2142_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1)))
=> ( ! [K3: int,L5: int] :
( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K3),L5)))
=> ( ( ~ ( pp(member(int,K3,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
& pp(member(int,L5,aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K3),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L5),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,K3),L5)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% and_int.pinduct
tff(fact_2143_mask__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat,M: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,ord_min(nat,M),N))),one_one(A)) ) ).
% mask_mod_exp
tff(fact_2144_pair__leqI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S),T2))
=> pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_leq)) ) ) ).
% pair_leqI2
tff(fact_2145_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N: A] :
( ( aa(A,A,ord_min(A,M),N) = M )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),N)) ) ) ).
% min_eq_arg(1)
tff(fact_2146_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N: A] :
( ( aa(A,A,ord_min(A,M),N) = N )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),M)) ) ) ).
% min_eq_arg(2)
tff(fact_2147_min__arg__le_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [N: A,M: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,ord_min(A,M),N)))
<=> ( aa(A,A,ord_min(A,M),N) = N ) ) ) ).
% min_arg_le(1)
tff(fact_2148_min__arg__le_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [M: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(A,A,ord_min(A,M),N)))
<=> ( aa(A,A,ord_min(A,M),N) = M ) ) ) ).
% min_arg_le(2)
tff(fact_2149_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,ord_min(A,M),N)),M))
<=> ( aa(A,A,ord_min(A,M),N) = M ) ) ) ).
% min_arg_not_ge(1)
tff(fact_2150_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M: A,N: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,ord_min(A,M),N)),N))
<=> ( aa(A,A,ord_min(A,M),N) = N ) ) ) ).
% min_arg_not_ge(2)
tff(fact_2151_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,ord_min(A,A3),B2)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% min_less_self_conv(1)
tff(fact_2152_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,ord_min(A,A3),B2)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% min_less_self_conv(2)
tff(fact_2153_min__simps_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( aa(A,A,ord_min(A,A3),B2) = A3 ) ) ) ).
% min_simps(1)
tff(fact_2154_min__simps_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,ord_min(A,A3),B2) = B2 ) ) ) ).
% min_simps(2)
tff(fact_2155_min__0__1_I2_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,ord_min(A,one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).
% min_0_1(2)
tff(fact_2156_min__0__1_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,ord_min(A,zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).
% min_0_1(1)
tff(fact_2157_min__0__1_I5_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X2: num] : aa(A,A,ord_min(A,one_one(A)),aa(num,A,numeral_numeral(A),X2)) = one_one(A) ) ).
% min_0_1(5)
tff(fact_2158_min__0__1_I6_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X2: num] : aa(A,A,ord_min(A,aa(num,A,numeral_numeral(A),X2)),one_one(A)) = one_one(A) ) ).
% min_0_1(6)
tff(fact_2159_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A,C2: A] :
( ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),insert2(A,C2),bot_bot(set(A))) )
<=> ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
tff(fact_2160_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ) ).
% atLeastAtMost_singleton
tff(fact_2161_min__Suc__gt_I2_J,axiom,
! [A3: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> ( aa(nat,nat,ord_min(nat,B2),aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).
% min_Suc_gt(2)
tff(fact_2162_min__Suc__gt_I1_J,axiom,
! [A3: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> ( aa(nat,nat,ord_min(nat,aa(nat,nat,suc,A3)),B2) = aa(nat,nat,suc,A3) ) ) ).
% min_Suc_gt(1)
tff(fact_2163_case__prod__Pair,axiom,
! [B: $tType,A: $tType] : aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)) = id(product_prod(A,B)) ).
% case_prod_Pair
tff(fact_2164_single__Diff__lessThan,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert2(A,K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),insert2(A,K),bot_bot(set(A))) ) ).
% single_Diff_lessThan
tff(fact_2165_min__add__distrib__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X2: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(A,A,ord_min(A,Y),Z2)) = aa(A,A,ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Z2)) ) ).
% min_add_distrib_right
tff(fact_2166_min__add__distrib__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X2: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,ord_min(A,X2),Y)),Z2) = aa(A,A,ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).
% min_add_distrib_left
tff(fact_2167_min__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,ord_min(A,X2),Y)),Z2) = aa(A,A,ord_min(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).
% min_diff_distrib_left
tff(fact_2168_bot__option__def,axiom,
! [A: $tType] :
( order(A)
=> ( bot_bot(option(A)) = none(A) ) ) ).
% bot_option_def
tff(fact_2169_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),bot_bot(set(product_prod(A,B))))) ) ).
% bot_empty_eq2
tff(fact_2170_map__fun_Oidentity,axiom,
! [B: $tType,A: $tType] : map_fun(A,A,B,B,aTP_Lamp_ae(A,A),aTP_Lamp_hm(B,B)) = id(fun(A,B)) ).
% map_fun.identity
tff(fact_2171_map__option_Oidentity,axiom,
! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_ae(A,A)) = id(option(A)) ).
% map_option.identity
tff(fact_2172_minus__max__eq__min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [X2: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,ord_max(A,X2),Y)) = aa(A,A,ord_min(A,aa(A,A,uminus_uminus(A),X2)),aa(A,A,uminus_uminus(A),Y)) ) ).
% minus_max_eq_min
tff(fact_2173_minus__min__eq__max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [X2: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,ord_min(A,X2),Y)) = aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),X2)),aa(A,A,uminus_uminus(A),Y)) ) ).
% minus_min_eq_max
tff(fact_2174_insert__minus__eq,axiom,
! [A: $tType,X2: A,Y: A,A4: set(A)] :
( ( X2 != Y )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),insert2(A,X2),A4)),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) = aa(set(A),set(A),insert2(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))) ) ) ).
% insert_minus_eq
tff(fact_2175_set__minus__singleton__eq,axiom,
! [A: $tType,X2: A,X7: set(A)] :
( ~ pp(member(A,X2,X7))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X7),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))) = X7 ) ) ).
% set_minus_singleton_eq
tff(fact_2176_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( set_or1337092689740270186AtMost(A,A3,B2) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ) ) ) ).
% atLeastAtMost_singleton'
tff(fact_2177_remove__subset,axiom,
! [A: $tType,X2: A,S2: set(A)] :
( pp(member(A,X2,S2))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))),S2)) ) ).
% remove_subset
tff(fact_2178_nat__of__integer__def,axiom,
code_nat_of_integer = aa(fun(int,nat),fun(code_integer,nat),map_fun(code_integer,int,nat,nat,code_int_of_integer,id(nat)),nat2) ).
% nat_of_integer_def
tff(fact_2179_bit__integer__def,axiom,
bit_se5641148757651400278ts_bit(code_integer) = aa(fun(int,fun(nat,bool)),fun(code_integer,fun(nat,bool)),map_fun(code_integer,int,fun(nat,bool),fun(nat,bool),code_int_of_integer,id(fun(nat,bool))),bit_se5641148757651400278ts_bit(int)) ).
% bit_integer_def
tff(fact_2180_mask__integer__def,axiom,
bit_se2239418461657761734s_mask(code_integer) = aa(fun(nat,int),fun(nat,code_integer),map_fun(nat,nat,int,code_integer,id(nat),code_integer_of_int),bit_se2239418461657761734s_mask(int)) ).
% mask_integer_def
tff(fact_2181_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] : set_or7035219750837199246ssThan(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_2182_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
=> ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),insert2(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M,N)) ) ) ).
% atLeastAtMostPlus1_int_conv
tff(fact_2183_simp__from__to,axiom,
! [J: int,I: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( set_or1337092689740270186AtMost(int,I,J) = bot_bot(set(int)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( set_or1337092689740270186AtMost(int,I,J) = aa(set(int),set(int),insert2(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ) ).
% simp_from_to
tff(fact_2184_Code__Numeral_Osub__def,axiom,
code_sub = aa(fun(num,fun(num,int)),fun(num,fun(num,code_integer)),map_fun(num,num,fun(num,int),fun(num,code_integer),id(num),map_fun(num,num,int,code_integer,id(num),code_integer_of_int)),aTP_Lamp_hn(num,fun(num,int))) ).
% Code_Numeral.sub_def
tff(fact_2185_num__of__integer__def,axiom,
code_num_of_integer = aa(fun(int,num),fun(code_integer,num),map_fun(code_integer,int,num,num,code_int_of_integer,id(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)) ).
% num_of_integer_def
tff(fact_2186_take__bit__integer__def,axiom,
bit_se2584673776208193580ke_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se2584673776208193580ke_bit(int)) ).
% take_bit_integer_def
tff(fact_2187_drop__bit__integer__def,axiom,
bit_se4197421643247451524op_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se4197421643247451524op_bit(int)) ).
% drop_bit_integer_def
tff(fact_2188_push__bit__integer__def,axiom,
bit_se4730199178511100633sh_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se4730199178511100633sh_bit(int)) ).
% push_bit_integer_def
tff(fact_2189_unset__bit__integer__def,axiom,
bit_se2638667681897837118et_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se2638667681897837118et_bit(int)) ).
% unset_bit_integer_def
tff(fact_2190_set__bit__integer__def,axiom,
bit_se5668285175392031749et_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se5668285175392031749et_bit(int)) ).
% set_bit_integer_def
tff(fact_2191_flip__bit__integer__def,axiom,
bit_se8732182000553998342ip_bit(code_integer) = aa(fun(nat,fun(int,int)),fun(nat,fun(code_integer,code_integer)),map_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),id(nat),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),bit_se8732182000553998342ip_bit(int)) ).
% flip_bit_integer_def
tff(fact_2192_pair__leqI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2)),fun_pair_leq)) ) ).
% pair_leqI1
tff(fact_2193_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
=> ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X2,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X2),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C2)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Y))
=> ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X2,Y)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Y))
=> ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X2,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_2194_card__atLeastAtMost__int,axiom,
! [L: int,U: int] : finite_card(int,set_or1337092689740270186AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)),one_one(int))) ).
% card_atLeastAtMost_int
tff(fact_2195_range__mod,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_hp(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).
% range_mod
tff(fact_2196_Some__image__these__eq,axiom,
! [A: $tType,A4: set(option(A))] : aa(set(A),set(option(A)),image2(A,option(A),some(A)),these(A,A4)) = aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_hq(set(option(A)),fun(option(A),bool),A4)) ).
% Some_image_these_eq
tff(fact_2197_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [B5: set(A),A4: set(A),F: fun(A,B)] :
( finite_finite2(A,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
=> ( ! [B3: A] :
( pp(member(A,B3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,B3))) )
=> ( ! [A5: A] :
( pp(member(A,A5,A4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A5))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B5))) ) ) ) ) ) ).
% prod_mono2
tff(fact_2198_the__dflt__None__nonempty,axiom,
! [A: $tType,S2: set(A)] :
( ( S2 != bot_bot(set(A)) )
=> ( dflt_None_set(A,S2) = aa(set(A),option(set(A)),some(set(A)),S2) ) ) ).
% the_dflt_None_nonempty
tff(fact_2199_the__dflt__None__empty,axiom,
! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).
% the_dflt_None_empty
tff(fact_2200_atMost__UNIV__triv,axiom,
! [A: $tType] : aa(set(A),set(set(A)),set_ord_atMost(set(A)),top_top(set(A))) = top_top(set(set(A))) ).
% atMost_UNIV_triv
tff(fact_2201_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : finite_finite2(nat,set_or1337092689740270186AtMost(nat,L,U)) ).
% finite_atLeastAtMost
tff(fact_2202_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : finite_finite2(nat,set_or7035219750837199246ssThan(nat,L,U)) ).
% finite_atLeastLessThan
tff(fact_2203_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,ord_min(code_integer,K),L)) = aa(int,int,ord_min(int,aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ).
% int_of_integer_min
tff(fact_2204_finite__atMost,axiom,
! [K: nat] : finite_finite2(nat,aa(nat,set(nat),set_ord_atMost(nat),K)) ).
% finite_atMost
tff(fact_2205_finite__lessThan,axiom,
! [K: nat] : finite_finite2(nat,aa(nat,set(nat),set_ord_lessThan(nat),K)) ).
% finite_lessThan
tff(fact_2206_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : finite_finite2(int,set_or1337092689740270186AtMost(int,L,U)) ).
% finite_atLeastAtMost_int
tff(fact_2207_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : finite_finite2(int,set_or7035219750837199246ssThan(int,L,U)) ).
% finite_atLeastLessThan_int
tff(fact_2208_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : finite_finite2(code_integer,set_or7035219750837199246ssThan(code_integer,L,U)) ).
% finite_atLeastLessThan_integer
tff(fact_2209_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : finite_finite2(code_integer,set_or1337092689740270186AtMost(code_integer,L,U)) ).
% finite_atLeastAtMost_integer
tff(fact_2210_finite__option__UNIV,axiom,
! [A: $tType] :
( finite_finite2(option(A),top_top(set(option(A))))
<=> finite_finite2(A,top_top(set(A))) ) ).
% finite_option_UNIV
tff(fact_2211_card__eq__UNIV,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S2: set(A)] :
( ( finite_card(A,S2) = finite_card(A,top_top(set(A))) )
<=> ( S2 = top_top(set(A)) ) ) ) ).
% card_eq_UNIV
tff(fact_2212_card__eq__UNIV2,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S2: set(A)] :
( ( finite_card(A,top_top(set(A))) = finite_card(A,S2) )
<=> ( S2 = top_top(set(A)) ) ) ) ).
% card_eq_UNIV2
tff(fact_2213_these__empty,axiom,
! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).
% these_empty
tff(fact_2214_these__insert__None,axiom,
! [A: $tType,A4: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),A4)) = these(A,A4) ).
% these_insert_None
tff(fact_2215_the__dflt__None__set,axiom,
! [A: $tType,X2: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,X2)) = X2 ).
% the_dflt_None_set
tff(fact_2216_surj__plus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),top_top(set(A))) = top_top(set(A)) ) ).
% surj_plus
tff(fact_2217_infinite__Icc__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A3,B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% infinite_Icc_iff
tff(fact_2218_infinite__Ico__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A3,B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% infinite_Ico_iff
tff(fact_2219_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( ~ finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = one_one(A) ) ) ) ).
% prod.infinite
tff(fact_2220_card__ge__UNIV,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [S2: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,top_top(set(A)))),finite_card(A,S2)))
<=> ( S2 = top_top(set(A)) ) ) ) ).
% card_ge_UNIV
tff(fact_2221_surj__diff__right,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).
% surj_diff_right
tff(fact_2222_prod__eq__1__iff,axiom,
! [A: $tType,A4: set(A),F: fun(A,nat)] :
( finite_finite2(A,A4)
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A4) = one_one(nat) )
<=> ! [X4: A] :
( pp(member(A,X4,A4))
=> ( aa(A,nat,F,X4) = one_one(nat) ) ) ) ) ).
% prod_eq_1_iff
tff(fact_2223_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),A3: B,B2: fun(B,A)] :
( finite_finite2(B,S2)
=> ( ( pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_hr(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = aa(B,A,B2,A3) ) )
& ( ~ pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_hr(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = one_one(A) ) ) ) ) ) ).
% prod.delta
tff(fact_2224_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),A3: B,B2: fun(B,A)] :
( finite_finite2(B,S2)
=> ( ( pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_hs(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = aa(B,A,B2,A3) ) )
& ( ~ pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_hs(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = one_one(A) ) ) ) ) ) ).
% prod.delta'
tff(fact_2225_these__insert__Some,axiom,
! [A: $tType,X2: A,A4: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert2(option(A),aa(A,option(A),some(A),X2)),A4)) = aa(set(A),set(A),insert2(A,X2),these(A,A4)) ).
% these_insert_Some
tff(fact_2226_these__image__Some__eq,axiom,
! [A: $tType,A4: set(A)] : these(A,aa(set(A),set(option(A)),image2(A,option(A),some(A)),A4)) = A4 ).
% these_image_Some_eq
tff(fact_2227_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),X2: B,G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( ~ pp(member(B,X2,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),insert2(B,X2),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)) ) ) ) ) ).
% sum.insert
tff(fact_2228_atLeastLessThan__singleton,axiom,
! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),insert2(nat,M),bot_bot(set(nat))) ).
% atLeastLessThan_singleton
tff(fact_2229_atMost__0,axiom,
aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))) ).
% atMost_0
tff(fact_2230_card__atLeastLessThan__int,axiom,
! [L: int,U: int] : finite_card(int,set_or7035219750837199246ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).
% card_atLeastLessThan_int
tff(fact_2231_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( finite_card(A,aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
<=> ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
tff(fact_2232_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X2: A] :
( ( aa(A,set(A),set_ord_atMost(A),X2) = top_top(set(A)) )
<=> ( X2 = top_top(A) ) ) ) ).
% atMost_eq_UNIV_iff
tff(fact_2233_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set(A),A4: set(B),R: fun(B,fun(A,bool))] :
( finite_finite2(A,B5)
=> ( ! [A5: B] :
( pp(member(B,A5,A4))
=> ? [B6: A] :
( pp(member(A,B6,B5))
& pp(aa(A,bool,aa(B,fun(A,bool),R,A5),B6)) ) )
=> ( ! [A12: B,A23: B,B3: A] :
( pp(member(B,A12,A4))
=> ( pp(member(B,A23,A4))
=> ( pp(member(A,B3,B5))
=> ( pp(aa(A,bool,aa(B,fun(A,bool),R,A12),B3))
=> ( pp(aa(A,bool,aa(B,fun(A,bool),R,A23),B3))
=> ( A12 = A23 ) ) ) ) ) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(B,A4)),finite_card(A,B5))) ) ) ) ).
% card_le_if_inj_on_rel
tff(fact_2234_top__option__def,axiom,
! [A: $tType] :
( order_top(A)
=> ( top_top(option(A)) = aa(A,option(A),some(A),top_top(A)) ) ) ).
% top_option_def
tff(fact_2235_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S2: set(A),S3: set(A)] :
( finite_finite2(A,S2)
=> finite_finite2(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_ht(set(A),fun(set(A),fun(set(A),bool)),S2),S3))) ) ).
% finite_if_eq_beyond_finite
tff(fact_2236_finite__range__Some,axiom,
! [A: $tType] :
( finite_finite2(option(A),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A))))
<=> finite_finite2(A,top_top(set(A))) ) ).
% finite_range_Some
tff(fact_2237_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S2: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),finite_card(A,S2)))
=> ~ ! [T3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S2))
=> ( ( finite_card(A,T3) = N )
=> ~ finite_finite2(A,T3) ) ) ) ).
% obtain_subset_with_card_n
tff(fact_2238_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( bounded_lattice(A)
=> ! [X2: A,Y: A] :
( ( set_or1337092689740270186AtMost(A,X2,Y) = top_top(set(A)) )
<=> ( ( X2 = bot_bot(A) )
& ( Y = top_top(A) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
tff(fact_2239_UNIV__option__conv,axiom,
! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))) ).
% UNIV_option_conv
tff(fact_2240_bounded__nat__set__is__finite,axiom,
! [N5: set(nat),N: nat] :
( ! [X3: nat] :
( pp(member(nat,X3,N5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),N)) )
=> finite_finite2(nat,N5) ) ).
% bounded_nat_set_is_finite
tff(fact_2241_finite__nat__set__iff__bounded,axiom,
! [N5: set(nat)] :
( finite_finite2(nat,N5)
<=> ? [M6: nat] :
! [X4: nat] :
( pp(member(nat,X4,N5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M6)) ) ) ).
% finite_nat_set_iff_bounded
tff(fact_2242_finite__nat__set__iff__bounded__le,axiom,
! [N5: set(nat)] :
( finite_finite2(nat,N5)
<=> ? [M6: nat] :
! [X4: nat] :
( pp(member(nat,X4,N5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M6)) ) ) ).
% finite_nat_set_iff_bounded_le
tff(fact_2243_surjD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Y: A] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ? [X3: B] : Y = aa(B,A,F,X3) ) ).
% surjD
tff(fact_2244_surjE,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Y: A] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ~ ! [X3: B] : Y != aa(B,A,F,X3) ) ).
% surjE
tff(fact_2245_surjI,axiom,
! [B: $tType,A: $tType,G: fun(B,A),F: fun(A,B)] :
( ! [X3: A] : aa(B,A,G,aa(A,B,F,X3)) = X3
=> ( aa(set(B),set(A),image2(B,A,G),top_top(set(B))) = top_top(set(A)) ) ) ).
% surjI
tff(fact_2246_surj__def,axiom,
! [A: $tType,B: $tType,F: fun(B,A)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
<=> ! [Y3: A] :
? [X4: B] : Y3 = aa(B,A,F,X4) ) ).
% surj_def
tff(fact_2247_these__empty__eq,axiom,
! [A: $tType,B5: set(option(A))] :
( ( these(A,B5) = bot_bot(set(A)) )
<=> ( ( B5 = bot_bot(set(option(A))) )
| ( B5 = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).
% these_empty_eq
tff(fact_2248_these__not__empty__eq,axiom,
! [A: $tType,B5: set(option(A))] :
( ( these(A,B5) != bot_bot(set(A)) )
<=> ( ( B5 != bot_bot(set(option(A))) )
& ( B5 != aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).
% these_not_empty_eq
tff(fact_2249_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( no_top(A)
=> ! [L4: A,H3: A] : top_top(set(A)) != set_or1337092689740270186AtMost(A,L4,H3) ) ).
% not_UNIV_eq_Icc
tff(fact_2250_infinite__Iic,axiom,
! [A: $tType] :
( ( linorder(A)
& no_bot(A) )
=> ! [A3: A] : ~ finite_finite2(A,aa(A,set(A),set_ord_atMost(A),A3)) ) ).
% infinite_Iic
tff(fact_2251_infinite__Iio,axiom,
! [A: $tType] :
( ( linorder(A)
& no_bot(A) )
=> ! [A3: A] : ~ finite_finite2(A,aa(A,set(A),set_ord_lessThan(A),A3)) ) ).
% infinite_Iio
tff(fact_2252_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( no_top(A)
=> ! [H3: A] : top_top(set(A)) != aa(A,set(A),set_ord_atMost(A),H3) ) ).
% not_UNIV_eq_Iic
tff(fact_2253_card__1__singletonI,axiom,
! [A: $tType,S2: set(A),X2: A] :
( finite_finite2(A,S2)
=> ( ( finite_card(A,S2) = one_one(nat) )
=> ( pp(member(A,X2,S2))
=> ( S2 = aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))) ) ) ) ) ).
% card_1_singletonI
tff(fact_2254_finite__M__bounded__by__nat,axiom,
! [P: fun(nat,bool),I: nat] : finite_finite2(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_hu(fun(nat,bool),fun(nat,fun(nat,bool)),P),I))) ).
% finite_M_bounded_by_nat
tff(fact_2255_finite__less__ub,axiom,
! [F: fun(nat,nat),U: nat] :
( ! [N3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),aa(nat,nat,F,N3)))
=> finite_finite2(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_hv(fun(nat,nat),fun(nat,fun(nat,bool)),F),U))) ) ).
% finite_less_ub
tff(fact_2256_card__insert__disjoint_H,axiom,
! [A: $tType,A4: set(A),X2: A] :
( finite_finite2(A,A4)
=> ( ~ pp(member(A,X2,A4))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,aa(set(A),set(A),insert2(A,X2),A4))),aa(nat,nat,suc,zero_zero(nat))) = finite_card(A,A4) ) ) ) ).
% card_insert_disjoint'
tff(fact_2257_in__these__eq,axiom,
! [A: $tType,X2: A,A4: set(option(A))] :
( pp(member(A,X2,these(A,A4)))
<=> pp(member(option(A),aa(A,option(A),some(A),X2),A4)) ) ).
% in_these_eq
tff(fact_2258_infinite__Icc,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A3,B2)) ) ) ).
% infinite_Icc
tff(fact_2259_infinite__Ico,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A3,B2)) ) ) ).
% infinite_Ico
tff(fact_2260_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set(nat),N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
=> finite_finite2(nat,N5) ) ).
% subset_eq_atLeast0_atMost_finite
tff(fact_2261_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set(nat),N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
=> finite_finite2(nat,N5) ) ).
% subset_eq_atLeast0_lessThan_finite
tff(fact_2262_atMost__Suc,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert2(nat,aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).
% atMost_Suc
tff(fact_2263_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,A),X7: set(B),G1: fun(A,C),G22: fun(A,C)] :
( ( aa(set(B),set(A),image2(B,A,F),X7) = top_top(set(A)) )
=> ( ! [X3: B] :
( pp(member(B,X3,X7))
=> ( aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G1),F),X3) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G22),F),X3) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
tff(fact_2264_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(A,C)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( ( aa(set(A),set(C),image2(A,C,G),top_top(set(A))) = top_top(set(C)) )
=> ( aa(set(B),set(C),image2(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G),F)),top_top(set(B))) = top_top(set(C)) ) ) ) ).
% comp_surj
tff(fact_2265_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: fun(A,B),V: fun(D,A)] : aa(set(D),set(B),image2(D,B,aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),V)),top_top(set(D))) = aa(set(A),set(B),image2(A,B,F),aa(set(D),set(A),image2(D,A,V),top_top(set(D)))) ).
% fun.set_map
tff(fact_2266_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),Ya: fun(D,A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(member(A,Z3,aa(set(D),set(A),image2(D,A,Ya),top_top(set(D)))))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),Ya) ) ) ) ).
% fun.map_cong
tff(fact_2267_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),F: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(member(A,Z3,aa(set(D),set(A),image2(D,A,X2),top_top(set(D)))))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),X2) ) ) ).
% fun.map_cong0
tff(fact_2268_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),Xa: fun(D,A),F: fun(A,B),Fa: fun(A,B)] :
( ! [Z3: A,Za: A] :
( pp(member(A,Z3,aa(set(D),set(A),image2(D,A,X2),top_top(set(D)))))
=> ( pp(member(A,Za,aa(set(D),set(A),image2(D,A,Xa),top_top(set(D)))))
=> ( ( aa(A,B,F,Z3) = aa(A,B,Fa,Za) )
=> ( Z3 = Za ) ) ) )
=> ( ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),Fa),Xa) )
=> ( X2 = Xa ) ) ) ).
% fun.inj_map_strong
tff(fact_2269_lessThan__Suc,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert2(nat,K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).
% lessThan_Suc
tff(fact_2270_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( no_top(A)
=> ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).
% not_UNIV_le_Icc
tff(fact_2271_surj__id,axiom,
! [A: $tType] : aa(set(A),set(A),image2(A,A,id(A)),top_top(set(A))) = top_top(set(A)) ).
% surj_id
tff(fact_2272_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( no_top(A)
=> ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H))) ) ).
% not_UNIV_le_Iic
tff(fact_2273_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I4: set(B),X2: fun(B,A),Y: fun(B,A)] :
( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,A),fun(B,bool)),I4),X2)))
=> ( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,A),fun(B,bool)),I4),Y)))
=> finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_hx(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I4),X2),Y))) ) ) ) ).
% sum.finite_Collect_op
tff(fact_2274_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I4: set(B),X2: fun(B,A),Y: fun(B,A)] :
( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),X2)))
=> ( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),Y)))
=> finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_hz(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I4),X2),Y))) ) ) ) ).
% prod.finite_Collect_op
tff(fact_2275_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : finite_finite2(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).
% finite_atLeastZeroLessThan_int
tff(fact_2276_card__eq__sum,axiom,
! [A: $tType,A4: set(A)] : finite_card(A,A4) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ia(A,nat)),A4) ).
% card_eq_sum
tff(fact_2277_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A),P: fun(B,bool)] :
( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_ib(set(B),fun(fun(B,bool),fun(B,bool)),A4),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_ic(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A4) ) ) ) ).
% prod.inter_filter
tff(fact_2278_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),A3: B,B2: fun(B,A),C2: A] :
( finite_finite2(B,S2)
=> ( ( pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_id(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(B,S2)),one_one(nat)))) ) )
& ( ~ pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_id(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),finite_card(B,S2)) ) ) ) ) ) ).
% prod_gen_delta
tff(fact_2279_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : finite_finite2(code_integer,set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).
% finite_atLeastZeroLessThan_integer
tff(fact_2280_card__2__iff_H,axiom,
! [A: $tType,S2: set(A)] :
( ( finite_card(A,S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
<=> ? [X4: A] :
( pp(member(A,X4,S2))
& ? [Xa2: A] :
( pp(member(A,Xa2,S2))
& ( X4 != Xa2 )
& ! [Xb4: A] :
( pp(member(A,Xb4,S2))
=> ( ( Xb4 = X4 )
| ( Xb4 = Xa2 ) ) ) ) ) ) ).
% card_2_iff'
tff(fact_2281_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [R4: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),R4,zero_zero(A)),zero_zero(A)))
=> ( ! [X12: A,Y12: A,X23: A,Y22: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X12),X23))
& pp(aa(A,bool,aa(A,fun(A,bool),R4,Y12),Y22)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(A,A,aa(A,fun(A,A),plus_plus(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y22))) )
=> ( finite_finite2(B,S2)
=> ( ! [X3: B] :
( pp(member(B,X3,S2))
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(B,A,H,X3)),aa(B,A,G,X3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2))) ) ) ) ) ) ).
% sum.related
tff(fact_2282_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [R4: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),R4,one_one(A)),one_one(A)))
=> ( ! [X12: A,Y12: A,X23: A,Y22: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X12),X23))
& pp(aa(A,bool,aa(A,fun(A,bool),R4,Y12),Y22)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(A,A,aa(A,fun(A,A),times_times(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y22))) )
=> ( finite_finite2(B,S2)
=> ( ! [X3: B] :
( pp(member(B,X3,S2))
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(B,A,H,X3)),aa(B,A,G,X3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2))) ) ) ) ) ) ).
% prod.related
tff(fact_2283_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),X2: B,G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( ( pp(member(B,X2,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),insert2(B,X2),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) ) )
& ( ~ pp(member(B,X2,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),insert2(B,X2),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)) ) ) ) ) ) ).
% sum.insert_if
tff(fact_2284_atLeast0__atMost__Suc,axiom,
! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).
% atLeast0_atMost_Suc
tff(fact_2285_atLeast0__lessThan__Suc,axiom,
! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).
% atLeast0_lessThan_Suc
tff(fact_2286_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( set_or1337092689740270186AtMost(nat,M,N) = aa(set(nat),set(nat),insert2(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).
% Icc_eq_insert_lb_nat
tff(fact_2287_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% atLeastAtMostSuc_conv
tff(fact_2288_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),set(nat),insert2(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).
% atLeastAtMost_insertL
tff(fact_2289_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S5: set(B),T4: set(C),S2: set(B),I: fun(C,B),J: fun(B,C),T5: set(C),G: fun(B,A),H: fun(C,A)] :
( finite_finite2(B,S5)
=> ( finite_finite2(C,T4)
=> ( ! [A5: B] :
( pp(member(B,A5,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
=> ( aa(C,B,I,aa(B,C,J,A5)) = A5 ) )
=> ( ! [A5: B] :
( pp(member(B,A5,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
=> pp(member(C,aa(B,C,J,A5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))) )
=> ( ! [B3: C] :
( pp(member(C,B3,aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
=> ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
=> ( ! [B3: C] :
( pp(member(C,B3,aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4)))
=> pp(member(B,aa(C,B,I,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5))) )
=> ( ! [A5: B] :
( pp(member(B,A5,S5))
=> ( aa(B,A,G,A5) = one_one(A) ) )
=> ( ! [B3: C] :
( pp(member(C,B3,T4))
=> ( aa(C,A,H,B3) = one_one(A) ) )
=> ( ! [A5: B] :
( pp(member(B,A5,S2))
=> ( aa(C,A,H,aa(B,C,J,A5)) = aa(B,A,G,A5) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H),T5) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_2290_dflt__None__set__def,axiom,
! [A: $tType,S2: set(A)] :
( ( ( S2 = bot_bot(set(A)) )
=> ( dflt_None_set(A,S2) = none(set(A)) ) )
& ( ( S2 != bot_bot(set(A)) )
=> ( dflt_None_set(A,S2) = aa(set(A),option(set(A)),some(set(A)),S2) ) ) ) ).
% dflt_None_set_def
tff(fact_2291_sum__eq__1__iff,axiom,
! [A: $tType,A4: set(A),F: fun(A,nat)] :
( finite_finite2(A,A4)
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4) = one_one(nat) )
<=> ? [X4: A] :
( pp(member(A,X4,A4))
& ( aa(A,nat,F,X4) = one_one(nat) )
& ! [Xa2: A] :
( pp(member(A,Xa2,A4))
=> ( ( X4 != Xa2 )
=> ( aa(A,nat,F,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_2292_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F),A4))),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),uminus_uminus(set(B)),A4)))) ) ).
% surj_Compl_image_subset
tff(fact_2293_card__atLeastZeroLessThan__int,axiom,
! [U: int] : finite_card(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ).
% card_atLeastZeroLessThan_int
tff(fact_2294_atMost__nat__numeral,axiom,
! [K: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert2(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).
% atMost_nat_numeral
tff(fact_2295_lessThan__nat__numeral,axiom,
! [K: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert2(nat,pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).
% lessThan_nat_numeral
tff(fact_2296_sum__Suc,axiom,
! [A: $tType,F: fun(A,nat),A4: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ie(fun(A,nat),fun(A,nat),F)),A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4)),finite_card(A,A4)) ).
% sum_Suc
tff(fact_2297_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_if(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) ) ) ) ).
% prod.setdiff_irrelevant
tff(fact_2298_notin__range__Some,axiom,
! [A: $tType,X2: option(A)] :
( ~ pp(member(option(A),X2,aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))))
<=> ( X2 = none(A) ) ) ).
% notin_range_Some
tff(fact_2299_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I4: set(A),I: A,F: fun(A,B)] :
( finite_finite2(A,I4)
=> ( pp(member(A,I,I4))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I)))
=> ( ! [I2: A] :
( pp(member(A,I2,I4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,I2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I4))) ) ) ) ) ) ).
% less_1_prod2
tff(fact_2300_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I4: set(A),F: fun(A,B)] :
( finite_finite2(A,I4)
=> ( ( I4 != bot_bot(set(A)) )
=> ( ! [I2: A] :
( pp(member(A,I2,I4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),I4))) ) ) ) ) ).
% less_1_prod
tff(fact_2301_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),H: fun(B,C),G: fun(C,A)] :
( finite_finite2(B,A4)
=> ( ! [X3: B,Y4: B] :
( pp(member(B,X3,A4))
=> ( pp(member(B,Y4,A4))
=> ( ( X3 != Y4 )
=> ( ( aa(B,C,H,X3) = aa(B,C,H,Y4) )
=> ( aa(C,A,G,aa(B,C,H,X3)) = zero_zero(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A4) ) ) ) ) ).
% sum.reindex_nontrivial
tff(fact_2302_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),H: fun(B,C),G: fun(C,A)] :
( finite_finite2(B,A4)
=> ( ! [X3: B,Y4: B] :
( pp(member(B,X3,A4))
=> ( pp(member(B,Y4,A4))
=> ( ( X3 != Y4 )
=> ( ( aa(B,C,H,X3) = aa(B,C,H,Y4) )
=> ( aa(C,A,G,aa(B,C,H,X3)) = one_one(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A4) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_2303_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [B5: set(B),A4: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A4))
=> ( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ).
% sum.subset_diff
tff(fact_2304_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C5: set(B),A4: set(B),B5: set(B),G: fun(B,A),H: fun(B,A)] :
( finite_finite2(B,C5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C5))
=> ( ! [A5: B] :
( pp(member(B,A5,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A4)))
=> ( aa(B,A,G,A5) = one_one(A) ) )
=> ( ! [B3: B] :
( pp(member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B5)))
=> ( aa(B,A,H,B3) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B5) )
<=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C5) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_2305_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C5: set(B),A4: set(B),B5: set(B),G: fun(B,A),H: fun(B,A)] :
( finite_finite2(B,C5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),C5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C5))
=> ( ! [A5: B] :
( pp(member(B,A5,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),A4)))
=> ( aa(B,A,G,A5) = one_one(A) ) )
=> ( ! [B3: B] :
( pp(member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C5),B5)))
=> ( aa(B,A,H,B3) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C5) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B5) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_2306_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S2: set(B),G: fun(B,A)] :
( finite_finite2(B,T5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) ) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_2307_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S2: set(B),G: fun(B,A)] :
( finite_finite2(B,T5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) ) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_2308_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
( finite_finite2(B,T5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,H,X3) = one_one(A) ) )
=> ( ! [X3: B] :
( pp(member(B,X3,S2))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T5) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_2309_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
( finite_finite2(B,T5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( ! [X3: B] :
( pp(member(B,X3,S2))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_2310_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,N),set_or7035219750837199246ssThan(nat,M,N)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).
% atLeastLessThanSuc
tff(fact_2311_Option_Othese__def,axiom,
! [A: $tType,A4: set(option(A))] : these(A,A4) = aa(set(option(A)),set(A),image2(option(A),A,the2(A)),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_hq(set(option(A)),fun(option(A),bool),A4))) ).
% Option.these_def
tff(fact_2312_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).
% atLeast0_atMost_Suc_eq_insert_0
tff(fact_2313_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).
% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_2314_atMost__Suc__eq__insert__0,axiom,
! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N))) ).
% atMost_Suc_eq_insert_0
tff(fact_2315_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N))) ).
% lessThan_Suc_eq_insert_0
tff(fact_2316_card__2__iff,axiom,
! [A: $tType,S2: set(A)] :
( ( finite_card(A,S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
<=> ? [X4: A,Y3: A] :
( ( S2 = aa(set(A),set(A),insert2(A,X4),aa(set(A),set(A),insert2(A,Y3),bot_bot(set(A)))) )
& ( X4 != Y3 ) ) ) ).
% card_2_iff
tff(fact_2317_card__3__iff,axiom,
! [A: $tType,S2: set(A)] :
( ( finite_card(A,S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
<=> ? [X4: A,Y3: A,Z4: A] :
( ( S2 = aa(set(A),set(A),insert2(A,X4),aa(set(A),set(A),insert2(A,Y3),aa(set(A),set(A),insert2(A,Z4),bot_bot(set(A))))) )
& ( X4 != Y3 )
& ( Y3 != Z4 )
& ( X4 != Z4 ) ) ) ).
% card_3_iff
tff(fact_2318_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ordere6911136660526730532id_add(B)
=> ! [I4: set(C),G: fun(A,B),F: fun(C,A)] :
( finite_finite2(C,I4)
=> ( ! [I2: C] :
( pp(member(C,I2,I4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F,I2)))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(C),set(A),image2(C,A,F),I4))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),G),F)),I4))) ) ) ) ).
% sum_image_le
tff(fact_2319_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),X2: B,G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( pp(member(B,X2,A4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert2(B,X2),bot_bot(set(B)))))) ) ) ) ) ).
% sum.remove
tff(fact_2320_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),G: fun(B,A),X2: B] :
( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),insert2(B,X2),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert2(B,X2),bot_bot(set(B)))))) ) ) ) ).
% sum.insert_remove
tff(fact_2321_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atMost(nat),N)),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat)))) ).
% atLeast1_atMost_eq_remove0
tff(fact_2322_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_lessThan(nat),N)),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat)))) ).
% atLeast1_lessThan_eq_remove0
tff(fact_2323_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
=> ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert2(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
=> ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = bot_bot(set(nat)) ) ) ) ).
% atLeastLessThan_nat_numeral
tff(fact_2324_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S2: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
( finite_finite2(B,S2)
=> ( ( pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ig(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))) ) )
& ( ~ pp(member(B,A3,S2))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ig(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).
% sum.delta_remove
tff(fact_2325_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( linordered_semidom(A)
=> ! [A4: set(B),F: fun(B,A),N: A,K: nat] :
( ! [I2: B] :
( pp(member(B,I2,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I2)),N)) ) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(B,A4)),K))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A4)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).
% prod_le_power
tff(fact_2326_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set(A),B5: set(A)] :
( pp(member(A,A3,A4))
=> ( ~ pp(member(A,A3,B5))
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert2(A,A3),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))),one_one(nat)) ) ) ) ).
% card_Diff_insert
tff(fact_2327_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),insert2(A,X2),Y))),N)) ) ) ).
% card_insert_le_m1
tff(fact_2328_card__Diff__singleton__if,axiom,
! [A: $tType,X2: A,A4: set(A)] :
( ( pp(member(A,X2,A4))
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,A4)),one_one(nat)) ) )
& ( ~ pp(member(A,X2,A4))
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) = finite_card(A,A4) ) ) ) ).
% card_Diff_singleton_if
tff(fact_2329_card__Diff__singleton,axiom,
! [A: $tType,X2: A,A4: set(A)] :
( pp(member(A,X2,A4))
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,A4)),one_one(nat)) ) ) ).
% card_Diff_singleton
tff(fact_2330_in__finite__psubset,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A4),B5),finite_psubset(A)))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5))
& finite_finite2(A,B5) ) ) ).
% in_finite_psubset
tff(fact_2331_card__1__singletonE,axiom,
! [A: $tType,A4: set(A)] :
( ( finite_card(A,A4) = one_one(nat) )
=> ~ ! [X3: A] : A4 != aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))) ) ).
% card_1_singletonE
tff(fact_2332_integer__of__nat__def,axiom,
code_integer_of_nat = aa(fun(nat,int),fun(nat,code_integer),map_fun(nat,nat,int,code_integer,id(nat),code_integer_of_int),semiring_1_of_nat(int)) ).
% integer_of_nat_def
tff(fact_2333_card__UNIV__unit,axiom,
finite_card(product_unit,top_top(set(product_unit))) = one_one(nat) ).
% card_UNIV_unit
tff(fact_2334_card__lessThan,axiom,
! [U: nat] : finite_card(nat,aa(nat,set(nat),set_ord_lessThan(nat),U)) = U ).
% card_lessThan
tff(fact_2335_card__atLeastLessThan,axiom,
! [L: nat,U: nat] : finite_card(nat,set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).
% card_atLeastLessThan
tff(fact_2336_card__atMost,axiom,
! [U: nat] : finite_card(nat,aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).
% card_atMost
tff(fact_2337_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: bool] :
( ( pp(P)
=> ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ih(bool,fun(A,fun(B,bool)),P))) = top_top(set(product_prod(A,B))) ) )
& ( ~ pp(P)
=> ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ih(bool,fun(A,fun(B,bool)),P))) = bot_bot(set(product_prod(A,B))) ) ) ) ).
% Collect_const_case_prod
tff(fact_2338_nat__of__integer__integer__of__nat,axiom,
! [N: nat] : aa(code_integer,nat,code_nat_of_integer,aa(nat,code_integer,code_integer_of_nat,N)) = N ).
% nat_of_integer_integer_of_nat
tff(fact_2339_card__atLeastAtMost,axiom,
! [L: nat,U: nat] : finite_card(nat,set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).
% card_atLeastAtMost
tff(fact_2340_integer__of__nat_Orep__eq,axiom,
! [X2: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,X2)) = aa(nat,int,semiring_1_of_nat(int),X2) ).
% integer_of_nat.rep_eq
tff(fact_2341_int__of__integer__integer__of__nat,axiom,
! [N: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,N)) = aa(nat,int,semiring_1_of_nat(int),N) ).
% int_of_integer_integer_of_nat
tff(fact_2342_top__empty__eq2,axiom,
! [B: $tType,A: $tType,X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),top_top(set(product_prod(A,B))))) ) ).
% top_empty_eq2
tff(fact_2343_int__of__integer__induct,axiom,
! [Y: int,P: fun(int,bool)] :
( pp(member(int,Y,top_top(set(int))))
=> ( ! [X3: code_integer] : pp(aa(int,bool,P,aa(code_integer,int,code_int_of_integer,X3)))
=> pp(aa(int,bool,P,Y)) ) ) ).
% int_of_integer_induct
tff(fact_2344_int__of__integer__cases,axiom,
! [Y: int] :
( pp(member(int,Y,top_top(set(int))))
=> ~ ! [X3: code_integer] : Y != aa(code_integer,int,code_int_of_integer,X3) ) ).
% int_of_integer_cases
tff(fact_2345_int__of__integer,axiom,
! [X2: code_integer] : pp(member(int,aa(code_integer,int,code_int_of_integer,X2),top_top(set(int)))) ).
% int_of_integer
tff(fact_2346_integer__of__int__inject,axiom,
! [X2: int,Y: int] :
( pp(member(int,X2,top_top(set(int))))
=> ( pp(member(int,Y,top_top(set(int))))
=> ( ( aa(int,code_integer,code_integer_of_int,X2) = aa(int,code_integer,code_integer_of_int,Y) )
<=> ( X2 = Y ) ) ) ) ).
% integer_of_int_inject
tff(fact_2347_integer__of__int__induct,axiom,
! [P: fun(code_integer,bool),X2: code_integer] :
( ! [Y4: int] :
( pp(member(int,Y4,top_top(set(int))))
=> pp(aa(code_integer,bool,P,aa(int,code_integer,code_integer_of_int,Y4))) )
=> pp(aa(code_integer,bool,P,X2)) ) ).
% integer_of_int_induct
tff(fact_2348_integer__of__int__cases,axiom,
! [X2: code_integer] :
~ ! [Y4: int] :
( ( X2 = aa(int,code_integer,code_integer_of_int,Y4) )
=> ~ pp(member(int,Y4,top_top(set(int)))) ) ).
% integer_of_int_cases
tff(fact_2349_subset__card__intvl__is__intvl,axiom,
! [A4: set(nat),K: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),A4),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),finite_card(nat,A4)))))
=> ( A4 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),finite_card(nat,A4))) ) ) ).
% subset_card_intvl_is_intvl
tff(fact_2350_integer__of__int__inverse,axiom,
! [Y: int] :
( pp(member(int,Y,top_top(set(int))))
=> ( aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,Y)) = Y ) ) ).
% integer_of_int_inverse
tff(fact_2351_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set(nat),N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(nat,N5)),N)) ) ).
% subset_eq_atLeast0_lessThan_card
tff(fact_2352_integer__of__nat__eq__of__nat,axiom,
code_integer_of_nat = semiring_1_of_nat(code_integer) ).
% integer_of_nat_eq_of_nat
tff(fact_2353_card__less__Suc2,axiom,
! [M2: set(nat),I: nat] :
( ~ pp(member(nat,zero_zero(nat),M2))
=> ( finite_card(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ii(set(nat),fun(nat,fun(nat,bool)),M2),I))) = finite_card(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),M2),I))) ) ) ).
% card_less_Suc2
tff(fact_2354_card__less__Suc,axiom,
! [M2: set(nat),I: nat] :
( pp(member(nat,zero_zero(nat),M2))
=> ( aa(nat,nat,suc,finite_card(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ii(set(nat),fun(nat,fun(nat,bool)),M2),I)))) = finite_card(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),M2),I))) ) ) ).
% card_less_Suc
tff(fact_2355_card__less,axiom,
! [M2: set(nat),I: nat] :
( pp(member(nat,zero_zero(nat),M2))
=> ( finite_card(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),M2),I))) != zero_zero(nat) ) ) ).
% card_less
tff(fact_2356_integer__of__nat__0,axiom,
aa(nat,code_integer,code_integer_of_nat,zero_zero(nat)) = zero_zero(code_integer) ).
% integer_of_nat_0
tff(fact_2357_integer__of__nat_Oabs__eq,axiom,
! [X2: nat] : aa(nat,code_integer,code_integer_of_nat,X2) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X2)) ).
% integer_of_nat.abs_eq
tff(fact_2358_integer__of__nat__numeral,axiom,
! [N: num] : aa(nat,code_integer,code_integer_of_nat,aa(num,nat,numeral_numeral(nat),N)) = aa(num,code_integer,numeral_numeral(code_integer),N) ).
% integer_of_nat_numeral
tff(fact_2359_integer__of__nat__1,axiom,
aa(nat,code_integer,code_integer_of_nat,one_one(nat)) = one_one(code_integer) ).
% integer_of_nat_1
tff(fact_2360_card__sum__le__nat__sum,axiom,
! [S2: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bx(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(nat,S2)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_bx(nat,nat)),S2))) ).
% card_sum_le_nat_sum
tff(fact_2361_set__encode__insert,axiom,
! [A4: set(nat),N: nat] :
( finite_finite2(nat,A4)
=> ( ~ pp(member(nat,N,A4))
=> ( nat_set_encode(aa(set(nat),set(nat),insert2(nat,N),A4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),nat_set_encode(A4)) ) ) ) ).
% set_encode_insert
tff(fact_2362_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] : finite_card(int,set_or5935395276787703475ssThan(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).
% card_greaterThanLessThan_int
tff(fact_2363_wmax__insertI,axiom,
! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X2: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
( pp(member(product_prod(nat,nat),Y,YS))
=> ( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y),fun_pair_leq))
=> ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_max_weak))
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),X2),XS)),YS),fun_max_weak)) ) ) ) ).
% wmax_insertI
tff(fact_2364_wmin__insertI,axiom,
! [X2: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
( pp(member(product_prod(nat,nat),X2,XS))
=> ( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y),fun_pair_leq))
=> ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_min_weak))
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),Y),YS)),fun_min_weak)) ) ) ) ).
% wmin_insertI
tff(fact_2365_smin__insertI,axiom,
! [X2: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
( pp(member(product_prod(nat,nat),X2,XS))
=> ( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y),fun_pair_less))
=> ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS),fun_min_strict))
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),Y),YS)),fun_min_strict)) ) ) ) ).
% smin_insertI
tff(fact_2366_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I4: set(B),P3: fun(B,A),I: B] :
( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),P3)))
=> ( ( pp(member(B,I,I4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),aa(set(B),set(B),insert2(B,I),I4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I4) ) )
& ( ~ pp(member(B,I,I4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),aa(set(B),set(B),insert2(B,I),I4)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P3,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I4)) ) ) ) ) ) ).
% prod.insert'
tff(fact_2367_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,L: A,U: A] :
( pp(member(A,I,set_or5935395276787703475ssThan(A,L,U)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),U)) ) ) ) ).
% greaterThanLessThan_iff
tff(fact_2368_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : finite_finite2(int,set_or5935395276787703475ssThan(int,L,U)) ).
% finite_greaterThanLessThan_int
tff(fact_2369_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B2) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% greaterThanLessThan_empty_iff2
tff(fact_2370_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ( set_or5935395276787703475ssThan(A,A3,B2) = bot_bot(set(A)) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% greaterThanLessThan_empty_iff
tff(fact_2371_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
=> ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanLessThan_empty
tff(fact_2372_infinite__Ioo__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A3,B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% infinite_Ioo_iff
tff(fact_2373_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or5935395276787703475ssThan(A,X2,Y)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_greaterThanLessThan
tff(fact_2374_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [P3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),bot_bot(set(B))) = one_one(A) ) ).
% prod.empty'
tff(fact_2375_smin__emptyI,axiom,
! [X7: set(product_prod(nat,nat))] :
( ( X7 != bot_bot(set(product_prod(nat,nat))) )
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X7),bot_bot(set(product_prod(nat,nat)))),fun_min_strict)) ) ).
% smin_emptyI
tff(fact_2376_wmax__emptyI,axiom,
! [X7: set(product_prod(nat,nat))] :
( finite_finite2(product_prod(nat,nat),X7)
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X7),fun_max_weak)) ) ).
% wmax_emptyI
tff(fact_2377_wmin__emptyI,axiom,
! [X7: set(product_prod(nat,nat))] : pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X7),bot_bot(set(product_prod(nat,nat)))),fun_min_weak)) ).
% wmin_emptyI
tff(fact_2378_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),I4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_ik(fun(B,A),fun(set(B),fun(B,bool)),G),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I4) ) ).
% prod.non_neutral'
tff(fact_2379_infinite__Ioo,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A3,B2)) ) ) ).
% infinite_Ioo
tff(fact_2380_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_2381_prod__decode__aux_Ocases,axiom,
! [X2: product_prod(nat,nat)] :
~ ! [K3: nat,M4: nat] : X2 != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K3),M4) ).
% prod_decode_aux.cases
tff(fact_2382_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_2383_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),T5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) ) ) ) ) ).
% prod.mono_neutral_left'
tff(fact_2384_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),T5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S2) ) ) ) ) ).
% prod.mono_neutral_right'
tff(fact_2385_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),T5: set(B),H: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [I2: B] :
( pp(member(B,I2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,H,I2) = one_one(A) ) )
=> ( ! [X3: B] :
( pp(member(B,X3,S2))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),T5) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
tff(fact_2386_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S2: set(B),T5: set(B),G: fun(B,A),H: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T5))
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( ! [X3: B] :
( pp(member(B,X3,S2))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),S2) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
tff(fact_2387_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I4: set(B),G: fun(B,A),H: fun(B,A)] :
( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),G)))
=> ( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),H)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_il(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I4) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I4)) ) ) ) ) ).
% prod.distrib'
tff(fact_2388_prod_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [I4: set(B),P3: fun(B,A)] :
( ( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),P3)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),P3))) ) )
& ( ~ finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),I4),P3)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I4) = one_one(A) ) ) ) ) ).
% prod.G_def
tff(fact_2389_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_2390_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_2391_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B2) ) ).
% atLeastAtMost_diff_ends
tff(fact_2392_smax__insertI,axiom,
! [Y: product_prod(nat,nat),Y7: set(product_prod(nat,nat)),X2: product_prod(nat,nat),X7: set(product_prod(nat,nat))] :
( pp(member(product_prod(nat,nat),Y,Y7))
=> ( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y),fun_pair_less))
=> ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X7),Y7),fun_max_strict))
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert2(product_prod(nat,nat),X2),X7)),Y7),fun_max_strict)) ) ) ) ).
% smax_insertI
tff(fact_2393_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ pp(member(nat,N,nat_set_decode(Z2)))
=> ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),Z2)) = aa(set(nat),set(nat),insert2(nat,N),nat_set_decode(Z2)) ) ) ).
% set_decode_plus_power_2
tff(fact_2394_prod__decode__aux_Osimps,axiom,
! [M: nat,K: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
=> ( nat_prod_decode_aux(K,M) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
=> ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K))) ) ) ) ).
% prod_decode_aux.simps
tff(fact_2395_prod__decode__aux_Oelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod(nat,nat)] :
( ( nat_prod_decode_aux(X2,Xa) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X2))
=> ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X2),Xa)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X2))
=> ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X2))) ) ) ) ) ).
% prod_decode_aux.elims
tff(fact_2396_min__rpair__set,axiom,
fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_min_strict),fun_min_weak)) ).
% min_rpair_set
tff(fact_2397_triangle__Suc,axiom,
! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).
% triangle_Suc
tff(fact_2398_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : finite_finite2(nat,set_or5935395276787703475ssThan(nat,L,U)) ).
% finite_greaterThanLessThan
tff(fact_2399_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : finite_finite2(code_integer,set_or5935395276787703475ssThan(code_integer,L,U)) ).
% finite_greaterThanLessThan_integer
tff(fact_2400_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] : finite_card(nat,set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).
% card_greaterThanLessThan
tff(fact_2401_max__rpair__set,axiom,
fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_max_strict),fun_max_weak)) ).
% max_rpair_set
tff(fact_2402_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).
% atLeastSucLessThan_greaterThanLessThan
tff(fact_2403_smax__emptyI,axiom,
! [Y7: set(product_prod(nat,nat))] :
( finite_finite2(product_prod(nat,nat),Y7)
=> ( ( Y7 != bot_bot(set(product_prod(nat,nat))) )
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y7),fun_max_strict)) ) ) ).
% smax_emptyI
tff(fact_2404_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_2405_prod__decode__aux_Opelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod(nat,nat)] :
( ( nat_prod_decode_aux(X2,Xa) = Y )
=> ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Xa)))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X2))
=> ( Y = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X2),Xa)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X2))
=> ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X2))) ) ) )
=> ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Xa))) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_2406_total__on__singleton,axiom,
! [A: $tType,X2: A] : total_on(A,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),bot_bot(set(product_prod(A,A))))) ).
% total_on_singleton
tff(fact_2407_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M) ).
% prod_encode_prod_decode_aux
tff(fact_2408_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I4: set(B),P3: fun(B,A),I: B] :
( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,A),fun(B,bool)),I4),P3)))
=> ( ( pp(member(B,I,I4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),aa(set(B),set(B),insert2(B,I),I4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),I4) ) )
& ( ~ pp(member(B,I,I4))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),aa(set(B),set(B),insert2(B,I),I4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P3,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),I4)) ) ) ) ) ) ).
% sum.insert'
tff(fact_2409_max__ext_Omax__extI,axiom,
! [A: $tType,X7: set(A),Y7: set(A),R4: set(product_prod(A,A))] :
( finite_finite2(A,X7)
=> ( finite_finite2(A,Y7)
=> ( ( Y7 != bot_bot(set(A)) )
=> ( ! [X3: A] :
( pp(member(A,X3,X7))
=> ? [Xa3: A] :
( pp(member(A,Xa3,Y7))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R4)) ) )
=> pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X7),Y7),max_ext(A,R4))) ) ) ) ) ).
% max_ext.max_extI
tff(fact_2410_total__on__def,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( total_on(A,A4,R)
<=> ! [X4: A] :
( pp(member(A,X4,A4))
=> ! [Xa2: A] :
( pp(member(A,Xa2,A4))
=> ( ( X4 != Xa2 )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R))
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R)) ) ) ) ) ) ).
% total_on_def
tff(fact_2411_total__onI,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,A4))
=> ( pp(member(A,Y4,A4))
=> ( ( X3 != Y4 )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R))
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R)) ) ) ) )
=> total_on(A,A4,R) ) ).
% total_onI
tff(fact_2412_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I4: set(B),G: fun(B,A),H: fun(B,A)] :
( finite_finite2(B,I4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bz(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),I4)) ) ) ) ).
% sum.distrib_triv'
tff(fact_2413_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2)))) ).
% le_prod_encode_1
tff(fact_2414_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2)))) ).
% le_prod_encode_2
tff(fact_2415_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I4: set(B),G: fun(B,A),H: fun(B,A)] :
( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,A),fun(B,bool)),I4),G)))
=> ( finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,A),fun(B,bool)),I4),H)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_bz(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),I4)) ) ) ) ) ).
% sum.distrib'
tff(fact_2416_prod__encode__def,axiom,
nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_im(nat,fun(nat,nat))) ).
% prod_encode_def
tff(fact_2417_max__ext_Ocases,axiom,
! [A: $tType,A1: set(A),A22: set(A),R4: set(product_prod(A,A))] :
( pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R4)))
=> ~ ( finite_finite2(A,A1)
=> ( finite_finite2(A,A22)
=> ( ( A22 != bot_bot(set(A)) )
=> ~ ! [X: A] :
( pp(member(A,X,A1))
=> ? [Xa4: A] :
( pp(member(A,Xa4,A22))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4),R4)) ) ) ) ) ) ) ).
% max_ext.cases
tff(fact_2418_max__ext_Osimps,axiom,
! [A: $tType,A1: set(A),A22: set(A),R4: set(product_prod(A,A))] :
( pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22),max_ext(A,R4)))
<=> ( finite_finite2(A,A1)
& finite_finite2(A,A22)
& ( A22 != bot_bot(set(A)) )
& ! [X4: A] :
( pp(member(A,X4,A1))
=> ? [Xa2: A] :
( pp(member(A,Xa2,A22))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R4)) ) ) ) ) ).
% max_ext.simps
tff(fact_2419_max__extp__eq,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X2: set(A),Y: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R),X2),Y))
<=> pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X2),Y),max_ext(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% max_extp_eq
tff(fact_2420_max__extp__max__ext__eq,axiom,
! [A: $tType,R4: set(product_prod(A,A)),X: set(A),Xa3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R4)),X),Xa3))
<=> pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X),Xa3),max_ext(A,R4))) ) ).
% max_extp_max_ext_eq
tff(fact_2421_one__div__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).
% one_div_numeral
tff(fact_2422_sqr_Osimps_I3_J,axiom,
! [N: num] : sqr(aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(N)),N))) ).
% sqr.simps(3)
tff(fact_2423_one__mod__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).
% one_mod_numeral
tff(fact_2424_card__UNION,axiom,
! [A: $tType,A4: set(set(A))] :
( finite_finite2(set(A),A4)
=> ( ! [X3: set(A)] :
( pp(member(set(A),X3,A4))
=> finite_finite2(A,X3) )
=> ( finite_card(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_io(set(set(A)),int)),aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_ip(set(set(A)),fun(set(set(A)),bool),A4)))) ) ) ) ).
% card_UNION
tff(fact_2425_Sup__atMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),Y)) = Y ) ).
% Sup_atMost
tff(fact_2426_Sup__lessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),Y)) = Y ) ).
% Sup_lessThan
tff(fact_2427_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),X2: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),X2)) = aa(product_prod(A,C),A,product_fst(A,C),X2) ).
% fst_apsnd
tff(fact_2428_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),aa(fun(product_prod(A,C),A),fun(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A)),comp(product_prod(A,C),A,product_prod(A,B)),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F)) = product_fst(A,B) ).
% fst_comp_apsnd
tff(fact_2429_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),X2: product_prod(A,C),G: fun(C,B)] :
( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),X2) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G),X2) )
<=> ( aa(C,B,F,aa(product_prod(A,C),C,product_snd(A,C),X2)) = aa(C,B,G,aa(product_prod(A,C),C,product_snd(A,C),X2)) ) ) ).
% apsnd_eq_conv
tff(fact_2430_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),X2: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F),X2)) = aa(C,A,F,aa(product_prod(B,C),C,product_snd(B,C),X2)) ).
% snd_apsnd
tff(fact_2431_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),aa(fun(product_prod(A,C),C),fun(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C)),comp(product_prod(A,C),C,product_prod(A,B)),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),aa(fun(B,C),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),C)),comp(B,C,product_prod(A,B)),F),product_snd(A,B)) ).
% snd_comp_apsnd
tff(fact_2432_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L) ).
% fst_divmod_integer
tff(fact_2433_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L) ).
% snd_divmod_integer
tff(fact_2434_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X2,Y)) = Y ) ) ) ).
% Sup_atLeastAtMost
tff(fact_2435_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> ( complete_Inf_Inf(A,set_or1337092689740270186AtMost(A,X2,Y)) = X2 ) ) ) ).
% Inf_atLeastAtMost
tff(fact_2436_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X2,Y)) = Y ) ) ) ).
% Sup_atLeastLessThan
tff(fact_2437_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> ( complete_Inf_Inf(A,set_or7035219750837199246ssThan(A,X2,Y)) = X2 ) ) ) ).
% Inf_atLeastLessThan
tff(fact_2438_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).
% prod.collapse
tff(fact_2439_Inf__atMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A] : complete_Inf_Inf(A,aa(A,set(A),set_ord_atMost(A),X2)) = bot_bot(A) ) ).
% Inf_atMost
tff(fact_2440_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X2,Y)) = Y ) ) ) ).
% Sup_greaterThanLessThan
tff(fact_2441_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S2: set(product_prod(A,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S2))
=> pp(member(A,A3,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S2))) ) ).
% img_fst
tff(fact_2442_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> ( complete_Inf_Inf(A,set_or5935395276787703475ssThan(A,X2,Y)) = X2 ) ) ) ).
% Inf_greaterThanLessThan
tff(fact_2443_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S2: set(product_prod(A,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),S2))
=> pp(member(B,B2,aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S2))) ) ).
% img_snd
tff(fact_2444_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).
% fst_divmod_abs
tff(fact_2445_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).
% snd_divmod_abs
tff(fact_2446_Inf__atMostLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X2))
=> ( complete_Inf_Inf(A,aa(A,set(A),set_ord_lessThan(A),X2)) = bot_bot(A) ) ) ) ).
% Inf_atMostLessThan
tff(fact_2447_split__comp__eq,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,F: fun(A,fun(B,C)),G: fun(D,A)] : aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_iq(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),F),G) = aa(fun(D,fun(B,C)),fun(product_prod(D,B),C),product_case_prod(D,B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_ir(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),F),G)) ).
% split_comp_eq
tff(fact_2448_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),X: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),X) = aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),X)),aa(product_prod(A,B),B,product_snd(A,B),X)) ).
% case_prod_beta'
tff(fact_2449_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType,X: fun(A,fun(B,C)),Xa3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X),Xa3) = aa(B,C,aa(A,fun(B,C),X,aa(product_prod(A,B),A,product_fst(A,B),Xa3)),aa(product_prod(A,B),B,product_snd(A,B),Xa3)) ).
% case_prod_unfold
tff(fact_2450_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),A4: fun(A,fun(B,bool))] :
( pp(member(product_prod(A,B),X2,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A4))))
=> pp(aa(B,bool,aa(A,fun(B,bool),A4,aa(product_prod(A,B),A,product_fst(A,B),X2)),aa(product_prod(A,B),B,product_snd(A,B),X2))) ) ).
% Product_Type.Collect_case_prodD
tff(fact_2451_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,A)),P3: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),P3) = aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),P3)),aa(product_prod(B,C),C,product_snd(B,C),P3)) ).
% case_prod_beta
tff(fact_2452_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod) = aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).
% split_beta
tff(fact_2453_effect__heapI,axiom,
! [A: $tType,N: nat,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A] :
( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( H3 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) )
=> heap_Time_effect(A,heap_Time_heap(A,F),H,H3,R,N) ) ) ) ).
% effect_heapI
tff(fact_2454_effect__heapE,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_heap(A,F),H,H3,R,N)
=> ~ ( ( H3 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( R != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) ) ) ) ) ).
% effect_heapE
tff(fact_2455_effect__guardI,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),H3: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),N: nat,R: A] :
( pp(aa(heap_ext(product_unit),bool,P,H))
=> ( ( H3 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) )
=> heap_Time_effect(A,heap_Time_guard(A,P,F),H,H3,R,N) ) ) ) ) ).
% effect_guardI
tff(fact_2456_effect__guardE,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_guard(A,P,F),H,H3,R,N)
=> ~ ( ( H3 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
=> ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) )
=> ~ pp(aa(heap_ext(product_unit),bool,P,H)) ) ) ) ) ).
% effect_guardE
tff(fact_2457_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,C2,H,H3,R,N)
=> ~ ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H))) )
=> ( ( H3 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H)))) )
=> ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H)))) )
=> ~ heap_Time_success(A,C2,H) ) ) ) ) ).
% effectE
tff(fact_2458_prod__eq__iff,axiom,
! [B: $tType,A: $tType,S: product_prod(A,B),T2: product_prod(A,B)] :
( ( S = T2 )
<=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S) = aa(product_prod(A,B),A,product_fst(A,B),T2) )
& ( aa(product_prod(A,B),B,product_snd(A,B),S) = aa(product_prod(A,B),B,product_snd(A,B),T2) ) ) ) ).
% prod_eq_iff
tff(fact_2459_prod__eqI,axiom,
! [B: $tType,A: $tType,P3: product_prod(A,B),Q3: product_prod(A,B)] :
( ( aa(product_prod(A,B),A,product_fst(A,B),P3) = aa(product_prod(A,B),A,product_fst(A,B),Q3) )
=> ( ( aa(product_prod(A,B),B,product_snd(A,B),P3) = aa(product_prod(A,B),B,product_snd(A,B),Q3) )
=> ( P3 = Q3 ) ) ) ).
% prod_eqI
tff(fact_2460_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
& ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
tff(fact_2461_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
( ! [A7: A,X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A7),X_1))
<=> ! [Z4: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),Z4)),aa(product_prod(A,B),B,product_snd(A,B),Z4))) ) ).
% All_prod_contract
tff(fact_2462_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
( ? [A7: A,X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A7),X_1))
<=> ? [Z4: product_prod(A,B)] : pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),Z4)),aa(product_prod(A,B),B,product_snd(A,B),Z4))) ) ).
% Ex_prod_contract
tff(fact_2463_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
( pp(aa(C,bool,P,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod)))
<=> ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
=> pp(aa(C,bool,P,aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).
% prod.split_sel
tff(fact_2464_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
( pp(aa(C,bool,P,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod)))
<=> ~ ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
& ~ pp(aa(C,bool,P,aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).
% prod.split_sel_asm
tff(fact_2465_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod(A,B)] : T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ).
% surjective_pairing
tff(fact_2466_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).
% prod.exhaust_sel
tff(fact_2467_exI__realizer,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Y: A,X2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,Y),X2))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y)))) ) ).
% exI_realizer
tff(fact_2468_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),P3: A,Q: fun(B,bool),Q3: B] :
( pp(aa(A,bool,P,P3))
=> ( pp(aa(B,bool,Q,Q3))
=> ( pp(aa(A,bool,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3))))
& pp(aa(B,bool,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3)))) ) ) ) ).
% conjI_realizer
tff(fact_2469_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
( heap_Time_success(A,F,H)
=> ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,G)),H) = heap_Time_timeFrame(B,aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H)))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),aa(A,heap_Time_Heap(B),G,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H))))),aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H)))))) ) ) ).
% execute_bind_success
tff(fact_2470_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),Y: A,Z2: B] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
=> ( aa(product_prod(A,B),B,product_snd(A,B),X2) = Z2 ) ) ).
% sndI
tff(fact_2471_sndE,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),A3: A,B2: B,P: fun(B,bool)] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> ( pp(aa(B,bool,P,aa(product_prod(A,B),B,product_snd(A,B),X2)))
=> pp(aa(B,bool,P,B2)) ) ) ).
% sndE
tff(fact_2472_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y: A,A3: A] :
( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y)) = A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
tff(fact_2473_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] : aa(product_prod(Aa,A),A,product_snd(Aa,A),aa(A,product_prod(Aa,A),aa(Aa,fun(A,product_prod(Aa,A)),product_Pair(Aa,A),X1),X22)) = X22 ).
% snd_conv
tff(fact_2474_Some__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] : aa(A,option(A),some(A),complete_Inf_Inf(A,A4)) = complete_Inf_Inf(option(A),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A4)) ) ).
% Some_Inf
tff(fact_2475_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),Y: A,Z2: B] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z2) )
=> ( aa(product_prod(A,B),A,product_fst(A,B),X2) = Y ) ) ).
% fstI
tff(fact_2476_fstE,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),A3: A,B2: B,P: fun(A,bool)] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> ( pp(aa(A,bool,P,aa(product_prod(A,B),A,product_fst(A,B),X2)))
=> pp(aa(A,bool,P,A3)) ) ) ).
% fstE
tff(fact_2477_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,A3: A] :
( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)) = A3 )
=> ( X2 = A3 ) ) ).
% fst_eqD
tff(fact_2478_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).
% fst_conv
tff(fact_2479_snd__fst__flip,axiom,
! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),aa(fun(product_prod(A,B),A),fun(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A)),comp(product_prod(A,B),A,product_prod(B,A)),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))))),Xy) ).
% snd_fst_flip
tff(fact_2480_fst__snd__flip,axiom,
! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),aa(fun(product_prod(B,A),A),fun(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A)),comp(product_prod(B,A),A,product_prod(A,B)),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_it(A,fun(B,product_prod(B,A))))),Xy) ).
% fst_snd_flip
tff(fact_2481_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B5: fun(B,set(A)),A4: set(B),A3: B] :
( finite_finite2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A4)))
=> ( pp(member(B,A3,A4))
=> finite_finite2(A,aa(B,set(A),B5,A3)) ) ) ).
% finite_UNION_then_finite
tff(fact_2482_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(D,fun(C,A)),G: fun(B,D),X2: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),aa(fun(D,fun(C,A)),fun(fun(B,D),fun(B,fun(C,A))),comp(D,fun(C,A),B),F),G)),X2) = aa(C,A,aa(D,fun(C,A),F,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),X2))),aa(product_prod(B,C),C,product_snd(B,C),X2)) ).
% case_prod_comp
tff(fact_2483_fn__snd__conv,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aTP_Lamp_iu(fun(B,C),fun(product_prod(A,B),C),F) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_iv(fun(B,C),fun(A,fun(B,C)),F)) ).
% fn_snd_conv
tff(fact_2484_snd__def,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,aa(fun(A,fun(B,B)),fun(product_prod(A,B),B),product_case_prod(A,B,B),aTP_Lamp_iw(A,fun(B,B))),Prod) ).
% snd_def
tff(fact_2485_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] : aTP_Lamp_ix(fun(A,C),fun(product_prod(A,B),C),F) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_iy(fun(A,C),fun(A,fun(B,C)),F)) ).
% fn_fst_conv
tff(fact_2486_fst__def,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_iz(A,fun(B,A))),Prod) ).
% fst_def
tff(fact_2487_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X7: set(A),A4: set(product_prod(A,B)),Y7: set(B),P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
( ( X7 = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4) )
=> ( ( Y7 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A4) )
=> ( ! [X3: A] :
( pp(member(A,X3,X7))
=> ! [Xa4: B] :
( pp(member(B,Xa4,Y7))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Xa4))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Xa4)) ) ) )
=> ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P))))
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q)))) ) ) ) ) ).
% Collect_split_mono_strong
tff(fact_2488_Some__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(A)] :
( ( A4 != bot_bot(set(A)) )
=> ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A4)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),A4)) ) ) ) ).
% Some_Sup
tff(fact_2489_Inf__option__def,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(option(A))] :
( ( pp(member(option(A),none(A),A4))
=> ( complete_Inf_Inf(option(A),A4) = none(A) ) )
& ( ~ pp(member(option(A),none(A),A4))
=> ( complete_Inf_Inf(option(A),A4) = aa(A,option(A),some(A),complete_Inf_Inf(A,these(A,A4))) ) ) ) ) ).
% Inf_option_def
tff(fact_2490_Some__INF,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A4: set(B)] : aa(A,option(A),some(A),complete_Inf_Inf(A,aa(set(B),set(A),image2(B,A,F),A4))) = complete_Inf_Inf(option(A),aa(set(B),set(option(A)),image2(B,option(A),aTP_Lamp_ja(fun(B,A),fun(B,option(A)),F)),A4)) ) ).
% Some_INF
tff(fact_2491_INF__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( complete_Inf(A)
=> ! [G: fun(B,A),F: fun(C,B),A4: set(C)] : complete_Inf_Inf(A,aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A4))) = complete_Inf_Inf(A,aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A4)) ) ).
% INF_image
tff(fact_2492_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( complete_Sup(A)
=> ! [G: fun(B,A),F: fun(C,B),A4: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A4)) ) ).
% SUP_image
tff(fact_2493_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S2: set(product_prod(B,A))] :
( pp(member(A,Y,aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S2)))
=> ~ ! [X3: B] : ~ pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y),S2)) ) ).
% in_snd_imageE
tff(fact_2494_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X2: A,S2: set(product_prod(A,B))] :
( pp(member(A,X2,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S2)))
=> ~ ! [Y4: B] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4),S2)) ) ).
% in_fst_imageE
tff(fact_2495_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [A4: set(A),F: fun(A,B)] :
( ( A4 != bot_bot(set(A)) )
=> ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A4))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image2(A,option(B),aTP_Lamp_jb(fun(A,B),fun(A,option(B)),F)),A4)) ) ) ) ).
% Some_SUP
tff(fact_2496_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,product_prod(A,C))] :
( finite_finite2(A,aa(set(B),set(A),image2(B,A,aa(fun(B,product_prod(A,C)),fun(B,A),aa(fun(product_prod(A,C),A),fun(fun(B,product_prod(A,C)),fun(B,A)),comp(product_prod(A,C),A,B),product_fst(A,C)),F)),top_top(set(B))))
=> ( finite_finite2(C,aa(set(B),set(C),image2(B,C,aa(fun(B,product_prod(A,C)),fun(B,C),aa(fun(product_prod(A,C),C),fun(fun(B,product_prod(A,C)),fun(B,C)),comp(product_prod(A,C),C,B),product_snd(A,C)),F)),top_top(set(B))))
=> finite_finite2(product_prod(A,C),aa(set(B),set(product_prod(A,C)),image2(B,product_prod(A,C),F),top_top(set(B)))) ) ) ).
% finite_range_prod
tff(fact_2497_UN__atMost__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).
% UN_atMost_UNIV
tff(fact_2498_UN__lessThan__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).
% UN_lessThan_UNIV
tff(fact_2499_Sup__option__def,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(option(A))] :
( ( ( ( A4 = bot_bot(set(option(A))) )
| ( A4 = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) )
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A4) = none(A) ) )
& ( ~ ( ( A4 = bot_bot(set(option(A))) )
| ( A4 = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) )
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),A4) = aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),these(A,A4))) ) ) ) ) ).
% Sup_option_def
tff(fact_2500_snd__image__mp,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(B,A)),B5: set(A),X2: B,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A4)),B5))
=> ( pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y),A4))
=> pp(member(A,Y,B5)) ) ) ).
% snd_image_mp
tff(fact_2501_fst__image__mp,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B5: set(A),X2: A,Y: B] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4)),B5))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),A4))
=> pp(member(A,X2,B5)) ) ) ).
% fst_image_mp
tff(fact_2502_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [B5: set(set(B)),G: fun(B,A)] :
( ! [X3: set(B)] :
( pp(member(set(B),X3,B5))
=> finite_finite2(B,X3) )
=> ( ! [A13: set(B)] :
( pp(member(set(B),A13,B5))
=> ! [A24: set(B)] :
( pp(member(set(B),A24,B5))
=> ( ( A13 != A24 )
=> ! [X3: B] :
( pp(member(B,X3,A13))
=> ( pp(member(B,X3,A24))
=> ( aa(B,A,G,X3) = one_one(A) ) ) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B5)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),B5) ) ) ) ) ).
% prod.Union_comp
tff(fact_2503_UN__UN__finite__eq,axiom,
! [A: $tType,A4: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_jc(fun(nat,set(A)),fun(nat,set(A)),A4)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat)))) ).
% UN_UN_finite_eq
tff(fact_2504_UN__le__add__shift,axiom,
! [A: $tType,M2: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_jd(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M2),K)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).
% UN_le_add_shift
tff(fact_2505_UN__le__add__shift__strict,axiom,
! [A: $tType,M2: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_jd(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M2),K)),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).
% UN_le_add_shift_strict
tff(fact_2506_rat__sgn__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P3)))),one_one(int)) ).
% rat_sgn_code
tff(fact_2507_bezw__non__0,axiom,
! [Y: nat,X2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
=> ( bezw(X2,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X2),Y))))) ) ) ).
% bezw_non_0
tff(fact_2508_UN__finite__subset,axiom,
! [A: $tType,A4: fun(nat,set(A)),C5: set(A)] :
( ! [N3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),C5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat))))),C5)) ) ).
% UN_finite_subset
tff(fact_2509_UN__finite2__eq,axiom,
! [A: $tType,A4: fun(nat,set(A)),B5: fun(nat,set(A)),K: nat] :
( ! [N3: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),K))))
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),top_top(set(nat)))) ) ) ).
% UN_finite2_eq
tff(fact_2510_bezw_Oelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X2,Xa) = Y )
=> ( ( ( Xa = zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
& ( ( Xa != zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X2),Xa))))) ) ) ) ) ).
% bezw.elims
tff(fact_2511_bezw_Osimps,axiom,
! [Y: nat,X2: nat] :
( ( ( Y = zero_zero(nat) )
=> ( bezw(X2,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
& ( ( Y != zero_zero(nat) )
=> ( bezw(X2,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X2),Y))))) ) ) ) ).
% bezw.simps
tff(fact_2512_max__ext__def,axiom,
! [A: $tType,X: set(product_prod(A,A))] : max_ext(A,X) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),max_extp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),X)))) ).
% max_ext_def
tff(fact_2513_UN__finite2__subset,axiom,
! [A: $tType,A4: fun(nat,set(A)),B5: fun(nat,set(A)),K: nat] :
( ! [N3: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),K))))))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B5),top_top(set(nat)))))) ) ).
% UN_finite2_subset
tff(fact_2514_bezw_Opelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod(int,int)] :
( ( bezw(X2,Xa) = Y )
=> ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Xa)))
=> ~ ( ( ( ( Xa = zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
& ( ( Xa != zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X2),Xa))))) ) ) )
=> ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Xa))) ) ) ) ).
% bezw.pelims
tff(fact_2515_snd__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),aa(fun(product_prod(A,A),A),fun(fun(A,product_prod(A,A)),fun(A,A)),comp(product_prod(A,A),A,A),product_snd(A,A)),aTP_Lamp_je(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% snd_diag_id
tff(fact_2516_fst__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),aa(fun(product_prod(A,A),A),fun(fun(A,product_prod(A,A)),fun(A,A)),comp(product_prod(A,A),A,A),product_fst(A,A)),aTP_Lamp_je(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% fst_diag_id
tff(fact_2517_size__prod__simp,axiom,
! [A: $tType,B: $tType,F: fun(A,nat),G: fun(B,nat),P3: product_prod(A,B)] : basic_BNF_size_prod(A,B,F,G,P3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,aa(product_prod(A,B),A,product_fst(A,B),P3))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ).
% size_prod_simp
tff(fact_2518_fst__diag__snd,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),aa(fun(product_prod(B,B),B),fun(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B)),comp(product_prod(B,B),B,product_prod(A,B)),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),aa(fun(B,product_prod(B,B)),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B))),comp(B,product_prod(B,B),product_prod(A,B)),aTP_Lamp_jf(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).
% fst_diag_snd
tff(fact_2519_snd__diag__fst,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),aa(fun(product_prod(A,A),A),fun(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A)),comp(product_prod(A,A),A,product_prod(A,B)),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A))),comp(A,product_prod(A,A),product_prod(A,B)),aTP_Lamp_je(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).
% snd_diag_fst
tff(fact_2520_None__in__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A4: set(option(A))] :
( pp(member(option(A),none(A),A4))
=> ( complete_Inf_Inf(option(A),A4) = none(A) ) ) ) ).
% None_in_Inf
tff(fact_2521_empty__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),bot_bot(set(option(A)))) = none(A) ) ) ).
% empty_Sup
tff(fact_2522_singleton__None__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ( aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),bot_bot(set(option(A))))) = none(A) ) ) ).
% singleton_None_Sup
tff(fact_2523_INF__INT__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S2: set(C),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),complete_Inf_Inf(fun(A,fun(B,bool)),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_jg(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R)),S2)),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),complete_Inf_Inf(set(product_prod(A,B)),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R),S2)))) ) ).
% INF_INT_eq2
tff(fact_2524_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),complete_Inf_Inf(fun(A,fun(B,bool)),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool)))),S2)),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),complete_Inf_Inf(set(product_prod(A,B)),S2))) ) ).
% INF_Int_eq2
tff(fact_2525_SUP__UN__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S2: set(C),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),image2(C,fun(A,fun(B,bool)),aTP_Lamp_jg(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R)),S2)),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R),S2)))) ) ).
% SUP_UN_eq2
tff(fact_2526_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),image2(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool)))),S2)),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S2))) ) ).
% SUP_Sup_eq2
tff(fact_2527_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType,S2: set(fun(A,fun(B,bool))),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),complete_Inf_Inf(fun(A,fun(B,bool)),S2),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),complete_Inf_Inf(set(product_prod(A,B)),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S2))))) ) ).
% Inf_INT_eq2
tff(fact_2528_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType,S2: set(fun(A,fun(B,bool))),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),S2),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool)),S2))))) ) ).
% Sup_SUP_eq2
tff(fact_2529_fst__diag__fst,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),aa(fun(product_prod(A,A),A),fun(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A)),comp(product_prod(A,A),A,product_prod(A,B)),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A))),comp(A,product_prod(A,A),product_prod(A,B)),aTP_Lamp_je(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).
% fst_diag_fst
tff(fact_2530_snd__diag__snd,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),aa(fun(product_prod(B,B),B),fun(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B)),comp(product_prod(B,B),B,product_prod(A,B)),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),aa(fun(B,product_prod(B,B)),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B))),comp(B,product_prod(B,B),product_prod(A,B)),aTP_Lamp_jf(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).
% snd_diag_snd
tff(fact_2531_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X2: A,Y: B,A3: product_prod(A,B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X2),Y))
=> ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3))) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_2532_normalize__def,axiom,
! [P3: product_prod(int,int)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)))
=> ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)))
=> ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int) )
=> ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ) )
& ( ( aa(product_prod(int,int),int,product_snd(int,int),P3) != zero_zero(int) )
=> ( normalize(P3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))))) ) ) ) ) ) ).
% normalize_def
tff(fact_2533_range__prod,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),F),top_top(set(C)))),product_Sigma(A,B,aa(set(C),set(A),image2(C,A,aa(fun(C,product_prod(A,B)),fun(C,A),aa(fun(product_prod(A,B),A),fun(fun(C,product_prod(A,B)),fun(C,A)),comp(product_prod(A,B),A,C),product_fst(A,B)),F)),top_top(set(C))),aTP_Lamp_jh(fun(C,product_prod(A,B)),fun(A,set(B)),F)))) ).
% range_prod
tff(fact_2534_fun_Oin__rel,axiom,
! [A: $tType,B: $tType,D: $tType,R4: fun(A,fun(B,bool)),A3: fun(D,A),B2: fun(D,B)] :
( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R4),A3),B2))
<=> ? [Z4: fun(D,product_prod(A,B))] :
( pp(member(fun(D,product_prod(A,B)),Z4,aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_ji(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R4))))
& ( aa(fun(D,product_prod(A,B)),fun(D,A),aa(fun(product_prod(A,B),A),fun(fun(D,product_prod(A,B)),fun(D,A)),comp(product_prod(A,B),A,D),product_fst(A,B)),Z4) = A3 )
& ( aa(fun(D,product_prod(A,B)),fun(D,B),aa(fun(product_prod(A,B),B),fun(fun(D,product_prod(A,B)),fun(D,B)),comp(product_prod(A,B),B,D),product_snd(A,B)),Z4) = B2 ) ) ) ).
% fun.in_rel
tff(fact_2535_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X2: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X2))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X2))) ).
% apfst_apsnd
tff(fact_2536_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),X2: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),X2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X2))),aa(C,B,F,aa(product_prod(D,C),C,product_snd(D,C),X2))) ).
% apsnd_apfst
tff(fact_2537_eq__snd__iff,axiom,
! [B: $tType,A: $tType,B2: A,P3: product_prod(B,A)] :
( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P3) )
<=> ? [A7: B] : P3 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B2) ) ).
% eq_snd_iff
tff(fact_2538_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set(A),B2: B,B5: fun(A,set(B))] :
( pp(member(A,A3,A4))
=> ( pp(member(B,B2,aa(A,set(B),B5,A3)))
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,A4,B5))) ) ) ).
% SigmaI
tff(fact_2539_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B5: fun(A,set(B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,A4,B5)))
<=> ( pp(member(A,A3,A4))
& pp(member(B,B2,aa(A,set(B),B5,A3))) ) ) ).
% mem_Sigma_iff
tff(fact_2540_gcd__add1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).
% gcd_add1
tff(fact_2541_gcd__add2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).
% gcd_add2
tff(fact_2542_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A3) = one_one(A) ) ).
% gcd.bottom_left_bottom
tff(fact_2543_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),one_one(A)) = one_one(A) ) ).
% gcd.bottom_right_bottom
tff(fact_2544_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(C,A),X2: C,Y: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X2),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X2)),Y) ).
% apfst_conv
tff(fact_2545_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,A),X2: product_prod(C,B),G: fun(C,A)] :
( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),X2) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G),X2) )
<=> ( aa(C,A,F,aa(product_prod(C,B),C,product_fst(C,B),X2)) = aa(C,A,G,aa(product_prod(C,B),C,product_fst(C,B),X2)) ) ) ).
% apfst_eq_conv
tff(fact_2546_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,A),X2: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),X2)) = aa(C,A,F,aa(product_prod(C,B),C,product_fst(C,B),X2)) ).
% fst_apfst
tff(fact_2547_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,B),X2: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F),X2)) = aa(product_prod(C,A),A,product_snd(C,A),X2) ).
% snd_apfst
tff(fact_2548_Collect__case__prod,axiom,
! [A: $tType,B: $tType,P: fun(A,bool),Q: fun(B,bool)] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_jj(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),P),Q))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P),aTP_Lamp_jk(fun(B,bool),fun(A,set(B)),Q)) ).
% Collect_case_prod
tff(fact_2549_is__unit__gcd__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),one_one(A)))
<=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).
% is_unit_gcd_iff
tff(fact_2550_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B5: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B5) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty1
tff(fact_2551_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A4: set(A)] : product_Sigma(A,B,A4,aTP_Lamp_jl(A,set(B))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty2
tff(fact_2552_Times__empty,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ( product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)) = bot_bot(set(product_prod(A,B))) )
<=> ( ( A4 = bot_bot(set(A)) )
| ( B5 = bot_bot(set(B)) ) ) ) ).
% Times_empty
tff(fact_2553_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A4: set(A),X7: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),X7))),product_Sigma(A,B,A4,aTP_Lamp_jn(A,set(B)))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_UNIV_cancel
tff(fact_2554_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A4: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_jn(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A4),aTP_Lamp_jn(A,set(B))) ).
% Compl_Times_UNIV2
tff(fact_2555_Compl__Times__UNIV1,axiom,
! [A: $tType,B: $tType,A4: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_jm(set(B),fun(A,set(B)),A4))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_jo(set(B),fun(A,set(B)),A4)) ).
% Compl_Times_UNIV1
tff(fact_2556_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),aa(fun(product_prod(C,B),B),fun(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B)),comp(product_prod(C,B),B,product_prod(A,B)),product_snd(C,B)),product_apfst(A,C,B,F)) = product_snd(A,B) ).
% snd_comp_apfst
tff(fact_2557_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_jn(A,set(B))) = top_top(set(product_prod(A,B))) ).
% UNIV_Times_UNIV
tff(fact_2558_apfst__id,axiom,
! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).
% apfst_id
tff(fact_2559_pairself__image__cart,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A4: set(B),B5: set(B)] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),product_Sigma(B,B,A4,aTP_Lamp_jp(set(B),fun(B,set(B)),B5))) = product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),A4),aa(set(B),fun(A,set(A)),aTP_Lamp_jq(fun(B,A),fun(set(B),fun(A,set(A))),F),B5)) ).
% pairself_image_cart
tff(fact_2560_fst__image__times,axiom,
! [B: $tType,A: $tType,B5: set(B),A4: set(A)] :
( ( ( B5 = bot_bot(set(B)) )
=> ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5))) = bot_bot(set(A)) ) )
& ( ( B5 != bot_bot(set(B)) )
=> ( aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5))) = A4 ) ) ) ).
% fst_image_times
tff(fact_2561_snd__image__times,axiom,
! [B: $tType,A: $tType,A4: set(B),B5: set(A)] :
( ( ( A4 = bot_bot(set(B)) )
=> ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,aTP_Lamp_jr(set(A),fun(B,set(A)),B5))) = bot_bot(set(A)) ) )
& ( ( A4 != bot_bot(set(B)) )
=> ( aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,aTP_Lamp_jr(set(A),fun(B,set(A)),B5))) = B5 ) ) ) ).
% snd_image_times
tff(fact_2562_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),aa(fun(product_prod(C,B),C),fun(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C)),comp(product_prod(C,B),C,product_prod(A,B)),product_fst(C,B)),product_apfst(A,C,B,F)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)) ).
% fst_comp_apfst
tff(fact_2563_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G2: $tType,F4: $tType,Rb2: fun(A,fun(F4,bool)),Sd: fun(B,fun(G2,bool))] : pp(aa(fun(fun(F4,G2),fun(fun(D,F4),fun(D,G2))),bool,aa(fun(fun(A,B),fun(fun(D,A),fun(D,B))),fun(fun(fun(F4,G2),fun(fun(D,F4),fun(D,G2))),bool),bNF_rel_fun(fun(A,B),fun(F4,G2),fun(fun(D,A),fun(D,B)),fun(fun(D,F4),fun(D,G2)),bNF_rel_fun(A,F4,B,G2,Rb2,Sd),bNF_rel_fun(fun(D,A),fun(D,F4),fun(D,B),fun(D,G2),bNF_rel_fun(D,D,A,F4,fequal(D),Rb2),bNF_rel_fun(D,D,B,G2,fequal(D),Sd))),comp(A,B,D)),comp(F4,G2,D))) ).
% fun.map_transfer
tff(fact_2564_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,B5: fun(A,fun(C,bool)),C5: fun(B,fun(D,bool)),A4: fun(E,fun(F4,bool))] : pp(aa(fun(fun(C,D),fun(fun(F4,C),fun(F4,D))),bool,aa(fun(fun(A,B),fun(fun(E,A),fun(E,B))),fun(fun(fun(C,D),fun(fun(F4,C),fun(F4,D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,A),fun(E,B)),fun(fun(F4,C),fun(F4,D)),bNF_rel_fun(A,C,B,D,B5,C5),bNF_rel_fun(fun(E,A),fun(F4,C),fun(E,B),fun(F4,D),bNF_rel_fun(E,F4,A,C,A4,B5),bNF_rel_fun(E,F4,B,D,A4,C5))),comp(A,B,E)),comp(C,D,F4))) ).
% comp_transfer
tff(fact_2565_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( power(B)
& power(A) )
=> ! [R4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R4,bNF_rel_fun(A,B,A,B,R4,R4)),times_times(A)),times_times(B)))
=> pp(aa(fun(B,fun(nat,B)),bool,aa(fun(A,fun(nat,A)),fun(fun(B,fun(nat,B)),bool),bNF_rel_fun(A,B,fun(nat,A),fun(nat,B),R4,bNF_rel_fun(nat,nat,A,B,fequal(nat),R4)),power_power(A)),power_power(B))) ) ) ) ).
% power_transfer
tff(fact_2566_times__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),times_times(int)),times_times(int))) ).
% times_integer.rsp
tff(fact_2567_times__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),times_times(nat)),times_times(nat))) ).
% times_natural.rsp
tff(fact_2568_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C5: set(A),A4: set(B),B5: set(B)] :
( pp(member(A,X2,C5))
=> ( ( product_Sigma(B,A,A4,aTP_Lamp_jr(set(A),fun(B,set(A)),C5)) = product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),C5)) )
<=> ( A4 = B5 ) ) ) ).
% Times_eq_cancel2
tff(fact_2569_Sigma__cong,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(A),C5: fun(A,set(B)),D4: fun(A,set(B))] :
( ( A4 = B5 )
=> ( ! [X3: A] :
( pp(member(A,X3,B5))
=> ( aa(A,set(B),C5,X3) = aa(A,set(B),D4,X3) ) )
=> ( product_Sigma(A,B,A4,C5) = product_Sigma(A,B,B5,D4) ) ) ) ).
% Sigma_cong
tff(fact_2570_divide__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),divide_divide(nat)),divide_divide(nat))) ).
% divide_natural.rsp
tff(fact_2571_divide__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),divide_divide(int)),divide_divide(int))) ).
% divide_integer.rsp
tff(fact_2572_or__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se1065995026697491101ons_or(nat)),bit_se1065995026697491101ons_or(nat))) ).
% or_natural.rsp
tff(fact_2573_or__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se1065995026697491101ons_or(int)),bit_se1065995026697491101ons_or(int))) ).
% or_integer.rsp
tff(fact_2574_set__bit__integer_Orsp,axiom,
pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(int))) ).
% set_bit_integer.rsp
tff(fact_2575_set__bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5668285175392031749et_bit(nat)),bit_se5668285175392031749et_bit(nat))) ).
% set_bit_natural.rsp
tff(fact_2576_xor__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5824344971392196577ns_xor(int)),bit_se5824344971392196577ns_xor(int))) ).
% xor_integer.rsp
tff(fact_2577_xor__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5824344971392196577ns_xor(nat)),bit_se5824344971392196577ns_xor(nat))) ).
% xor_natural.rsp
tff(fact_2578_and__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5824344872417868541ns_and(int)),bit_se5824344872417868541ns_and(int))) ).
% and_integer.rsp
tff(fact_2579_and__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5824344872417868541ns_and(nat)),bit_se5824344872417868541ns_and(nat))) ).
% and_natural.rsp
tff(fact_2580_modulo__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),modulo_modulo(int)),modulo_modulo(int))) ).
% modulo_integer.rsp
tff(fact_2581_modulo__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),modulo_modulo(nat)),modulo_modulo(nat))) ).
% modulo_natural.rsp
tff(fact_2582_less__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less(int)),ord_less(int))) ).
% less_integer.rsp
tff(fact_2583_less__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less(nat)),ord_less(nat))) ).
% less_natural.rsp
tff(fact_2584_unset__bit__integer_Orsp,axiom,
pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(int))) ).
% unset_bit_integer.rsp
tff(fact_2585_take__bit__integer_Orsp,axiom,
pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2584673776208193580ke_bit(int)),bit_se2584673776208193580ke_bit(int))) ).
% take_bit_integer.rsp
tff(fact_2586_flip__bit__integer_Orsp,axiom,
pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(int))) ).
% flip_bit_integer.rsp
tff(fact_2587_flip__bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se8732182000553998342ip_bit(nat)),bit_se8732182000553998342ip_bit(nat))) ).
% flip_bit_natural.rsp
tff(fact_2588_push__bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se4730199178511100633sh_bit(nat)),bit_se4730199178511100633sh_bit(nat))) ).
% push_bit_natural.rsp
tff(fact_2589_push__bit__integer_Orsp,axiom,
pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se4730199178511100633sh_bit(int)),bit_se4730199178511100633sh_bit(int))) ).
% push_bit_integer.rsp
tff(fact_2590_plus__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),plus_plus(int)),plus_plus(int))) ).
% plus_integer.rsp
tff(fact_2591_plus__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),plus_plus(nat)),plus_plus(nat))) ).
% plus_natural.rsp
tff(fact_2592_drop__bit__integer_Orsp,axiom,
pp(aa(fun(nat,fun(int,int)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se4197421643247451524op_bit(int)),bit_se4197421643247451524op_bit(int))) ).
% drop_bit_integer.rsp
tff(fact_2593_minus__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),minus_minus(nat)),minus_minus(nat))) ).
% minus_natural.rsp
tff(fact_2594_minus__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),minus_minus(int)),minus_minus(int))) ).
% minus_integer.rsp
tff(fact_2595_less__eq__integer_Orsp,axiom,
pp(aa(fun(int,fun(int,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(int,int,fun(int,bool),fun(int,bool),fequal(int),bNF_rel_fun(int,int,bool,bool,fequal(int),fequal(bool))),ord_less_eq(int)),ord_less_eq(int))) ).
% less_eq_integer.rsp
tff(fact_2596_less__eq__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),bNF_rel_fun(nat,nat,bool,bool,fequal(nat),fequal(bool))),ord_less_eq(nat)),ord_less_eq(nat))) ).
% less_eq_natural.rsp
tff(fact_2597_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod(A,B),A4: set(A),B5: fun(A,set(B))] :
( pp(member(product_prod(A,B),C2,product_Sigma(A,B,A4,B5)))
=> ~ ! [X3: A] :
( pp(member(A,X3,A4))
=> ! [Y4: B] :
( pp(member(B,Y4,aa(A,set(B),B5,X3)))
=> ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4) ) ) ) ) ).
% SigmaE
tff(fact_2598_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B5: fun(A,set(B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,A4,B5)))
=> pp(member(A,A3,A4)) ) ).
% SigmaD1
tff(fact_2599_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B5: fun(A,set(B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,A4,B5)))
=> pp(member(B,B2,aa(A,set(B),B5,A3))) ) ).
% SigmaD2
tff(fact_2600_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set(A),B5: fun(A,set(B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),product_Sigma(A,B,A4,B5)))
=> ~ ( pp(member(A,A3,A4))
=> ~ pp(member(B,B2,aa(A,set(B),B5,A3))) ) ) ).
% SigmaE2
tff(fact_2601_Suc_Orsp,axiom,
pp(aa(fun(nat,nat),bool,aa(fun(nat,nat),fun(fun(nat,nat),bool),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),suc),suc)) ).
% Suc.rsp
tff(fact_2602_sub_Orsp,axiom,
pp(aa(fun(num,fun(num,int)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),bool),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_hn(num,fun(num,int))),aTP_Lamp_hn(num,fun(num,int)))) ).
% sub.rsp
tff(fact_2603_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& semiring_numeral(B)
& monoid_add(A)
& semiring_numeral(A) )
=> ! [R4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R4,bNF_rel_fun(A,B,A,B,R4,R4)),plus_plus(A)),plus_plus(B)))
=> pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R4),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).
% transfer_rule_numeral
tff(fact_2604_uminus__integer_Orsp,axiom,
pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),uminus_uminus(int)),uminus_uminus(int))) ).
% uminus_integer.rsp
tff(fact_2605_integer__of__natural_Orsp,axiom,
pp(aa(fun(nat,int),bool,aa(fun(nat,int),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),semiring_1_of_nat(int)),semiring_1_of_nat(int))) ).
% integer_of_natural.rsp
tff(fact_2606_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I4: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I4),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I4,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Diff_distrib1
tff(fact_2607_natural__of__integer_Orsp,axiom,
pp(aa(fun(int,nat),bool,aa(fun(int,nat),fun(fun(int,nat),bool),bNF_rel_fun(int,int,nat,nat,fequal(int),fequal(nat)),nat2),nat2)) ).
% natural_of_integer.rsp
tff(fact_2608_abs__integer_Orsp,axiom,
pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),abs_abs(int)),abs_abs(int))) ).
% abs_integer.rsp
tff(fact_2609_bit__integer_Orsp,axiom,
pp(aa(fun(int,fun(nat,bool)),bool,aa(fun(int,fun(nat,bool)),fun(fun(int,fun(nat,bool)),bool),bNF_rel_fun(int,int,fun(nat,bool),fun(nat,bool),fequal(int),fequal(fun(nat,bool))),bit_se5641148757651400278ts_bit(int)),bit_se5641148757651400278ts_bit(int))) ).
% bit_integer.rsp
tff(fact_2610_not__integer_Orsp,axiom,
pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),bit_ri4277139882892585799ns_not(int)),bit_ri4277139882892585799ns_not(int))) ).
% not_integer.rsp
tff(fact_2611_sgn__integer_Orsp,axiom,
pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),sgn_sgn(int)),sgn_sgn(int))) ).
% sgn_integer.rsp
tff(fact_2612_mask__integer_Orsp,axiom,
pp(aa(fun(nat,int),bool,aa(fun(nat,int),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),bit_se2239418461657761734s_mask(int)),bit_se2239418461657761734s_mask(int))) ).
% mask_integer.rsp
tff(fact_2613_mask__natural_Orsp,axiom,
pp(aa(fun(nat,nat),bool,aa(fun(nat,nat),fun(fun(nat,nat),bool),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),bit_se2239418461657761734s_mask(nat)),bit_se2239418461657761734s_mask(nat))) ).
% mask_natural.rsp
tff(fact_2614_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ring_1(B)
& ring_1(A) )
=> ! [R4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R4,bNF_rel_fun(A,B,A,B,R4,R4)),plus_plus(A)),plus_plus(B)))
=> ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R4,R4),uminus_uminus(A)),uminus_uminus(B)))
=> pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),bNF_rel_fun(int,int,A,B,fequal(int),R4),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).
% transfer_rule_of_int
tff(fact_2615_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X2: fun(D,A),Y: fun(D,B)] :
( pp(aa(fun(D,B),bool,aa(fun(D,C),fun(fun(D,B),bool),bNF_rel_fun(D,D,C,B,fequal(D),Sb),aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),I),X2)),Y))
<=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_js(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X2),Y)) ) ).
% fun.rel_map(1)
tff(fact_2616_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: fun(A,fun(C,bool)),X2: fun(D,A),G: fun(B,C),Y: fun(D,B)] :
( pp(aa(fun(D,C),bool,aa(fun(D,A),fun(fun(D,C),bool),bNF_rel_fun(D,D,A,C,fequal(D),Sa),X2),aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),G),Y)))
<=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_jt(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X2),Y)) ) ).
% fun.rel_map(2)
tff(fact_2617_times__eq__iff,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B),C5: set(A),D4: set(B)] :
( ( product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)) = product_Sigma(A,B,C5,aTP_Lamp_jm(set(B),fun(A,set(B)),D4)) )
<=> ( ( ( A4 = C5 )
& ( B5 = D4 ) )
| ( ( ( A4 = bot_bot(set(A)) )
| ( B5 = bot_bot(set(B)) ) )
& ( ( C5 = bot_bot(set(A)) )
| ( D4 = bot_bot(set(B)) ) ) ) ) ) ).
% times_eq_iff
tff(fact_2618_dup_Orsp,axiom,
pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_hf(int,int)),aTP_Lamp_hf(int,int))) ).
% dup.rsp
tff(fact_2619_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ju(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I4,A4)),product_Sigma(A,B,I4,B5)) ).
% Sigma_Diff_distrib2
tff(fact_2620_Times__Diff__distrib1,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5),aTP_Lamp_jm(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))) ).
% Times_Diff_distrib1
tff(fact_2621_Sigma__Union,axiom,
! [B: $tType,A: $tType,X7: set(set(A)),B5: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X7),B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),image2(set(A),set(product_prod(A,B)),aTP_Lamp_jv(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B5)),X7)) ).
% Sigma_Union
tff(fact_2622_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),Q: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_jw(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P),aTP_Lamp_jx(fun(A,fun(B,bool)),fun(A,set(B)),Q)) ).
% Collect_case_prod_Sigma
tff(fact_2623_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,C),X2: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G),X2)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),aa(fun(C,A),fun(fun(D,C),fun(D,A)),comp(C,A,D),F),G)),X2) ).
% apfst_compose
tff(fact_2624_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),P3: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),P3)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F),P3)) ).
% apsnd_apfst_commute
tff(fact_2625_gcd__add__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [M: A,K: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).
% gcd_add_mult
tff(fact_2626_Sigma__mono,axiom,
! [B: $tType,A: $tType,A4: set(A),C5: set(A),B5: fun(A,set(B)),D4: fun(A,set(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C5))
=> ( ! [X3: A] :
( pp(member(A,X3,A4))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B5,X3)),aa(A,set(B),D4,X3))) )
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,B5)),product_Sigma(A,B,C5,D4))) ) ) ).
% Sigma_mono
tff(fact_2627_bezout__int,axiom,
! [X2: int,Y: int] :
? [U2: int,V4: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U2),X2)),aa(int,int,aa(int,fun(int,int),times_times(int),V4),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X2),Y) ).
% bezout_int
tff(fact_2628_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C5: set(A),A4: set(B),B5: set(B)] :
( pp(member(A,X2,C5))
=> ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A4,aTP_Lamp_jr(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),C5))))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B5)) ) ) ).
% Times_subset_cancel2
tff(fact_2629_UN__Times__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,E4: fun(C,set(A)),F5: fun(D,set(B)),A4: set(C),B5: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),image2(product_prod(C,D),set(product_prod(A,B)),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_jz(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E4),F5))),product_Sigma(C,D,A4,aTP_Lamp_ka(set(D),fun(C,set(D)),B5)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),E4),A4)),aa(set(D),fun(A,set(B)),aTP_Lamp_kb(fun(D,set(B)),fun(set(D),fun(A,set(B))),F5),B5)) ).
% UN_Times_distrib
tff(fact_2630_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),A4: set(A),B5: set(B)] :
( pp(member(product_prod(A,B),X2,product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5))))
<=> ( pp(member(A,aa(product_prod(A,B),A,product_fst(A,B),X2),A4))
& pp(member(B,aa(product_prod(A,B),B,product_snd(A,B),X2),B5)) ) ) ).
% mem_Times_iff
tff(fact_2631_in__prod__fst__sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),A4: set(A),B5: set(B)] :
( pp(member(A,aa(product_prod(A,B),A,product_fst(A,B),X2),A4))
=> ( pp(member(B,aa(product_prod(A,B),B,product_snd(A,B),X2),B5))
=> pp(member(product_prod(A,B),X2,product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))) ) ) ).
% in_prod_fst_sndI
tff(fact_2632_swap__product,axiom,
! [A: $tType,B: $tType,A4: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A4,aTP_Lamp_jr(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),A4)) ).
% swap_product
tff(fact_2633_Sigma__empty__iff,axiom,
! [A: $tType,B: $tType,I4: set(A),X7: fun(A,set(B))] :
( ( product_Sigma(A,B,I4,X7) = bot_bot(set(product_prod(A,B))) )
<=> ! [X4: A] :
( pp(member(A,X4,I4))
=> ( aa(A,set(B),X7,X4) = bot_bot(set(B)) ) ) ) ).
% Sigma_empty_iff
tff(fact_2634_gcd__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_mult_unit2
tff(fact_2635_gcd__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_mult_unit1
tff(fact_2636_gcd__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_div_unit1
tff(fact_2637_gcd__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).
% gcd_div_unit2
tff(fact_2638_less__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).
% less_int.transfer
tff(fact_2639_less__eq__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).
% less_eq_int.transfer
tff(fact_2640_plus__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).
% plus_int.transfer
tff(fact_2641_minus__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ct(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).
% minus_int.transfer
tff(fact_2642_times__subset__iff,axiom,
! [A: $tType,B: $tType,A4: set(A),C5: set(B),B5: set(A),D4: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),D4))))
<=> ( ( A4 = bot_bot(set(A)) )
| ( C5 = bot_bot(set(B)) )
| ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),D4)) ) ) ) ).
% times_subset_iff
tff(fact_2643_image__paired__Times,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A4: set(C),B5: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_kc(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),product_Sigma(C,D,A4,aTP_Lamp_ka(set(D),fun(C,set(D)),B5))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A4),aa(set(D),fun(A,set(B)),aTP_Lamp_kd(fun(D,B),fun(set(D),fun(A,set(B))),G),B5)) ).
% image_paired_Times
tff(fact_2644_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,B5)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_ke(set(A),fun(fun(A,set(B)),fun(A,bool)),A4),B5)) ).
% fst_image_Sigma
tff(fact_2645_Sigma__def,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: fun(A,set(B))] : product_Sigma(A,B,A4,B5) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),image2(A,set(product_prod(A,B)),aTP_Lamp_kg(fun(A,set(B)),fun(A,set(product_prod(A,B))),B5)),A4)) ).
% Sigma_def
tff(fact_2646_times__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).
% times_int.transfer
tff(fact_2647_num__of__integer_Orsp,axiom,
pp(aa(fun(int,num),bool,aa(fun(int,num),fun(fun(int,num),bool),bNF_rel_fun(int,int,num,num,fequal(int),fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2))) ).
% num_of_integer.rsp
tff(fact_2648_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A4: set(B),B5: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,B5)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B5),A4)) ).
% snd_image_Sigma
tff(fact_2649_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A4),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4),aTP_Lamp_kh(set(product_prod(A,B)),fun(A,set(B)),A4)))) ).
% subset_fst_snd
tff(fact_2650_int__transfer,axiom,
pp(aa(fun(nat,int),bool,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),bool),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_ki(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).
% int_transfer
tff(fact_2651_uminus__int_Otransfer,axiom,
pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).
% uminus_int.transfer
tff(fact_2652_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P3: product_prod(A,B)] :
( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P3) )
<=> ? [B7: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B7) ) ).
% eq_fst_iff
tff(fact_2653_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& semiring_1(A) )
=> ! [R4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R4,bNF_rel_fun(A,B,A,B,R4,R4)),plus_plus(A)),plus_plus(B)))
=> pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),bNF_rel_fun(nat,nat,A,B,fequal(nat),R4),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).
% transfer_rule_of_nat
tff(fact_2654_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( zero_neq_one(B)
& zero_neq_one(A) )
=> ! [R4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,one_one(A)),one_one(B)))
=> pp(aa(fun(bool,B),bool,aa(fun(bool,A),fun(fun(bool,B),bool),bNF_rel_fun(bool,bool,A,B,fequal(bool),R4),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).
% transfer_rule_of_bool
tff(fact_2655_plus__rat_Otransfer,axiom,
pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_kj(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).
% plus_rat.transfer
tff(fact_2656_inverse__rat_Otransfer,axiom,
pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_kk(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).
% inverse_rat.transfer
tff(fact_2657_Id__on__def,axiom,
! [A: $tType,A4: set(A)] : id_on(A,A4) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_kl(A,set(product_prod(A,A)))),A4)) ).
% Id_on_def
tff(fact_2658_times__rat_Otransfer,axiom,
pp(aa(fun(rat,fun(rat,rat)),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(rat,fun(rat,rat)),bool),bNF_rel_fun(product_prod(int,int),rat,fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),pcr_rat,bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat)),aTP_Lamp_km(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).
% times_rat.transfer
tff(fact_2659_gcd__1__nat,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),one_one(nat)) = one_one(nat) ).
% gcd_1_nat
tff(fact_2660_Id__onI,axiom,
! [A: $tType,A3: A,A4: set(A)] :
( pp(member(A,A3,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),id_on(A,A4))) ) ).
% Id_onI
tff(fact_2661_unset__bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se2638667681897837118et_bit(nat)),bit_se2638667681897837118et_bit(nat))) ).
% unset_bit_natural.rsp
tff(fact_2662_take__bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se2584673776208193580ke_bit(nat)),bit_se2584673776208193580ke_bit(nat))) ).
% take_bit_natural.rsp
tff(fact_2663_bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(nat,fun(nat,bool)),bool),bNF_rel_fun(nat,nat,fun(nat,bool),fun(nat,bool),fequal(nat),fequal(fun(nat,bool))),bit_se5641148757651400278ts_bit(nat)),bit_se5641148757651400278ts_bit(nat))) ).
% bit_natural.rsp
tff(fact_2664_drop__bit__natural_Orsp,axiom,
pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se4197421643247451524op_bit(nat)),bit_se4197421643247451524op_bit(nat))) ).
% drop_bit_natural.rsp
tff(fact_2665_Id__on__iff,axiom,
! [A: $tType,X2: A,Y: A,A4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),id_on(A,A4)))
<=> ( ( X2 = Y )
& pp(member(A,X2,A4)) ) ) ).
% Id_on_iff
tff(fact_2666_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A4: set(A)] :
( ( A3 = B2 )
=> ( pp(member(A,A3,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id_on(A,A4))) ) ) ).
% Id_on_eqI
tff(fact_2667_Id__onE,axiom,
! [A: $tType,C2: product_prod(A,A),A4: set(A)] :
( pp(member(product_prod(A,A),C2,id_on(A,A4)))
=> ~ ! [X3: A] :
( pp(member(A,X3,A4))
=> ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ) ) ).
% Id_onE
tff(fact_2668_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero(nat) )
=> ? [X3: nat,Y4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).
% bezout_nat
tff(fact_2669_one__rat_Otransfer,axiom,
pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),one_one(rat))) ).
% one_rat.transfer
tff(fact_2670_zero__rat_Otransfer,axiom,
pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),zero_zero(rat))) ).
% zero_rat.transfer
tff(fact_2671_bezw__aux,axiom,
! [X2: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X2),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(X2,Y))),aa(nat,int,semiring_1_of_nat(int),X2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X2,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).
% bezw_aux
tff(fact_2672_uminus__rat_Otransfer,axiom,
pp(aa(fun(rat,rat),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(rat,rat),bool),bNF_rel_fun(product_prod(int,int),rat,product_prod(int,int),rat,pcr_rat,pcr_rat),aTP_Lamp_kn(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).
% uminus_rat.transfer
tff(fact_2673_gcd__nat_Opelims,axiom,
! [X2: nat,Xa: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X2),Xa) = Y )
=> ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Xa)))
=> ~ ( ( ( ( Xa = zero_zero(nat) )
=> ( Y = X2 ) )
& ( ( Xa != zero_zero(nat) )
=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X2),Xa)) ) ) )
=> ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Xa))) ) ) ) ).
% gcd_nat.pelims
tff(fact_2674_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F4: $tType,D: $tType,C: $tType,R23: fun(A,fun(C,bool)),R32: fun(B,fun(D,bool)),R12: fun(E,fun(F4,bool))] : pp(aa(fun(fun(C,D),fun(fun(F4,C),fun(F4,D))),bool,aa(fun(fun(A,B),fun(fun(E,A),fun(E,B))),fun(fun(fun(C,D),fun(fun(F4,C),fun(F4,D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,A),fun(E,B)),fun(fun(F4,C),fun(F4,D)),bNF_rel_fun(A,C,B,D,R23,R32),bNF_rel_fun(fun(E,A),fun(F4,C),fun(E,B),fun(F4,D),bNF_rel_fun(E,F4,A,C,R12,R23),bNF_rel_fun(E,F4,B,D,R12,R32))),comp(A,B,E)),comp(C,D,F4))) ).
% o_rsp(1)
tff(fact_2675_o__rsp_I2_J,axiom,
! [E: $tType,F4: $tType,H6: $tType,G2: $tType,R12: fun(E,fun(F4,bool))] : pp(aa(fun(fun(G2,H6),fun(fun(F4,G2),fun(F4,H6))),bool,aa(fun(fun(G2,H6),fun(fun(E,G2),fun(E,H6))),fun(fun(fun(G2,H6),fun(fun(F4,G2),fun(F4,H6))),bool),bNF_rel_fun(fun(G2,H6),fun(G2,H6),fun(fun(E,G2),fun(E,H6)),fun(fun(F4,G2),fun(F4,H6)),fequal(fun(G2,H6)),bNF_rel_fun(fun(E,G2),fun(F4,G2),fun(E,H6),fun(F4,H6),bNF_rel_fun(E,F4,G2,G2,R12,fequal(G2)),bNF_rel_fun(E,F4,H6,H6,R12,fequal(H6)))),comp(G2,H6,E)),comp(G2,H6,F4))) ).
% o_rsp(2)
tff(fact_2676_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q3: product_prod(A,B),F: fun(C,A),P3: product_prod(C,B)] :
( ( Q3 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),P3) )
=> ~ ! [X3: C,Y4: B] :
( ( P3 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X3),Y4) )
=> ( Q3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X3)),Y4) ) ) ) ).
% apfst_convE
tff(fact_2677_UN__le__eq__Un0,axiom,
! [A: $tType,M2: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M2,zero_zero(nat))) ).
% UN_le_eq_Un0
tff(fact_2678_times__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% times_int.rsp
tff(fact_2679_sup__Some,axiom,
! [A: $tType] :
( sup(A)
=> ! [X2: A,Y: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),aa(A,option(A),some(A),X2)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y)) ) ).
% sup_Some
tff(fact_2680_intrel__iff,axiom,
! [X2: nat,Y: nat,U: nat,V: nat] :
( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V)))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).
% intrel_iff
tff(fact_2681_insert__Times__insert,axiom,
! [A: $tType,B: $tType,A3: A,A4: set(A),B2: B,B5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert2(A,A3),A4),aa(set(B),fun(A,set(B)),aTP_Lamp_ko(B,fun(set(B),fun(A,set(B))),B2),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A4,aa(set(B),fun(A,set(B)),aTP_Lamp_ko(B,fun(set(B),fun(A,set(B))),B2),B5))),product_Sigma(A,B,aa(set(A),set(A),insert2(A,A3),A4),aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))) ).
% insert_Times_insert
tff(fact_2682_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I4: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I4),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I4,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Un_distrib1
tff(fact_2683_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_kp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I4,A4)),product_Sigma(A,B,I4,B5)) ).
% Sigma_Un_distrib2
tff(fact_2684_Times__Un__distrib1,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5),aTP_Lamp_jm(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))) ).
% Times_Un_distrib1
tff(fact_2685_set__diff__diff__left,axiom,
! [A: $tType,A4: set(A),B5: set(A),C5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C5)) ).
% set_diff_diff_left
tff(fact_2686_sup__option__def,axiom,
! [A: $tType] :
( sup(A)
=> ! [X2: option(A),Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X2),Y) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),Y),aa(option(A),fun(A,option(A)),aTP_Lamp_kr(option(A),fun(option(A),fun(A,option(A))),X2),Y)),X2) ) ).
% sup_option_def
tff(fact_2687_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
tff(fact_2688_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(3)
tff(fact_2689_max__ext__additive,axiom,
! [A: $tType,A4: set(A),B5: set(A),R4: set(product_prod(A,A)),C5: set(A),D4: set(A)] :
( pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A4),B5),max_ext(A,R4)))
=> ( pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),C5),D4),max_ext(A,R4)))
=> pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),D4)),max_ext(A,R4))) ) ) ).
% max_ext_additive
tff(fact_2690_zero__int_Orsp,axiom,
pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),zero_zero(nat)))) ).
% zero_int.rsp
tff(fact_2691_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(7)
tff(fact_2692_card__Un__le,axiom,
! [A: $tType,A4: set(A),B5: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(A,B5)))) ).
% card_Un_le
tff(fact_2693_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).
% ivl_disj_un_one(2)
tff(fact_2694_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).
% atLeastLessThan_add_Un
tff(fact_2695_uminus__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat))))),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat)))))) ).
% uminus_int.rsp
tff(fact_2696_one__int_Orsp,axiom,
pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),one_one(nat)),zero_zero(nat)))) ).
% one_int.rsp
tff(fact_2697_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
tff(fact_2698_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(1)
tff(fact_2699_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).
% ivl_disj_un_singleton(2)
tff(fact_2700_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).
% ivl_disj_un_one(4)
tff(fact_2701_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).
% ivl_disj_un_one(1)
tff(fact_2702_intrel__def,axiom,
intrel = aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kt(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).
% intrel_def
tff(fact_2703_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(6)
tff(fact_2704_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_2705_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(4)
tff(fact_2706_less__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).
% less_int.rsp
tff(fact_2707_less__eq__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).
% less_eq_int.rsp
tff(fact_2708_plus__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% plus_int.rsp
tff(fact_2709_minus__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ct(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ct(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% minus_int.rsp
tff(fact_2710_max__weak__def,axiom,
fun_max_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),max_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).
% max_weak_def
tff(fact_2711_min__weak__def,axiom,
fun_min_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),min_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).
% min_weak_def
tff(fact_2712_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set(A),X2: B,A4: set(product_prod(B,A))] :
( finite_finite2(A,B5)
=> ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image2(A,set(product_prod(B,A)),aTP_Lamp_ku(B,fun(A,set(product_prod(B,A))),X2)),B5))),A4) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_kv(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X2),A4,B5) ) ) ).
% fold_union_pair
tff(fact_2713_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R)) ).
% Field_insert
tff(fact_2714_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod(A,B),S2: set(product_prod(A,B))] :
( ~ pp(member(A,aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S2)))
=> ( set_to_map(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),Kv),S2)) = fun_upd(A,option(B),set_to_map(A,B,S2),aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),Kv))) ) ) ).
% set_to_map_insert
tff(fact_2715_plus__rat_Orsp,axiom,
pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_kj(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_kj(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).
% plus_rat.rsp
tff(fact_2716_inverse__rat_Orsp,axiom,
pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_kk(product_prod(int,int),product_prod(int,int))),aTP_Lamp_kk(product_prod(int,int),product_prod(int,int)))) ).
% inverse_rat.rsp
tff(fact_2717_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: fun(A,B),X2: A,Y: B,Z2: B] : fun_upd(A,B,fun_upd(A,B,F,X2,Y),X2,Z2) = fun_upd(A,B,F,X2,Z2) ).
% fun_upd_upd
tff(fact_2718_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X2: A] : fun_upd(A,B,F,X2,aa(A,B,F,X2)) = F ).
% fun_upd_triv
tff(fact_2719_fun__upd__apply,axiom,
! [A: $tType,B: $tType,Z2: B,X2: B,F: fun(B,A),Y: A] :
( ( ( Z2 = X2 )
=> ( aa(B,A,fun_upd(B,A,F,X2,Y),Z2) = Y ) )
& ( ( Z2 != X2 )
=> ( aa(B,A,fun_upd(B,A,F,X2,Y),Z2) = aa(B,A,F,Z2) ) ) ) ).
% fun_upd_apply
tff(fact_2720_image__update,axiom,
! [B: $tType,A: $tType,X2: A,A4: set(A),F: fun(A,B),N: B] :
( ~ pp(member(A,X2,A4))
=> ( aa(set(A),set(B),image2(A,B,fun_upd(A,B,F,X2,N)),A4) = aa(set(A),set(B),image2(A,B,F),A4) ) ) ).
% image_update
tff(fact_2721_empty__upd__none,axiom,
! [A: $tType,B: $tType,X2: A,X: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_hj(A,option(B)),X2,none(B)),X) = none(B) ).
% empty_upd_none
tff(fact_2722_sup__None__1,axiom,
! [A: $tType] :
( sup(A)
=> ! [Y: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),none(A)),Y) = Y ) ).
% sup_None_1
tff(fact_2723_sup__None__2,axiom,
! [A: $tType] :
( sup(A)
=> ! [X2: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),sup_sup(option(A)),X2),none(A)) = X2 ) ).
% sup_None_2
tff(fact_2724_finite__Field__eq__finite,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( finite_finite2(A,field2(A,R4))
<=> finite_finite2(product_prod(A,A),R4) ) ).
% finite_Field_eq_finite
tff(fact_2725_image__map__upd,axiom,
! [B: $tType,A: $tType,X2: A,A4: set(A),M: fun(A,option(B)),Y: B] :
( ~ pp(member(A,X2,A4))
=> ( aa(set(A),set(option(B)),image2(A,option(B),fun_upd(A,option(B),M,X2,aa(B,option(B),some(B),Y))),A4) = aa(set(A),set(option(B)),image2(A,option(B),M),A4) ) ) ).
% image_map_upd
tff(fact_2726_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),M: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),fun_upd(A,option(C),M,A3,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),M),A3,aa(B,option(B),some(B),aa(C,B,F,B2))) ).
% map_option_o_map_upd
tff(fact_2727_fun__upd__def,axiom,
! [B: $tType,A: $tType,A3: A,F: fun(A,B),B2: B,X: A] :
( ( ( X = A3 )
=> ( aa(A,B,fun_upd(A,B,F,A3,B2),X) = B2 ) )
& ( ( X != A3 )
=> ( aa(A,B,fun_upd(A,B,F,A3,B2),X) = aa(A,B,F,X) ) ) ) ).
% fun_upd_def
tff(fact_2728_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: fun(A,B),X2: A,Y: B,G: fun(A,B),Z2: B] :
( ( fun_upd(A,B,F,X2,Y) = fun_upd(A,B,G,X2,Z2) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
tff(fact_2729_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B,Y: A] :
( ( aa(B,A,F,X2) = Y )
=> ( fun_upd(B,A,F,X2,Y) = F ) ) ).
% fun_upd_idem
tff(fact_2730_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: fun(B,A),X2: B,Y: A] : aa(B,A,fun_upd(B,A,F,X2,Y),X2) = Y ).
% fun_upd_same
tff(fact_2731_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z2: A,X2: A,F: fun(A,B),Y: B] :
( ( Z2 != X2 )
=> ( aa(A,B,fun_upd(A,B,F,X2,Y),Z2) = aa(A,B,F,Z2) ) ) ).
% fun_upd_other
tff(fact_2732_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A3: A,C2: A,M: fun(A,B),B2: B,D3: B] :
( ( A3 != C2 )
=> ( fun_upd(A,B,fun_upd(A,B,M,A3,B2),C2,D3) = fun_upd(A,B,fun_upd(A,B,M,C2,D3),A3,B2) ) ) ).
% fun_upd_twist
tff(fact_2733_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: fun(A,B),X2: A,Y: B] :
( ( fun_upd(A,B,F,X2,Y) = F )
<=> ( aa(A,B,F,X2) = Y ) ) ).
% fun_upd_idem_iff
tff(fact_2734_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),A3: B,B2: A,X2: B,Y: A] :
( ( aa(B,option(A),fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B2)),X2) = aa(A,option(A),some(A),Y) )
<=> ( ( ( X2 = A3 )
& ( B2 = Y ) )
| ( ( X2 != A3 )
& ( aa(B,option(A),M,X2) = aa(A,option(A),some(A),Y) ) ) ) ) ).
% map_upd_Some_unfold
tff(fact_2735_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: fun(B,option(A)),K: B,X2: A] :
( ( aa(B,option(A),T2,K) = aa(A,option(A),some(A),X2) )
=> ( fun_upd(B,option(A),T2,K,aa(A,option(A),some(A),X2)) = T2 ) ) ).
% map_upd_triv
tff(fact_2736_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,X2: B,N: fun(A,option(B)),Y: B] :
( ( fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),X2)) = fun_upd(A,option(B),N,A3,aa(B,option(B),some(B),Y)) )
=> ( X2 = Y ) ) ).
% map_upd_eqD1
tff(fact_2737_FieldI2,axiom,
! [A: $tType,I: A,J: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R4))
=> pp(member(A,J,field2(A,R4))) ) ).
% FieldI2
tff(fact_2738_FieldI1,axiom,
! [A: $tType,I: A,J: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R4))
=> pp(member(A,I,field2(A,R4))) ) ).
% FieldI1
tff(fact_2739_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B)),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R4)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R4),S2))) ) ).
% sup_Un_eq2
tff(fact_2740_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),G: fun(A,C),X2: A,Y: C] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),fun_upd(A,C,G,X2,Y)) = fun_upd(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G),X2,aa(C,B,F,Y)) ).
% fun_upd_comp
tff(fact_2741_map__upd__nonempty,axiom,
! [A: $tType,B: $tType,T2: fun(A,option(B)),K: A,X2: B] :
~ ! [X3: A] : aa(A,option(B),fun_upd(A,option(B),T2,K,aa(B,option(B),some(B),X2)),X3) = none(B) ).
% map_upd_nonempty
tff(fact_2742_one__rat_Orsp,axiom,
pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)))) ).
% one_rat.rsp
tff(fact_2743_R__subset__Field,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),product_Sigma(A,A,field2(A,R4),aTP_Lamp_kw(set(product_prod(A,A)),fun(A,set(A)),R4)))) ).
% R_subset_Field
tff(fact_2744_fst__in__Field,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_fst(A,A)),R4)),field2(A,R4))) ).
% fst_in_Field
tff(fact_2745_snd__in__Field,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),image2(product_prod(A,A),A,product_snd(A,A)),R4)),field2(A,R4))) ).
% snd_in_Field
tff(fact_2746_zero__rat_Orsp,axiom,
pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)))) ).
% zero_rat.rsp
tff(fact_2747_Fract_Orsp,axiom,
pp(aa(fun(int,fun(int,product_prod(int,int))),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,product_prod(int,int))),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,product_prod(int,int)),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),product_prod(int,int),fequal(int),ratrel)),aTP_Lamp_kx(int,fun(int,product_prod(int,int)))),aTP_Lamp_kx(int,fun(int,product_prod(int,int))))) ).
% Fract.rsp
tff(fact_2748_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),plus_plus(A)),G),zero_zero(A),A4) ) ).
% sum.eq_fold
tff(fact_2749_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),times_times(A)),G),one_one(A),A4) ) ).
% prod.eq_fold
tff(fact_2750_finite__range__updI,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),A3: B,B2: A] :
( finite_finite2(option(A),aa(set(B),set(option(A)),image2(B,option(A),F),top_top(set(B))))
=> finite_finite2(option(A),aa(set(B),set(option(A)),image2(B,option(A),fun_upd(B,option(A),F,A3,aa(A,option(A),some(A),B2))),top_top(set(B)))) ) ).
% finite_range_updI
tff(fact_2751_fun__upd__image,axiom,
! [A: $tType,B: $tType,X2: B,A4: set(B),F: fun(B,A),Y: A] :
( ( pp(member(B,X2,A4))
=> ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X2,Y)),A4) = aa(set(A),set(A),insert2(A,Y),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),aa(set(B),set(B),insert2(B,X2),bot_bot(set(B)))))) ) )
& ( ~ pp(member(B,X2,A4))
=> ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X2,Y)),A4) = aa(set(B),set(A),image2(B,A,F),A4) ) ) ) ).
% fun_upd_image
tff(fact_2752_times__rat_Orsp,axiom,
pp(aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool,aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(product_prod(int,int),product_prod(int,int)),ratrel,bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel)),aTP_Lamp_km(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_km(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).
% times_rat.rsp
tff(fact_2753_uminus__rat_Orsp,axiom,
pp(aa(fun(product_prod(int,int),product_prod(int,int)),bool,aa(fun(product_prod(int,int),product_prod(int,int)),fun(fun(product_prod(int,int),product_prod(int,int)),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),product_prod(int,int),product_prod(int,int),ratrel,ratrel),aTP_Lamp_kn(product_prod(int,int),product_prod(int,int))),aTP_Lamp_kn(product_prod(int,int),product_prod(int,int)))) ).
% uminus_rat.rsp
tff(fact_2754_product__fold,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( finite_finite2(A,A4)
=> ( finite_finite2(B,B5)
=> ( product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_kz(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B5),bot_bot(set(product_prod(A,B))),A4) ) ) ) ).
% product_fold
tff(fact_2755_Id__on__fold,axiom,
! [A: $tType,A4: set(A)] :
( finite_finite2(A,A4)
=> ( id_on(A,A4) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_la(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A4) ) ) ).
% Id_on_fold
tff(fact_2756_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod(int,int),X2: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa),Xa))
=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X2),X2))
=> ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),aa(product_prod(int,int),rat,abs_Rat,X2)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X2)),aa(product_prod(int,int),int,product_snd(int,int),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X2)))) ) ) ) ).
% plus_rat.abs_eq
tff(fact_2757_inverse__rat_Oabs__eq,axiom,
! [X2: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X2),X2))
=> ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X2)) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),X2)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),X2)),aa(product_prod(int,int),int,product_fst(int,int),X2)))) ) ) ).
% inverse_rat.abs_eq
tff(fact_2758_times__rat_Oabs__eq,axiom,
! [Xa: product_prod(int,int),X2: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa),Xa))
=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X2),X2))
=> ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa)),aa(product_prod(int,int),rat,abs_Rat,X2)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_fst(int,int),X2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X2)))) ) ) ) ).
% times_rat.abs_eq
tff(fact_2759_uminus__rat_Oabs__eq,axiom,
! [X2: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X2),X2))
=> ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X2)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),X2))),aa(product_prod(int,int),int,product_snd(int,int),X2))) ) ) ).
% uminus_rat.abs_eq
tff(fact_2760_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).
% graph_map_upd
tff(fact_2761_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),X2: product_prod(C,A),X7: set(product_prod(C,B))] :
( finite_finite2(product_prod(A,B),S2)
=> ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert2(product_prod(C,A),X2),bot_bot(set(product_prod(C,A)))),S2)),X7) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_lb(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X2)),X7,S2) ) ) ).
% insert_relcomp_union_fold
tff(fact_2762_graph__empty,axiom,
! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_hj(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% graph_empty
tff(fact_2763_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
( pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22),relcomp(A,B,C,R,S)))
=> ~ ! [B3: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B3),R))
=> ~ pp(member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),A22),S)) ) ) ).
% relcomp.cases
tff(fact_2764_relcomp_Osimps,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
( pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22),relcomp(A,B,C,R,S)))
<=> ? [A7: A,B7: B,C4: C] :
( ( A1 = A7 )
& ( A22 = C4 )
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7),R))
& pp(member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B7),C4),S)) ) ) ).
% relcomp.simps
tff(fact_2765_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(A,B)),C2: C,S: set(product_prod(B,C))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R))
=> ( pp(member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2),S))
=> pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2),relcomp(A,B,C,R,S))) ) ) ).
% relcomp.relcompI
tff(fact_2766_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R: set(product_prod(A,C)),S: set(product_prod(C,B))] :
( pp(member(product_prod(A,B),Xz,relcomp(A,C,B,R,S)))
=> ~ ! [X3: A,Y4: C,Z3: B] :
( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3) )
=> ( pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X3),Y4),R))
=> ~ pp(member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y4),Z3),S)) ) ) ) ).
% relcompE
tff(fact_2767_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set(product_prod(A,C)),S: set(product_prod(C,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C2),relcomp(A,C,B,R,S)))
=> ~ ! [B3: C] :
( pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B3),R))
=> ~ pp(member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B3),C2),S)) ) ) ).
% relcompEpair
tff(fact_2768_in__graphI,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V) )
=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V),graph(B,A,M))) ) ).
% in_graphI
tff(fact_2769_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: fun(A,option(B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,M)))
=> ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).
% in_graphD
tff(fact_2770_max__ext__compat,axiom,
! [A: $tType,R4: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R4,S2)),R4))
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R4),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S2)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert2(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R4))) ) ).
% max_ext_compat
tff(fact_2771_min__ext__compat,axiom,
! [A: $tType,R4: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R4,S2)),R4))
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R4),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S2)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert2(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R4))) ) ).
% min_ext_compat
tff(fact_2772_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
( finite_finite2(product_prod(A,B),R4)
=> ( finite_finite2(product_prod(B,C),S2)
=> ( relcomp(A,B,C,R4,S2) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_ld(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S2)),bot_bot(set(product_prod(A,C))),R4) ) ) ) ).
% relcomp_fold
tff(fact_2773_one__rat__def,axiom,
one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) ).
% one_rat_def
tff(fact_2774_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M,K,none(B))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_le(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M),K)) ).
% graph_fun_upd_None
tff(fact_2775_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),X2: product_prod(C,A),R4: set(product_prod(C,A))] :
( finite_finite2(product_prod(A,B),S2)
=> ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert2(product_prod(C,A),X2),R4),S2) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_lb(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X2)),relcomp(C,A,B,R4,S2),S2) ) ) ).
% insert_relcomp_fold
tff(fact_2776_zero__rat__def,axiom,
zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int))) ).
% zero_rat_def
tff(fact_2777_plus__rat__def,axiom,
plus_plus(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_kj(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% plus_rat_def
tff(fact_2778_inverse__rat__def,axiom,
inverse_inverse(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_kk(product_prod(int,int),product_prod(int,int))) ).
% inverse_rat_def
tff(fact_2779_times__rat__def,axiom,
times_times(rat) = aa(fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),fun(rat,fun(rat,rat)),map_fun(rat,product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),rep_Rat,map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat)),aTP_Lamp_km(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% times_rat_def
tff(fact_2780_uminus__rat__def,axiom,
uminus_uminus(rat) = aa(fun(product_prod(int,int),product_prod(int,int)),fun(rat,rat),map_fun(rat,product_prod(int,int),product_prod(int,int),rat,rep_Rat,abs_Rat),aTP_Lamp_kn(product_prod(int,int),product_prod(int,int))) ).
% uminus_rat_def
tff(fact_2781_comp__fun__commute__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B))] :
( finite_finite2(product_prod(A,B),S2)
=> finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_lg(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S2))) ) ).
% comp_fun_commute_relcomp_fold
tff(fact_2782_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X2: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X2,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) ).
% restrict_upd_same
tff(fact_2783_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B))] : restrict_map(A,B,F,top_top(set(A))) = F ).
% restrict_map_UNIV
tff(fact_2784_restrict__out,axiom,
! [A: $tType,B: $tType,X2: A,A4: set(A),M: fun(A,option(B))] :
( ~ pp(member(A,X2,A4))
=> ( aa(A,option(B),restrict_map(A,B,M,A4),X2) = none(B) ) ) ).
% restrict_out
tff(fact_2785_restrict__map__empty,axiom,
! [A: $tType,B: $tType,D4: set(A),X: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_hj(A,option(B)),D4),X) = none(B) ).
% restrict_map_empty
tff(fact_2786_restrict__map__to__empty,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),X: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X) = none(B) ).
% restrict_map_to_empty
tff(fact_2787_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X2: A,D4: set(A),M: fun(A,option(B))] :
( ( pp(member(A,X2,D4))
=> ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X2,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D4),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) ) )
& ( ~ pp(member(A,X2,D4))
=> ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X2,none(B)) = restrict_map(A,B,M,D4) ) ) ) ).
% fun_upd_None_restrict
tff(fact_2788_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [M: fun(A,option(B)),X7: set(A)] : pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M,X7)),M)) ) ).
% le_map_restrict
tff(fact_2789_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),R4: set(A),M7: fun(A,option(B)),R6: set(A)] :
( ( restrict_map(A,B,M,R4) = M7 )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R6),R4))
=> ( restrict_map(A,B,M,R6) = restrict_map(A,B,M7,R6) ) ) ) ).
% restrict_map_subset_eq
tff(fact_2790_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A4: set(B),K: B,V: A] :
( ( aa(B,option(A),restrict_map(B,A,M,A4),K) = aa(A,option(A),some(A),V) )
<=> ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V) )
& pp(member(B,K,A4)) ) ) ).
% restrict_map_eq(2)
tff(fact_2791_restrict__map__def,axiom,
! [A: $tType,B: $tType,A4: set(A),M: fun(A,option(B)),X: A] :
( ( pp(member(A,X,A4))
=> ( aa(A,option(B),restrict_map(A,B,M,A4),X) = aa(A,option(B),M,X) ) )
& ( ~ pp(member(A,X,A4))
=> ( aa(A,option(B),restrict_map(A,B,M,A4),X) = none(B) ) ) ) ).
% restrict_map_def
tff(fact_2792_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [Y4: A,X3: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y4))
=> finite6289374366891150609ommute(A,B,F) ) ).
% comp_fun_commute.intro
tff(fact_2793_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Y: A,X2: A] :
( finite6289374366891150609ommute(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X2)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,Y)) ) ) ).
% comp_fun_commute.comp_fun_commute
tff(fact_2794_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite6289374366891150609ommute(A,B,F)
<=> ! [Y3: A,X4: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y3)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y3)) ) ).
% comp_fun_commute_def
tff(fact_2795_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: fun(A,option(B)),A4: set(A)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,restrict_map(A,B,M,A4))))
=> pp(member(A,K,A4)) ) ).
% graph_restrictD(1)
tff(fact_2796_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),X2: B,S: set(B)] :
( ( aa(B,option(A),M,X2) = none(A) )
=> ( restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),insert2(B,X2),S))) = restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),S)) ) ) ).
% map_restrict_insert_none_simp
tff(fact_2797_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),S2: set(A),K: A,V: B] : fun_upd(A,option(B),restrict_map(A,B,F,S2),K,aa(B,option(B),some(B),V)) = restrict_map(A,B,fun_upd(A,option(B),F,K,aa(B,option(B),some(B),V)),aa(set(A),set(A),insert2(A,K),S2)) ).
% restrict_map_upd
tff(fact_2798_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: fun(A,option(B)),A4: set(A)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),graph(A,B,restrict_map(A,B,M,A4))))
=> ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).
% graph_restrictD(2)
tff(fact_2799_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X2: A] : restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) = fun_upd(A,option(B),F,X2,none(B)) ).
% restrict_complement_singleton_eq
tff(fact_2800_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X2: A] : fun_upd(A,option(B),M,X2,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))) ).
% map_upd_eq_restrict
tff(fact_2801_comp__fun__commute__product__fold,axiom,
! [B: $tType,A: $tType,B5: set(A)] :
( finite_finite2(A,B5)
=> finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_lh(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B5)) ) ).
% comp_fun_commute_product_fold
tff(fact_2802_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A)] : lattic5882676163264333800up_fin(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_li(A,fun(option(A),option(A))),none(A),A4)) ) ).
% Sup_fin.eq_fold'
tff(fact_2803_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
( ( aa(B,option(A),M,A3) = none(A) )
=> ( ran(B,A,fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),insert2(A,B2),ran(B,A,M)) ) ) ).
% ran_map_upd
tff(fact_2804_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( finite_finite2(A,A4)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A4) ) )
& ( ~ finite_finite2(A,A4)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ) ).
% Gcd_fin.eq_fold
tff(fact_2805_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : lattic643756798349783984er_Max(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_lj(A,fun(option(A),option(A))),none(A),A4)) ) ).
% Max.eq_fold'
tff(fact_2806_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] : lattic643756798350308766er_Min(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_lk(A,fun(option(A),option(A))),none(A),A4)) ) ).
% Min.eq_fold'
tff(fact_2807_pairself__image__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,fun(B,bool))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image2(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),aa(fun(product_prod(B,B),bool),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,bool)),fun(product_prod(B,B),bool),product_case_prod(B,B,bool),P))) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_ll(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),F),P)) ).
% pairself_image_eq
tff(fact_2808_eq__or__mem__image__simp,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A3: B,B5: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_lm(fun(B,A),fun(B,fun(set(B),fun(A,bool))),F),A3),B5)) = aa(set(A),set(A),insert2(A,aa(B,A,F,A3)),aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_ln(fun(B,A),fun(set(B),fun(A,bool)),F),B5))) ).
% eq_or_mem_image_simp
tff(fact_2809_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ~ finite_finite2(A,A4)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).
% Gcd_fin.infinite
tff(fact_2810_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( pp(dvd_dvd(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A4),one_one(A)))
<=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_2811_Eps__Opt__eq__None,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( eps_Opt(A,P) = none(A) )
<=> ~ ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ).
% Eps_Opt_eq_None
tff(fact_2812_ran__empty,axiom,
! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_bd(B,option(A))) = bot_bot(set(A)) ).
% ran_empty
tff(fact_2813_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V: B] :
( ( aa(A,option(B),M,K) = none(B) )
=> ( ran(A,B,aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_lo(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),M),K),V)) = aa(set(B),set(B),insert2(B,V),ran(A,B,M)) ) ) ).
% map_update_eta_repair(2)
tff(fact_2814_ran__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : ran(A,B,M) = aa(fun(B,bool),set(B),collect(B),aTP_Lamp_lp(fun(A,option(B)),fun(B,bool),M)) ).
% ran_def
tff(fact_2815_ranI,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B2) )
=> pp(member(A,B2,ran(B,A,M))) ) ).
% ranI
tff(fact_2816_fs__contract,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,fun(B,C)),S2: set(C)] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_lq(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),F),S2))) = aa(fun(A,bool),set(A),collect(A),aa(set(C),fun(A,bool),aTP_Lamp_lr(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),F),S2)) ).
% fs_contract
tff(fact_2817_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y: A,M: fun(B,option(A)),A4: set(B)] :
( pp(member(A,Y,ran(B,A,restrict_map(B,A,M,A4))))
=> ? [X3: B] :
( pp(member(B,X3,A4))
& ( aa(B,option(A),M,X3) = aa(A,option(A),some(A),Y) ) ) ) ).
% ran_restrictD
tff(fact_2818_Un__interval,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [B1: A,B22: A,B32: A,F: fun(A,B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B1),B22))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B22),B32))
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ls(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B22),F))),aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ls(A,fun(A,fun(fun(A,B),fun(B,bool))),B22),B32),F))) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ls(A,fun(A,fun(fun(A,B),fun(B,bool))),B1),B32),F)) ) ) ) ) ).
% Un_interval
tff(fact_2819_fold__atLeastAtMost__nat,axiom,
! [A: $tType,F: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc2: A] :
( finite6289374366891150609ommute(nat,A,F)
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = finite_fold(nat,A,F,Acc2,set_or1337092689740270186AtMost(nat,A3,B2)) ) ) ).
% fold_atLeastAtMost_nat
tff(fact_2820_Min_Oinfinite,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( ~ finite_finite2(A,A4)
=> ( lattic643756798350308766er_Min(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).
% Min.infinite
tff(fact_2821_relcomp__unfold,axiom,
! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R,S) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_lt(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R),S))) ).
% relcomp_unfold
tff(fact_2822_Max_Oinfinite,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A4: set(A)] :
( ~ finite_finite2(A,A4)
=> ( lattic643756798349783984er_Max(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).
% Max.infinite
tff(fact_2823_graph__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_lu(fun(A,option(B)),fun(product_prod(A,B),bool),M)) ).
% graph_def
tff(fact_2824_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A4: set(A)] :
( ~ finite_finite2(A,A4)
=> ( lattic5882676163264333800up_fin(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).
% Sup_fin.infinite
tff(fact_2825_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S2: set(B),F: fun(B,A),K: A] :
( finite_finite2(B,S2)
=> ( ( S2 != bot_bot(set(B)) )
=> ( lattic643756798350308766er_Min(A,aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_lv(fun(B,A),fun(A,fun(B,A)),F),K)),S2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798350308766er_Min(A,aa(set(B),set(A),image2(B,A,F),S2))),K) ) ) ) ) ).
% Min_add_commute
tff(fact_2826_card__le__Suc__Max,axiom,
! [S2: set(nat)] :
( finite_finite2(nat,S2)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),finite_card(nat,S2)),aa(nat,nat,suc,lattic643756798349783984er_Max(nat,S2)))) ) ).
% card_le_Suc_Max
tff(fact_2827_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S2: set(B),F: fun(B,A),K: A] :
( finite_finite2(B,S2)
=> ( ( S2 != bot_bot(set(B)) )
=> ( lattic643756798349783984er_Max(A,aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_lv(fun(B,A),fun(A,fun(B,A)),F),K)),S2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798349783984er_Max(A,aa(set(B),set(A),image2(B,A,F),S2))),K) ) ) ) ) ).
% Max_add_commute
tff(fact_2828_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S2: set(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,set_to_map(B,A,S2))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),S2))) ).
% set_to_map_ran
tff(fact_2829_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V: B] : map_to_set(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),map_to_set(A,B,M)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_lw(A,fun(product_prod(A,B),bool),K)))) ).
% map_to_set_upd
tff(fact_2830_brk__rel__def,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B))] : brk_rel(A,B,R4) = aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B))),aa(set(product_prod(product_prod(bool,A),product_prod(bool,B))),fun(set(product_prod(product_prod(bool,A),product_prod(bool,B))),set(product_prod(product_prod(bool,A),product_prod(bool,B)))),sup_sup(set(product_prod(product_prod(bool,A),product_prod(bool,B)))),aa(fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),set(product_prod(product_prod(bool,A),product_prod(bool,B))),collect(product_prod(product_prod(bool,A),product_prod(bool,B))),aTP_Lamp_lx(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),R4))),aa(fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),set(product_prod(product_prod(bool,A),product_prod(bool,B))),collect(product_prod(product_prod(bool,A),product_prod(bool,B))),aTP_Lamp_ly(product_prod(product_prod(bool,A),product_prod(bool,B)),bool))) ).
% brk_rel_def
tff(fact_2831_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType,A4: set(C),F: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A4,F,G) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_lz(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A4),F),G)) ).
% image2_def
tff(fact_2832_greaterThan__Suc,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),insert2(nat,aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).
% greaterThan_Suc
tff(fact_2833_refl__on__singleton,axiom,
! [A: $tType,X2: A] : refl_on(A,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),bot_bot(set(product_prod(A,A))))) ).
% refl_on_singleton
tff(fact_2834_time__refines,axiom,
! [A: $tType,C2: heap_Time_Heap(A),C6: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( refine_Imp_refines(A,C2,C6)
=> ( ~ time_fails(A,C6,H)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),time_time(A,C2,H)),time_time(A,C6,H))) ) ) ).
% time_refines
tff(fact_2835_greaterThan__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y: A] :
( ( aa(A,set(A),set_ord_greaterThan(A),X2) = aa(A,set(A),set_ord_greaterThan(A),Y) )
<=> ( X2 = Y ) ) ) ).
% greaterThan_eq_iff
tff(fact_2836_greaterThan__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,aa(A,set(A),set_ord_greaterThan(A),K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),I)) ) ) ).
% greaterThan_iff
tff(fact_2837_Inf__greaterThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X2: A] : complete_Inf_Inf(A,aa(A,set(A),set_ord_greaterThan(A),X2)) = X2 ) ).
% Inf_greaterThan
tff(fact_2838_greaterThan__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X2)),aa(A,set(A),set_ord_greaterThan(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X2)) ) ) ).
% greaterThan_subset_iff
tff(fact_2839_Compl__greaterThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_greaterThan(A),K)) = aa(A,set(A),set_ord_atMost(A),K) ) ).
% Compl_greaterThan
tff(fact_2840_Compl__atMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atMost(A),K)) = aa(A,set(A),set_ord_greaterThan(A),K) ) ).
% Compl_atMost
tff(fact_2841_map__to__set__empty,axiom,
! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_hj(A,option(B))) = bot_bot(set(product_prod(A,B))) ).
% map_to_set_empty
tff(fact_2842_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),top_top(A)))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X2)) = top_top(A) ) ) ) ).
% Sup_greaterThanAtLeast
tff(fact_2843_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_greaterThan(A),X2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_greaterThan
tff(fact_2844_image__uminus__lessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_lessThan(A),X2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_lessThan
tff(fact_2845_refl__onD2,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,Y: A] :
( refl_on(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> pp(member(A,Y,A4)) ) ) ).
% refl_onD2
tff(fact_2846_refl__onD1,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,Y: A] :
( refl_on(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> pp(member(A,X2,A4)) ) ) ).
% refl_onD1
tff(fact_2847_refl__onD,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A] :
( refl_on(A,A4,R)
=> ( pp(member(A,A3,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R)) ) ) ).
% refl_onD
tff(fact_2848_greaterThan__non__empty,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X2: A] : aa(A,set(A),set_ord_greaterThan(A),X2) != bot_bot(set(A)) ) ).
% greaterThan_non_empty
tff(fact_2849_infinite__Ioi,axiom,
! [A: $tType] :
( ( linorder(A)
& no_top(A) )
=> ! [A3: A] : ~ finite_finite2(A,aa(A,set(A),set_ord_greaterThan(A),A3)) ) ).
% infinite_Ioi
tff(fact_2850_greaterThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less(A),L)) ) ).
% greaterThan_def
tff(fact_2851_fails__refines,axiom,
! [A: $tType,C2: heap_Time_Heap(A),C6: heap_Time_Heap(A),H: heap_ext(product_unit)] :
( refine_Imp_refines(A,C2,C6)
=> ( time_fails(A,C2,H)
=> time_fails(A,C6,H) ) ) ).
% fails_refines
tff(fact_2852_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : set_to_map(A,B,map_to_set(A,B,M)) = M ).
% map_to_set_inverse
tff(fact_2853_map__to__set__empty__iff_I1_J,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B))] :
( ( map_to_set(A,B,M) = bot_bot(set(product_prod(A,B))) )
<=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).
% map_to_set_empty_iff(1)
tff(fact_2854_map__to__set__empty__iff_I2_J,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B))] :
( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M) )
<=> ! [X4: A] : aa(A,option(B),M,X4) = none(B) ) ).
% map_to_set_empty_iff(2)
tff(fact_2855_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: fun(B,A),X2: B,C2: C,G: fun(B,C),A4: set(B)] :
( ( B2 = aa(B,A,F,X2) )
=> ( ( C2 = aa(B,C,G,X2) )
=> ( pp(member(B,X2,A4))
=> pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C2),bNF_Greatest_image2(B,A,C,A4,F,G))) ) ) ) ).
% image2_eqI
tff(fact_2856_refl__on__def,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( refl_on(A,A4,R)
<=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4))))
& ! [X4: A] :
( pp(member(A,X4,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R)) ) ) ) ).
% refl_on_def
tff(fact_2857_refl__onI,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4))))
=> ( ! [X3: A] :
( pp(member(A,X3,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R)) )
=> refl_on(A,A4,R) ) ) ).
% refl_onI
tff(fact_2858_map__to__set__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : map_to_set(A,B,M) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_mb(fun(A,option(B)),fun(A,fun(B,bool)),M))) ).
% map_to_set_def
tff(fact_2859_INT__greaterThan__UNIV,axiom,
complete_Inf_Inf(set(nat),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).
% INT_greaterThan_UNIV
tff(fact_2860_map__to__set__ran,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M)) ).
% map_to_set_ran
tff(fact_2861_greaterThan__0,axiom,
aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat))) ).
% greaterThan_0
tff(fact_2862_refines__def,axiom,
! [A: $tType,P3: heap_Time_Heap(A),Q3: heap_Time_Heap(A)] :
( refine_Imp_refines(A,P3,Q3)
<=> ! [H7: heap_ext(product_unit)] : pp(aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool,aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),bool),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool),aa(bool,fun(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),bool),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool)),case_option(bool,product_prod(A,product_prod(heap_ext(product_unit),nat))),fTrue),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),bool)),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),bool),product_case_prod(A,product_prod(heap_ext(product_unit),nat),bool),aa(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),bool)),aTP_Lamp_md(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),bool))),P3),H7))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Q3),H7))) ) ).
% refines_def
tff(fact_2863_refines__empty,axiom,
! [A: $tType,M: heap_Time_Heap(A)] : refine_Imp_refines(A,M,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_ao(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).
% refines_empty
tff(fact_2864_refines__bind,axiom,
! [B: $tType,A: $tType,M: heap_Time_Heap(A),M7: heap_Time_Heap(A),F: fun(A,heap_Time_Heap(B)),F2: fun(A,heap_Time_Heap(B))] :
( refine_Imp_refines(A,M,M7)
=> ( ! [X3: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),F,X3),aa(A,heap_Time_Heap(B),F2,X3))
=> refine_Imp_refines(B,heap_Time_bind(A,B,M,F),heap_Time_bind(A,B,M7,F2)) ) ) ).
% refines_bind
tff(fact_2865_refl__on__domain,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
( refl_on(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( pp(member(A,A3,A4))
& pp(member(A,B2,A4)) ) ) ) ).
% refl_on_domain
tff(fact_2866_refines__option,axiom,
! [B: $tType,A: $tType,A3: option(A),A6: option(A),M1: heap_Time_Heap(B),M12: heap_Time_Heap(B),M22: fun(A,heap_Time_Heap(B)),M23: fun(A,heap_Time_Heap(B))] :
( ( A3 = A6 )
=> ( refine_Imp_refines(B,M1,M12)
=> ( ! [X3: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),M22,X3),aa(A,heap_Time_Heap(B),M23,X3))
=> refine_Imp_refines(B,aa(option(A),heap_Time_Heap(B),aa(fun(A,heap_Time_Heap(B)),fun(option(A),heap_Time_Heap(B)),aa(heap_Time_Heap(B),fun(fun(A,heap_Time_Heap(B)),fun(option(A),heap_Time_Heap(B))),case_option(heap_Time_Heap(B),A),M1),M22),A3),aa(option(A),heap_Time_Heap(B),aa(fun(A,heap_Time_Heap(B)),fun(option(A),heap_Time_Heap(B)),aa(heap_Time_Heap(B),fun(fun(A,heap_Time_Heap(B)),fun(option(A),heap_Time_Heap(B))),case_option(heap_Time_Heap(B),A),M12),M23),A6)) ) ) ) ).
% refines_option
tff(fact_2867_linear__order__on__singleton,axiom,
! [A: $tType,X2: A] : order_679001287576687338der_on(A,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),bot_bot(set(product_prod(A,A))))) ).
% linear_order_on_singleton
tff(fact_2868_Total__subset__Id,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( total_on(A,field2(A,R),R)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A)))
=> ( ( R = bot_bot(set(product_prod(A,A))) )
| ? [A5: A] : R = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).
% Total_subset_Id
tff(fact_2869_relation__of__def,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),A4: set(A)] : order_relation_of(A,P,A4) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_me(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P),A4))) ).
% relation_of_def
tff(fact_2870_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( ~ pp(dvd_dvd(A,A3,one_one(A)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),B2)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).
% euclidean_size_times_nonunit
tff(fact_2871_surj__swap,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).
% surj_swap
tff(fact_2872_euclidean__size__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,nat,euclid6346220572633701492n_size(code_integer),X2) = aa(int,nat,euclid6346220572633701492n_size(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% euclidean_size_integer.rep_eq
tff(fact_2873_swap__swap,axiom,
! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P3)) = P3 ).
% swap_swap
tff(fact_2874_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id2(A)))
<=> ( A3 = B2 ) ) ).
% pair_in_Id_conv
tff(fact_2875_IdI,axiom,
! [A: $tType,A3: A] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),id2(A))) ).
% IdI
tff(fact_2876_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X2) ).
% swap_simp
tff(fact_2877_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,fun(B,A)),P3: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_mf(fun(C,fun(B,A)),fun(B,fun(C,A)),F)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P3)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F),P3) ).
% case_swap
tff(fact_2878_swap__comp__swap,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B))),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B)),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).
% swap_comp_swap
tff(fact_2879_bijective__Id,axiom,
! [A: $tType] : bijective(A,A,id2(A)) ).
% bijective_Id
tff(fact_2880_euclidean__size__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) = one_one(nat) ) ) ).
% euclidean_size_1
tff(fact_2881_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),X2)) = aa(product_prod(B,A),A,product_snd(B,A),X2) ).
% fst_swap
tff(fact_2882_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),X2)) = aa(product_prod(A,B),A,product_fst(A,B),X2) ).
% snd_swap
tff(fact_2883_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X2: B,A4: set(product_prod(B,A))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X2),aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),A4)))
<=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y),A4)) ) ).
% pair_in_swap_image
tff(fact_2884_euclidean__size__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,nat,euclid6346220572633701492n_size(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,nat,euclid6346220572633701492n_size(int),X2) ).
% euclidean_size_integer.abs_eq
tff(fact_2885_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A3: A,B2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),id2(A)))
=> ( A3 = B2 ) ) ).
% BNF_Greatest_Fixpoint.IdD
tff(fact_2886_IdE,axiom,
! [A: $tType,P3: product_prod(A,A)] :
( pp(member(product_prod(A,A),P3,id2(A)))
=> ~ ! [X3: A] : P3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).
% IdE
tff(fact_2887_euclidean__size__natural_Orsp,axiom,
pp(aa(fun(nat,nat),bool,aa(fun(nat,nat),fun(fun(nat,nat),bool),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),euclid6346220572633701492n_size(nat)),euclid6346220572633701492n_size(nat))) ).
% euclidean_size_natural.rsp
tff(fact_2888_Linear__order__in__diff__Id,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( order_679001287576687338der_on(A,field2(A,R),R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
<=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) ) ) ) ) ).
% Linear_order_in_diff_Id
tff(fact_2889_euclidean__size__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) ) ) ) ).
% euclidean_size_unit
tff(fact_2890_euclidean__size__integer__def,axiom,
euclid6346220572633701492n_size(code_integer) = aa(fun(int,nat),fun(code_integer,nat),map_fun(code_integer,int,nat,nat,code_int_of_integer,id(nat)),euclid6346220572633701492n_size(int)) ).
% euclidean_size_integer_def
tff(fact_2891_euclidean__size__integer_Orsp,axiom,
pp(aa(fun(int,nat),bool,aa(fun(int,nat),fun(fun(int,nat),bool),bNF_rel_fun(int,int,nat,nat,fequal(int),fequal(nat)),euclid6346220572633701492n_size(int)),euclid6346220572633701492n_size(int))) ).
% euclidean_size_integer.rsp
tff(fact_2892_Id__def,axiom,
! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_mg(product_prod(A,A),bool)) ).
% Id_def
tff(fact_2893_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
<=> ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) )
& ( A3 != zero_zero(A) ) ) ) ) ).
% unit_iff_euclidean_size
tff(fact_2894_euclidean__size__times__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,nat,euclid6346220572633701492n_size(A),B2) ) ) ) ).
% euclidean_size_times_unit
tff(fact_2895_product__swap,axiom,
! [A: $tType,B: $tType,A4: set(B),B5: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),image2(product_prod(B,A),product_prod(A,B),product_swap(B,A)),product_Sigma(B,A,A4,aTP_Lamp_jr(set(A),fun(B,set(A)),B5))) = product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),A4)) ).
% product_swap
tff(fact_2896_prod_Oswap__def,axiom,
! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P3) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(product_prod(A,B),B,product_snd(A,B),P3)),aa(product_prod(A,B),A,product_fst(A,B),P3)) ).
% prod.swap_def
tff(fact_2897_reflcl__set__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,Xa3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),fequal(A)),X),Xa3))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),id2(A)))) ) ).
% reflcl_set_eq
tff(fact_2898_divmod__cases,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,A3: A] :
( ( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = zero_zero(A) )
=> ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),B2) ) ) )
=> ( ( ( B2 != zero_zero(A) )
=> ! [Q2: A,R2: A] :
( ( aa(A,A,euclid7384307370059645450egment(A),R2) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R2)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
=> ( ( R2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q2 )
=> ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = R2 )
=> ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R2) ) ) ) ) ) ) )
=> ( B2 = zero_zero(A) ) ) ) ) ).
% divmod_cases
tff(fact_2899_mod__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R: A,Q3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,euclid7384307370059645450egment(A),R) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R) = A3 )
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = R ) ) ) ) ) ) ).
% mod_eqI
tff(fact_2900_div__bounded,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R: A,Q3: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,euclid7384307370059645450egment(A),R) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R)),B2) = Q3 ) ) ) ) ) ).
% div_bounded
tff(fact_2901_div__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R: A,Q3: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,euclid7384307370059645450egment(A),R) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R) = A3 )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q3 ) ) ) ) ) ) ).
% div_eqI
tff(fact_2902_division__segment__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,euclid7384307370059645450egment(code_integer),X2)) = aa(int,int,euclid7384307370059645450egment(int),aa(code_integer,int,code_int_of_integer,X2)) ).
% division_segment_integer.rep_eq
tff(fact_2903_division__segment__1,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ( aa(A,A,euclid7384307370059645450egment(A),one_one(A)) = one_one(A) ) ) ).
% division_segment_1
tff(fact_2904_division__segment__numeral,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [K: num] : aa(A,A,euclid7384307370059645450egment(A),aa(num,A,numeral_numeral(A),K)) = one_one(A) ) ).
% division_segment_numeral
tff(fact_2905_division__segment__of__nat,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : aa(A,A,euclid7384307370059645450egment(A),aa(nat,A,semiring_1_of_nat(A),N)) = one_one(A) ) ).
% division_segment_of_nat
tff(fact_2906_division__segment__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_integer,euclid7384307370059645450egment(code_integer),aa(int,code_integer,code_integer_of_int,X2)) = aa(int,code_integer,code_integer_of_int,aa(int,int,euclid7384307370059645450egment(int),X2)) ).
% division_segment_integer.abs_eq
tff(fact_2907_division__segment__nat__def,axiom,
! [N: nat] : aa(nat,nat,euclid7384307370059645450egment(nat),N) = one_one(nat) ).
% division_segment_nat_def
tff(fact_2908_division__segment__integer_Orsp,axiom,
pp(aa(fun(int,int),bool,aa(fun(int,int),fun(fun(int,int),bool),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),euclid7384307370059645450egment(int)),euclid7384307370059645450egment(int))) ).
% division_segment_integer.rsp
tff(fact_2909_division__segment__natural_Orsp,axiom,
pp(aa(fun(nat,nat),bool,aa(fun(nat,nat),fun(fun(nat,nat),bool),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),euclid7384307370059645450egment(nat)),euclid7384307370059645450egment(nat))) ).
% division_segment_natural.rsp
tff(fact_2910_division__segment__integer__def,axiom,
euclid7384307370059645450egment(code_integer) = aa(fun(int,int),fun(code_integer,code_integer),map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int),euclid7384307370059645450egment(int)) ).
% division_segment_integer_def
tff(fact_2911_is__unit__division__segment,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A] : pp(dvd_dvd(A,aa(A,A,euclid7384307370059645450egment(A),A3),one_one(A))) ) ).
% is_unit_division_segment
tff(fact_2912_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,field2(A,R),R)
=> ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))
<=> ! [A8: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),field2(A,R)))
=> ( ( A8 != bot_bot(set(A)) )
=> ? [X4: A] :
( pp(member(A,X4,A8))
& ! [Xa2: A] :
( pp(member(A,Xa2,A8))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R)) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
tff(fact_2913_bsqr__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_mi(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R)))) ).
% bsqr_def
tff(fact_2914_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R)))
=> ( ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))))
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> ( ( ( A3 = B2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).
% wo_rel.cases_Total3
tff(fact_2915_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(5)
tff(fact_2916_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_2917_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : finite_finite2(nat,set_or3652927894154168847AtMost(nat,L,U)) ).
% finite_greaterThanAtMost
tff(fact_2918_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : finite_finite2(int,set_or3652927894154168847AtMost(int,L,U)) ).
% finite_greaterThanAtMost_int
tff(fact_2919_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : finite_finite2(code_integer,set_or3652927894154168847AtMost(code_integer,L,U)) ).
% finite_greaterThanAtMost_integer
tff(fact_2920_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] : finite_card(nat,set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),L) ).
% card_greaterThanAtMost
tff(fact_2921_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,L: A,U: A] :
( pp(member(A,I,set_or3652927894154168847AtMost(A,L,U)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),U)) ) ) ) ).
% greaterThanAtMost_iff
tff(fact_2922_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,K: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
=> ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).
% greaterThanAtMost_empty
tff(fact_2923_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).
% greaterThanAtMost_empty_iff2
tff(fact_2924_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [K: A,L: A] :
( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).
% greaterThanAtMost_empty_iff
tff(fact_2925_infinite__Ioc__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A3,B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
% infinite_Ioc_iff
tff(fact_2926_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [C2: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_or3652927894154168847AtMost(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ).
% image_add_greaterThanAtMost
tff(fact_2927_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X2,Y)) = Y ) ) ) ).
% Sup_greaterThanAtMost
tff(fact_2928_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> ( complete_Inf_Inf(A,set_or3652927894154168847AtMost(A,X2,Y)) = X2 ) ) ) ).
% Inf_greaterThanAtMost
tff(fact_2929_brk__rel__wf,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( wf(A,R4)
=> wf(product_prod(bool,A),brk_rel(A,A,R4)) ) ).
% brk_rel_wf
tff(fact_2930_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] : finite_card(int,set_or3652927894154168847AtMost(int,L,U)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L)) ).
% card_greaterThanAtMost_int
tff(fact_2931_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or3652927894154168847AtMost(A,A3,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ).
% image_minus_const_greaterThanAtMost
tff(fact_2932_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),set_or7035219750837199246ssThan(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ).
% image_diff_atLeastLessThan
tff(fact_2933_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or3652927894154168847AtMost(A,X2,Y)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_greaterThanAtMost
tff(fact_2934_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),set_or7035219750837199246ssThan(A,X2,Y)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atLeastLessThan
tff(fact_2935_reduction__pairI,axiom,
! [A: $tType,R4: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( wf(A,R4)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R4,S2)),R4))
=> fun_reduction_pair(A,aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R4),S2)) ) ) ).
% reduction_pairI
tff(fact_2936_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 )
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R)) ) ) ) ) ).
% wo_rel.max2_equals1
tff(fact_2937_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 )
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)) ) ) ) ) ).
% wo_rel.max2_equals2
tff(fact_2938_wo__rel_Omax2__greater,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2)),R))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2)),R)) ) ) ) ) ).
% wo_rel.max2_greater
tff(fact_2939_wf__induct__rule,axiom,
! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
( wf(A,R)
=> ( ! [X3: A] :
( ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R))
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% wf_induct_rule
tff(fact_2940_wf__eq__minimal,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ! [Q6: set(A)] :
( ? [X4: A] : pp(member(A,X4,Q6))
=> ? [X4: A] :
( pp(member(A,X4,Q6))
& ! [Y3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R))
=> ~ pp(member(A,Y3,Q6)) ) ) ) ) ).
% wf_eq_minimal
tff(fact_2941_wf__not__refl,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( wf(A,R)
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R)) ) ).
% wf_not_refl
tff(fact_2942_wf__not__sym,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,X2: A] :
( wf(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2),R))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),A3),R)) ) ) ).
% wf_not_sym
tff(fact_2943_wf__irrefl,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( wf(A,R)
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),R)) ) ).
% wf_irrefl
tff(fact_2944_wf__induct,axiom,
! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
( wf(A,R)
=> ( ! [X3: A] :
( ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R))
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% wf_induct
tff(fact_2945_wf__asym,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,X2: A] :
( wf(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2),R))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),A3),R)) ) ) ).
% wf_asym
tff(fact_2946_wfUNIVI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [P4: fun(A,bool),X3: A] :
( ! [Xa3: A] :
( ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Xa3),R))
=> pp(aa(A,bool,P4,Y4)) )
=> pp(aa(A,bool,P4,Xa3)) )
=> pp(aa(A,bool,P4,X3)) )
=> wf(A,R) ) ).
% wfUNIVI
tff(fact_2947_wfI__min,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( ! [X3: A,Q7: set(A)] :
( pp(member(A,X3,Q7))
=> ? [Xa3: A] :
( pp(member(A,Xa3,Q7))
& ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Xa3),R4))
=> ~ pp(member(A,Y4,Q7)) ) ) )
=> wf(A,R4) ) ).
% wfI_min
tff(fact_2948_wfE__min,axiom,
! [A: $tType,R4: set(product_prod(A,A)),X2: A,Q: set(A)] :
( wf(A,R4)
=> ( pp(member(A,X2,Q))
=> ~ ! [Z3: A] :
( pp(member(A,Z3,Q))
=> ~ ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3),R4))
=> ~ pp(member(A,Y5,Q)) ) ) ) ) ).
% wfE_min
tff(fact_2949_wf__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ! [P5: fun(A,bool)] :
( ! [X4: A] :
( ! [Y3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R))
=> pp(aa(A,bool,P5,Y3)) )
=> pp(aa(A,bool,P5,X4)) )
=> ! [X_1: A] : pp(aa(A,bool,P5,X_1)) ) ) ).
% wf_def
tff(fact_2950_wo__rel_Omax2__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 ) )
& ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 ) ) ) ) ).
% wo_rel.max2_def
tff(fact_2951_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R)) )
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% wo_rel.well_order_induct
tff(fact_2952_Ioc__inj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( set_or3652927894154168847AtMost(A,A3,B2) = set_or3652927894154168847AtMost(A,C2,D3) )
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C2)) )
| ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ).
% Ioc_inj
tff(fact_2953_wo__rel_OTOTALS,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Wellorder_wo_rel(A,R)
=> ! [X: A] :
( pp(member(A,X,field2(A,R)))
=> ! [Xa3: A] :
( pp(member(A,Xa3,field2(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa3),R))
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X),R)) ) ) ) ) ).
% wo_rel.TOTALS
tff(fact_2954_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2)),R))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2)),R))
& pp(member(A,bNF_We1388413361240627857o_max2(A,R,A3,B2),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))) ) ) ) ) ).
% wo_rel.max2_greater_among
tff(fact_2955_wfE__min_H,axiom,
! [A: $tType,R4: set(product_prod(A,A)),Q: set(A)] :
( wf(A,R4)
=> ( ( Q != bot_bot(set(A)) )
=> ~ ! [Z3: A] :
( pp(member(A,Z3,Q))
=> ~ ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3),R4))
=> ~ pp(member(A,Y5,Q)) ) ) ) ) ).
% wfE_min'
tff(fact_2956_infinite__Ioc,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A3,B2)) ) ) ).
% infinite_Ioc
tff(fact_2957_Ioc__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
| ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% Ioc_subset_iff
tff(fact_2958_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).
% atLeastSucAtMost_greaterThanAtMost
tff(fact_2959_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(6)
tff(fact_2960_well__order__induct__imp,axiom,
! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( ! [X3: A] :
( ! [Y5: A] :
( ( ( Y5 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R)) )
=> ( pp(member(A,Y5,field2(A,R)))
=> pp(aa(A,bool,P,Y5)) ) )
=> ( pp(member(A,X3,field2(A,R)))
=> pp(aa(A,bool,P,X3)) ) )
=> ( pp(member(A,A3,field2(A,R)))
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% well_order_induct_imp
tff(fact_2961_wf__bounded__measure,axiom,
! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F: fun(A,nat)] :
( ! [A5: A,B3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),R))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B3)),aa(A,nat,Ub,A5)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,B3)),aa(A,nat,Ub,A5)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,A5)),aa(A,nat,F,B3))) ) )
=> wf(A,R) ) ).
% wf_bounded_measure
tff(fact_2962_wfI,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B5: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),B5))))
=> ( ! [X3: A,P4: fun(A,bool)] :
( ! [Xa3: A] :
( ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Xa3),R))
=> pp(aa(A,bool,P4,Y4)) )
=> pp(aa(A,bool,P4,Xa3)) )
=> ( pp(member(A,X3,A4))
=> ( pp(member(A,X3,B5))
=> pp(aa(A,bool,P4,X3)) ) ) )
=> wf(A,R) ) ) ).
% wfI
tff(fact_2963_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(8)
tff(fact_2964_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).
% ivl_disj_un_one(3)
tff(fact_2965_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] : set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or3652927894154168847AtMost(int,L,U) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_2966_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).
% ivl_disj_un_one(5)
tff(fact_2967_wf__eq__minimal2,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ! [A8: set(A)] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),field2(A,R)))
& ( A8 != bot_bot(set(A)) ) )
=> ? [X4: A] :
( pp(member(A,X4,A8))
& ! [Xa2: A] :
( pp(member(A,Xa2,A8))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R)) ) ) ) ) ).
% wf_eq_minimal2
tff(fact_2968_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),Ub: fun(A,set(B)),F: fun(A,set(B))] :
( ! [A5: A,B3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),R))
=> ( finite_finite2(B,aa(A,set(B),Ub,A5))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B3)),aa(A,set(B),Ub,A5)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,B3)),aa(A,set(B),Ub,A5)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F,A5)),aa(A,set(B),F,B3))) ) )
=> wf(A,R) ) ).
% wf_bounded_set
tff(fact_2969_wo__rel_Ocases__Total,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R)))
=> ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).
% wo_rel.cases_Total
tff(fact_2970_sum_Ohead,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).
% sum.head
tff(fact_2971_prod_Ohead,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M: nat,N: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).
% prod.head
tff(fact_2972_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_2973_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D3)) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_2974_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
tff(fact_2975_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] : set_or3652927894154168847AtMost(A,A3,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A3,B2)),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_2976_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D3)) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_2977_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).
% ivl_disj_un_two(2)
tff(fact_2978_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,M: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
tff(fact_2979_wf__bounded__supset,axiom,
! [A: $tType,S2: set(A)] :
( finite_finite2(A,S2)
=> wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_mj(set(A),fun(set(A),fun(set(A),bool)),S2)))) ) ).
% wf_bounded_supset
tff(fact_2980_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_2981_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_2982_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ~ ? [F6: fun(nat,A)] :
! [I3: nat] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,aa(nat,nat,suc,I3))),aa(nat,A,F6,I3)),R)) ) ).
% wf_iff_no_infinite_down_chain
tff(fact_2983_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set(product_prod(A,A)),F: fun(nat,A)] :
( wf(A,R)
=> ~ ! [K3: nat] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F,aa(nat,nat,suc,K3))),aa(nat,A,F,K3)),R)) ) ).
% wf_no_infinite_down_chainE
tff(fact_2984_wo__rel_Ominim__least,axiom,
! [A: $tType,R: set(product_prod(A,A)),B5: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R)))
=> ( pp(member(A,B2,B5))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R,B5)),B2),R)) ) ) ) ).
% wo_rel.minim_least
tff(fact_2985_wo__rel_Oequals__minim,axiom,
! [A: $tType,R: set(product_prod(A,A)),B5: set(A),A3: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R)))
=> ( pp(member(A,A3,B5))
=> ( ! [B3: A] :
( pp(member(A,B3,B5))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B3),R)) )
=> ( A3 = bNF_We6954850376910717587_minim(A,R,B5) ) ) ) ) ) ).
% wo_rel.equals_minim
tff(fact_2986_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,bool)))] :
( wf(A,R4)
=> ( ! [F3: fun(A,B),G3: fun(A,B),X3: A,R2: B] :
( ! [Z5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),X3),R4))
=> ( aa(A,B,F3,Z5) = aa(A,B,G3,Z5) ) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X3),R2))
<=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G3),X3),R2)) ) )
=> ( ! [X3: A,F3: fun(A,B)] :
( ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R4))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),Y5),aa(A,B,F3,Y5))) )
=> ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X3),X_12)) )
=> ? [F3: fun(A,B)] :
! [X: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X),aa(A,B,F3,X))) ) ) ) ).
% dependent_wf_choice
tff(fact_2987_of__rat__neg__one,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_of_rat(A,aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% of_rat_neg_one
tff(fact_2988_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),V: A] : ref_update(A,R,V) = heap_Time_heap(product_unit,aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mk(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),R),V)) ) ).
% Ref_Time.update_def
tff(fact_2989_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A4: set(B),F: fun(B,A),B2: A] :
( finite_finite2(B,A4)
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_ml(fun(B,A),fun(A,fun(B,A)),F),B2)),aa(set(B),set(B),inf_inf(set(B),A4),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_mm(fun(B,A),fun(A,fun(B,bool)),F),B2))))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),inf_inf(set(B),A4),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_mn(fun(B,A),fun(A,fun(B,bool)),F),B2))))),B2)) ) ) ) ).
% sum_div_partition
tff(fact_2990_bsqr__max2,axiom,
! [A: $tType,R: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
( order_well_order_on(A,field2(A,R),R)
=> ( pp(member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),B22)),bNF_Wellorder_bsqr(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R,A1,A22)),bNF_We1388413361240627857o_max2(A,R,B1,B22)),R)) ) ) ).
% bsqr_max2
tff(fact_2991_UN__atLeast__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).
% UN_atLeast_UNIV
tff(fact_2992_atLeast__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [X2: A,Y: A] :
( ( aa(A,set(A),set_ord_atLeast(A),X2) = aa(A,set(A),set_ord_atLeast(A),Y) )
<=> ( X2 = Y ) ) ) ).
% atLeast_eq_iff
tff(fact_2993_unit__abs__eta__conv,axiom,
! [A: $tType,F: fun(product_unit,A)] : aTP_Lamp_mo(fun(product_unit,A),fun(product_unit,A),F) = F ).
% unit_abs_eta_conv
tff(fact_2994_inf__Some,axiom,
! [A: $tType] :
( inf(A)
=> ! [X2: A,Y: A] : aa(option(A),option(A),inf_inf(option(A),aa(A,option(A),some(A),X2)),aa(A,option(A),some(A),Y)) = aa(A,option(A),some(A),aa(A,A,inf_inf(A,X2),Y)) ) ).
% inf_Some
tff(fact_2995_atLeast__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,aa(A,set(A),set_ord_atLeast(A),K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),I)) ) ) ).
% atLeast_iff
tff(fact_2996_Inf__atLeast,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A] : complete_Inf_Inf(A,aa(A,set(A),set_ord_atLeast(A),X2)) = X2 ) ).
% Inf_atLeast
tff(fact_2997_atLeast__subset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),X2)),aa(A,set(A),set_ord_atLeast(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X2)) ) ) ).
% atLeast_subset_iff
tff(fact_2998_image__add__atLeast,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),aa(A,set(A),set_ord_atLeast(A),I)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ).
% image_add_atLeast
tff(fact_2999_Sup__atLeast,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X2: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atLeast(A),X2)) = top_top(A) ) ).
% Sup_atLeast
tff(fact_3000_inter__compl__diff__conv,axiom,
! [A: $tType,A4: set(A),B5: set(A)] : aa(set(A),set(A),inf_inf(set(A),A4),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) ).
% inter_compl_diff_conv
tff(fact_3001_Int__atMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,ord_min(A,A3),B2)) ) ).
% Int_atMost
tff(fact_3002_Compl__lessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_lessThan(A),K)) = aa(A,set(A),set_ord_atLeast(A),K) ) ).
% Compl_lessThan
tff(fact_3003_Compl__atLeast,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atLeast(A),K)) = aa(A,set(A),set_ord_lessThan(A),K) ) ).
% Compl_atLeast
tff(fact_3004_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] :
( ( one_one(A) = field_char_0_of_rat(A,A3) )
<=> ( one_one(rat) = A3 ) ) ) ).
% one_eq_of_rat_iff
tff(fact_3005_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] :
( ( field_char_0_of_rat(A,A3) = one_one(A) )
<=> ( A3 = one_one(rat) ) ) ) ).
% of_rat_eq_1_iff
tff(fact_3006_of__rat__1,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_of_rat(A,one_one(rat)) = one_one(A) ) ) ).
% of_rat_1
tff(fact_3007_atLeast__0,axiom,
aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).
% atLeast_0
tff(fact_3008_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A,H: A,L4: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L4)))
<=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L4),L)) ) ) ) ).
% Icc_subset_Ici_iff
tff(fact_3009_Int__atLeastAtMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),set_or1337092689740270186AtMost(A,A3,B2)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,ord_max(A,A3),C2),aa(A,A,ord_min(A,B2),D3)) ) ).
% Int_atLeastAtMost
tff(fact_3010_Int__atLeastLessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),set_or7035219750837199246ssThan(A,A3,B2)),set_or7035219750837199246ssThan(A,C2,D3)) = set_or7035219750837199246ssThan(A,aa(A,A,ord_max(A,A3),C2),aa(A,A,ord_min(A,B2),D3)) ) ).
% Int_atLeastLessThan
tff(fact_3011_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atMost(A),B2)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,C2,aa(A,A,ord_min(A,B2),D3)) ) ).
% Int_atLeastAtMostR1
tff(fact_3012_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atMost(A),D3)) = set_or1337092689740270186AtMost(A,A3,aa(A,A,ord_min(A,B2),D3)) ) ).
% Int_atLeastAtMostL1
tff(fact_3013_image__minus__const__atLeast,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),aa(A,set(A),set_ord_atLeast(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),A3)) ) ).
% image_minus_const_atLeast
tff(fact_3014_image__minus__const__AtMost,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C2: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),minus_minus(A),C2)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)) ) ).
% image_minus_const_AtMost
tff(fact_3015_image__uminus__atLeast,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_atLeast(A),X2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atLeast
tff(fact_3016_image__uminus__atMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),image2(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_atMost(A),X2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atMost
tff(fact_3017_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] : aa(set(A),set(A),inf_inf(set(A),set_or1337092689740270186AtMost(A,A3,B2)),aa(A,set(A),set_ord_atLeast(A),C2)) = set_or1337092689740270186AtMost(A,aa(A,A,ord_max(A,A3),C2),B2) ) ).
% Int_atLeastAtMostL2
tff(fact_3018_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atLeast(A),A3)),set_or1337092689740270186AtMost(A,C2,D3)) = set_or1337092689740270186AtMost(A,aa(A,A,ord_max(A,A3),C2),D3) ) ).
% Int_atLeastAtMostR2
tff(fact_3019_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = set_or3652927894154168847AtMost(A,aa(A,A,ord_max(A,A3),C2),aa(A,A,ord_min(A,B2),D3)) ) ).
% Int_greaterThanAtMost
tff(fact_3020_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] : aa(set(A),set(A),inf_inf(set(A),set_or5935395276787703475ssThan(A,A3,B2)),set_or5935395276787703475ssThan(A,C2,D3)) = set_or5935395276787703475ssThan(A,aa(A,A,ord_max(A,A3),C2),aa(A,A,ord_min(A,B2),D3)) ) ).
% Int_greaterThanLessThan
tff(fact_3021_one__le__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R)) ) ) ).
% one_le_of_rat_iff
tff(fact_3022_of__rat__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R)),one_one(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),one_one(rat))) ) ) ).
% of_rat_le_1_iff
tff(fact_3023_one__less__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),field_char_0_of_rat(A,R)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R)) ) ) ).
% one_less_of_rat_iff
tff(fact_3024_of__rat__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),field_char_0_of_rat(A,R)),one_one(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),one_one(rat))) ) ) ).
% of_rat_less_1_iff
tff(fact_3025_atLeastLessThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or7035219750837199246ssThan(A,L,U) = aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).
% atLeastLessThan_def
tff(fact_3026_well__order__on__domain,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
( order_well_order_on(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( pp(member(A,A3,A4))
& pp(member(A,B2,A4)) ) ) ) ).
% well_order_on_domain
tff(fact_3027_Inf__unit__def,axiom,
! [Uu: set(product_unit)] : complete_Inf_Inf(product_unit,Uu) = product_Unity ).
% Inf_unit_def
tff(fact_3028_Sup__unit__def,axiom,
! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Sup_Sup(product_unit),Uu) = product_Unity ).
% Sup_unit_def
tff(fact_3029_Times__Int__distrib1,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),inf_inf(set(A),A4),B5),aTP_Lamp_jm(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),C5))) ).
% Times_Int_distrib1
tff(fact_3030_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B5: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_mp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),product_Sigma(A,B,I4,A4)),product_Sigma(A,B,I4,B5)) ).
% Sigma_Int_distrib2
tff(fact_3031_Times__Int__Times,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B),C5: set(A),D4: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5))),product_Sigma(A,B,C5,aTP_Lamp_jm(set(B),fun(A,set(B)),D4))) = product_Sigma(A,B,aa(set(A),set(A),inf_inf(set(A),A4),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_mq(set(B),fun(set(B),fun(A,set(B))),B5),D4)) ).
% Times_Int_Times
tff(fact_3032_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I4: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),inf_inf(set(A),I4),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),product_Sigma(A,B,I4,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Int_distrib1
tff(fact_3033_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
tff(fact_3034_top__unit__def,axiom,
top_top(product_unit) = product_Unity ).
% top_unit_def
tff(fact_3035_uminus__unit__def,axiom,
! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).
% uminus_unit_def
tff(fact_3036_bot__unit__def,axiom,
bot_bot(product_unit) = product_Unity ).
% bot_unit_def
tff(fact_3037_ivl__disj__int__one_I8_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(8)
tff(fact_3038_sup__unit__def,axiom,
! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),sup_sup(product_unit),Uu),Uv) = product_Unity ).
% sup_unit_def
tff(fact_3039_atLeastAtMost__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or1337092689740270186AtMost(A,L,U) = aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).
% atLeastAtMost_def
tff(fact_3040_ivl__disj__int__one_I6_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(6)
tff(fact_3041_disjointI,axiom,
! [A: $tType,A3: set(A),B2: set(A)] :
( ! [X3: A] :
( pp(member(A,X3,A3))
=> ~ pp(member(A,X3,B2)) )
=> ( aa(set(A),set(A),inf_inf(set(A),A3),B2) = bot_bot(set(A)) ) ) ).
% disjointI
tff(fact_3042_inter__eq__subsetI,axiom,
! [A: $tType,S2: set(A),S5: set(A),A4: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),S5))
=> ( ( aa(set(A),set(A),inf_inf(set(A),A4),S5) = aa(set(A),set(A),inf_inf(set(A),B5),S5) )
=> ( aa(set(A),set(A),inf_inf(set(A),A4),S2) = aa(set(A),set(A),inf_inf(set(A),B5),S2) ) ) ) ).
% inter_eq_subsetI
tff(fact_3043_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atLeast(A),L) ) ).
% not_empty_eq_Ici_eq_empty
tff(fact_3044_infinite__Ici,axiom,
! [A: $tType] :
( ( linorder(A)
& no_top(A) )
=> ! [A3: A] : ~ finite_finite2(A,aa(A,set(A),set_ord_atLeast(A),A3)) ) ).
% infinite_Ici
tff(fact_3045_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [L4: A] : top_top(set(A)) != aa(A,set(A),set_ord_atLeast(A),L4) ) ).
% not_UNIV_eq_Ici
tff(fact_3046_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( no_top(A)
=> ! [L4: A,L: A,H: A] : aa(A,set(A),set_ord_atLeast(A),L4) != set_or1337092689740270186AtMost(A,L,H) ) ).
% not_Ici_eq_Icc
tff(fact_3047_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( no_top(A)
=> ! [H: A,L4: A] : aa(A,set(A),set_ord_atMost(A),H) != aa(A,set(A),set_ord_atLeast(A),L4) ) ).
% not_Iic_eq_Ici
tff(fact_3048_atMost__Int__atLeast,axiom,
! [A: $tType] :
( order(A)
=> ! [N: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atMost(A),N)),aa(A,set(A),set_ord_atLeast(A),N)) = aa(set(A),set(A),insert2(A,N),bot_bot(set(A))) ) ).
% atMost_Int_atLeast
tff(fact_3049_atLeast__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).
% atLeast_def
tff(fact_3050_inf__option__def,axiom,
! [A: $tType] :
( inf(A)
=> ! [X2: option(A),Y: option(A)] : aa(option(A),option(A),inf_inf(option(A),X2),Y) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),none(A)),aTP_Lamp_ms(option(A),fun(A,option(A)),Y)),X2) ) ).
% inf_option_def
tff(fact_3051_translation__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),inf_inf(set(A),S),T2)) = aa(set(A),set(A),inf_inf(set(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).
% translation_Int
tff(fact_3052_disjoint__mono,axiom,
! [A: $tType,A3: set(A),A6: set(A),B2: set(A),B4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B2),B4))
=> ( ( aa(set(A),set(A),inf_inf(set(A),A6),B4) = bot_bot(set(A)) )
=> ( aa(set(A),set(A),inf_inf(set(A),A3),B2) = bot_bot(set(A)) ) ) ) ) ).
% disjoint_mono
tff(fact_3053_disjoint__alt__simp2,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) != A4 )
<=> ( aa(set(A),set(A),inf_inf(set(A),A4),B5) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp2
tff(fact_3054_disjoint__alt__simp1,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5) = A4 )
<=> ( aa(set(A),set(A),inf_inf(set(A),A4),B5) = bot_bot(set(A)) ) ) ).
% disjoint_alt_simp1
tff(fact_3055_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(3)
tff(fact_3056_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(6)
tff(fact_3057_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_lessThan(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,ord_min(A,A3),B2)) ) ).
% greaterThan_Int_greaterThan
tff(fact_3058_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X2: A] :
( ( aa(A,set(A),set_ord_atLeast(A),X2) = top_top(set(A)) )
<=> ( X2 = bot_bot(A) ) ) ) ).
% atLeast_eq_UNIV_iff
tff(fact_3059_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L))) ) ).
% not_UNIV_le_Ici
tff(fact_3060_UNIV__unit,axiom,
top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),insert2(product_unit,product_Unity),bot_bot(set(product_unit))) ).
% UNIV_unit
tff(fact_3061_not__Ici__le__Icc,axiom,
! [A: $tType] :
( no_top(A)
=> ! [L: A,L4: A,H3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),set_or1337092689740270186AtMost(A,L4,H3))) ) ).
% not_Ici_le_Icc
tff(fact_3062_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,ord_max(A,A3),B2)) ) ).
% lessThan_Int_lessThan
tff(fact_3063_not__Iic__le__Ici,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [H: A,L4: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),aa(A,set(A),set_ord_atLeast(A),L4))) ) ).
% not_Iic_le_Ici
tff(fact_3064_not__Ici__le__Iic,axiom,
! [A: $tType] :
( no_top(A)
=> ! [L: A,H3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),H3))) ) ).
% not_Ici_le_Iic
tff(fact_3065_translation__subtract__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),aa(set(A),set(A),inf_inf(set(A),S),T2)) = aa(set(A),set(A),inf_inf(set(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),S)),aa(set(A),set(A),image2(A,A,aTP_Lamp_go(A,fun(A,A),A3)),T2)) ) ).
% translation_subtract_Int
tff(fact_3066_of__rat__add,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat,B2: rat] : field_char_0_of_rat(A,aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),field_char_0_of_rat(A,A3)),field_char_0_of_rat(A,B2)) ) ).
% of_rat_add
tff(fact_3067_Ioi__le__Ico,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_atLeast(A),A3))) ) ).
% Ioi_le_Ico
tff(fact_3068_atLeast__Suc__greaterThan,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_greaterThan(nat),K) ).
% atLeast_Suc_greaterThan
tff(fact_3069_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(7)
tff(fact_3070_disjoint__alt__simp3,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),A4))
<=> ( aa(set(A),set(A),inf_inf(set(A),A4),B5) != bot_bot(set(A)) ) ) ).
% disjoint_alt_simp3
tff(fact_3071_Ioc__disjoint,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( aa(set(A),set(A),inf_inf(set(A),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D3)) = bot_bot(set(A)) )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),A3)) ) ) ) ).
% Ioc_disjoint
tff(fact_3072_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(4)
tff(fact_3073_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(2)
tff(fact_3074_wait__def,axiom,
! [N: nat] : heap_Time_wait(N) = aa(fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(product_unit),heap_Time_Heap2(product_unit),aTP_Lamp_mt(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ).
% wait_def
tff(fact_3075_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(8)
tff(fact_3076_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(5)
tff(fact_3077_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(4)
tff(fact_3078_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(1)
tff(fact_3079_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(3)
tff(fact_3080_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(7)
tff(fact_3081_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(1)
tff(fact_3082_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,M: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_two(2)
tff(fact_3083_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,U: A] : aa(set(A),set(A),inf_inf(set(A),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).
% ivl_disj_int_one(5)
tff(fact_3084_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).
% ivl_disj_un_one(8)
tff(fact_3085_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),G: fun(B,A),B5: set(B)] :
( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),inf_inf(set(B),A4),B5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(B),fun(B,A),aTP_Lamp_mu(fun(B,A),fun(set(B),fun(B,A)),G),B5)),A4) ) ) ) ).
% prod.inter_restrict
tff(fact_3086_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or3652927894154168847AtMost(A,L,U) = aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).
% greaterThanAtMost_def
tff(fact_3087_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A] : set_or5935395276787703475ssThan(A,A3,B2) = aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_greaterThan(A),A3)),aa(A,set(A),set_ord_lessThan(A),B2)) ) ).
% greaterThanLessThan_eq
tff(fact_3088_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A,U: A] : set_or5935395276787703475ssThan(A,L,U) = aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).
% greaterThanLessThan_def
tff(fact_3089_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A3)),aa(A,set(A),set_ord_greaterThan(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ).
% Ici_subset_Ioi_iff
tff(fact_3090_execute__assert_H,axiom,
! [P: bool,H: heap_ext(product_unit)] :
( ( pp(P)
=> ( aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(product_unit),fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(product_unit),refine_Imp_assert(P)),H) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),zero_zero(nat)))) ) )
& ( ~ pp(P)
=> ( aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(product_unit),fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(product_unit),refine_Imp_assert(P)),H) = none(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))) ) ) ) ).
% execute_assert'
tff(fact_3091_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B5: set(B),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(B,B5)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),inf_inf(set(B),A4),B5))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ).
% sum.union_inter
tff(fact_3092_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),G: fun(B,A),B5: set(B)] :
( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),inf_inf(set(B),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))) ) ) ) ).
% sum.Int_Diff
tff(fact_3093_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
( finite_finite2(B,T5)
=> ( finite_finite2(B,S2)
=> ( ! [I2: B] :
( pp(member(B,I2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S2)))
=> ( aa(B,A,H,I2) = one_one(A) ) )
=> ( ! [I2: B] :
( pp(member(B,I2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),T5)))
=> ( aa(B,A,G,I2) = one_one(A) ) )
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),inf_inf(set(B),S2),T5)))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T5) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
tff(fact_3094_Iio__Int__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [X2: A,K: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),K))
=> ( aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))) = aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),K))
=> ( aa(set(A),set(A),inf_inf(set(A),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).
% Iio_Int_singleton
tff(fact_3095_card__Un__Int,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( finite_finite2(A,A4)
=> ( finite_finite2(A,B5)
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(A,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))),finite_card(A,aa(set(A),set(A),inf_inf(set(A),A4),B5))) ) ) ) ).
% card_Un_Int
tff(fact_3096_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mv(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),aa(set(B),set(B),inf_inf(set(B),A4),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),inf_inf(set(B),A4),aa(set(B),set(B),uminus_uminus(set(B)),aa(fun(B,bool),set(B),collect(B),P))))) ) ) ) ).
% sum.If_cases
tff(fact_3097_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).
% ivl_disj_un_singleton(1)
tff(fact_3098_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).
% ivl_disj_un_one(7)
tff(fact_3099_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: A,U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).
% ivl_disj_un_one(6)
tff(fact_3100_Linear__order__Well__order__iff,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,field2(A,R),R)
=> ( order_well_order_on(A,field2(A,R),R)
<=> ! [A8: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A8),field2(A,R)))
=> ( ( A8 != bot_bot(set(A)) )
=> ? [X4: A] :
( pp(member(A,X4,A8))
& ! [Xa2: A] :
( pp(member(A,Xa2,A8))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R)) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
tff(fact_3101_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B5: set(B),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(B,B5)
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),inf_inf(set(B),A4),B5)))
=> ( aa(B,A,G,X3) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ) ).
% sum.union_inter_neutral
tff(fact_3102_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [A4: set(B),B5: set(B),F: fun(B,A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(B,B5)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),inf_inf(set(B),A4),B5))) ) ) ) ) ).
% sum_Un
tff(fact_3103_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B5: set(B),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(B,B5)
=> ( ( aa(set(B),set(B),inf_inf(set(B),A4),B5) = bot_bot(set(B)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ) ).
% sum.union_disjoint
tff(fact_3104_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A4: set(B),B5: set(B),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(B,B5)
=> ( ! [X3: B] :
( pp(member(B,X3,aa(set(B),set(B),inf_inf(set(B),A4),B5)))
=> ( aa(B,A,G,X3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A4)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B5)) ) ) ) ) ) ).
% prod.union_inter_neutral
tff(fact_3105_execute__update,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),V: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(product_unit),fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(product_unit),ref_update(A,R,V)),H) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,R,V,H)),one_one(nat)))) ) ).
% execute_update
tff(fact_3106_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B5: set(B),G: fun(B,A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(B,B5)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A4),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A4)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),inf_inf(set(B),A4),B5))) ) ) ) ) ).
% sum.union_diff2
tff(fact_3107_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(B)
=> ! [A4: set(A),B5: set(A),F: fun(A,B)] :
( finite_finite2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),aa(set(A),set(A),inf_inf(set(A),A4),B5))) ) ) ) ).
% sum_Un2
tff(fact_3108_card__Un__disjoint,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( finite_finite2(A,A4)
=> ( finite_finite2(A,B5)
=> ( ( aa(set(A),set(A),inf_inf(set(A),A4),B5) = bot_bot(set(A)) )
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(A,B5)) ) ) ) ) ).
% card_Un_disjoint
tff(fact_3109_sum__Un__nat,axiom,
! [A: $tType,A4: set(A),B5: set(A),F: fun(A,nat)] :
( finite_finite2(A,A4)
=> ( finite_finite2(A,B5)
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),B5))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),inf_inf(set(A),A4),B5))) ) ) ) ).
% sum_Un_nat
tff(fact_3110_atLeast__Suc,axiom,
! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),insert2(nat,K),bot_bot(set(nat)))) ).
% atLeast_Suc
tff(fact_3111_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] : product_rec_unit(T,F1,product_Unity) = F1 ).
% old.unit.rec
tff(fact_3112_default__unit__def,axiom,
default_default(product_unit) = product_Unity ).
% default_unit_def
tff(fact_3113_CODE__ABORT__def,axiom,
! [A: $tType,F: fun(product_unit,A)] : cODE_ABORT(A,F) = aa(product_unit,A,F,product_Unity) ).
% CODE_ABORT_def
tff(fact_3114_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A)] : lattic7752659483105999362nf_fin(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_mw(A,fun(option(A),option(A))),none(A),A4)) ) ).
% Inf_fin.eq_fold'
tff(fact_3115_image__split__eq__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),A4: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_mx(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),A4) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A4),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_my(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F),G),A4)) ).
% image_split_eq_Sigma
tff(fact_3116_atLeast__empty__triv,axiom,
! [A: $tType] : aa(set(A),set(set(A)),set_ord_atLeast(set(A)),bot_bot(set(A))) = top_top(set(set(A))) ).
% atLeast_empty_triv
tff(fact_3117_inf__None__2,axiom,
! [A: $tType] :
( inf(A)
=> ! [X2: option(A)] : aa(option(A),option(A),inf_inf(option(A),X2),none(A)) = none(A) ) ).
% inf_None_2
tff(fact_3118_inf__None__1,axiom,
! [A: $tType] :
( inf(A)
=> ! [Y: option(A)] : aa(option(A),option(A),inf_inf(option(A),none(A)),Y) = none(A) ) ).
% inf_None_1
tff(fact_3119_vimage__id,axiom,
! [A: $tType] : vimage(A,A,id(A)) = id(set(A)) ).
% vimage_id
tff(fact_3120_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),G: fun(B,A)] : aa(fun(set(A),set(B)),fun(set(A),set(C)),aa(fun(set(B),set(C)),fun(fun(set(A),set(B)),fun(set(A),set(C))),comp(set(B),set(C),set(A)),vimage(C,B,F)),vimage(B,A,G)) = vimage(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)) ).
% set.comp
tff(fact_3121_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(C,B),G: fun(B,A),Set: set(A)] : aa(set(B),set(C),vimage(C,B,F),aa(set(A),set(B),vimage(B,A,G),Set)) = aa(set(A),set(C),vimage(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),Set) ).
% set.compositionality
tff(fact_3122_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C),X2: set(C)] : aa(set(B),set(A),vimage(A,B,F),aa(set(C),set(B),vimage(B,C,G),X2)) = aa(set(C),set(A),vimage(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),X2) ).
% vimage_comp
tff(fact_3123_inf__unit__def,axiom,
! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,inf_inf(product_unit,Uu),Uv) = product_Unity ).
% inf_unit_def
tff(fact_3124_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B)),X: A,Xa3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),inf_inf(fun(A,fun(B,bool)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R4)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)),X),Xa3))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),R4),S2))) ) ).
% inf_Int_eq2
tff(fact_3125_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),vimage(B,A,F),A4)) = A4 ) ) ).
% surj_image_vimage_eq
tff(fact_3126_set_Oidentity,axiom,
! [A: $tType] : vimage(A,A,aTP_Lamp_ae(A,A)) = id(set(A)) ).
% set.identity
tff(fact_3127_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,product_prod(B,C)),A4: set(B),B5: set(C)] : aa(set(product_prod(B,C)),set(A),vimage(A,product_prod(B,C),F),product_Sigma(B,C,A4,aTP_Lamp_mz(set(C),fun(B,set(C)),B5))) = aa(set(A),set(A),inf_inf(set(A),aa(set(B),set(A),vimage(A,B,aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F)),A4)),aa(set(C),set(A),vimage(A,C,aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F)),B5)) ).
% vimage_Times
tff(fact_3128_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X2: B,A4: set(B),F: fun(B,set(A))] :
( ( pp(member(B,X2,A4))
=> ( aa(set(product_prod(B,A)),set(A),vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2)),product_Sigma(B,A,A4,F)) = aa(B,set(A),F,X2) ) )
& ( ~ pp(member(B,X2,A4))
=> ( aa(set(product_prod(B,A)),set(A),vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2)),product_Sigma(B,A,A4,F)) = bot_bot(set(A)) ) ) ) ).
% Pair_vimage_Sigma
tff(fact_3129_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(A)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( ( aa(set(A),set(B),vimage(B,A,F),A4) = bot_bot(set(B)) )
<=> ( A4 = bot_bot(set(A)) ) ) ) ).
% surj_vimage_empty
tff(fact_3130_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),B5: set(A),A4: set(B)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),vimage(B,A,F),B5)),A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F),A4))) ) ) ).
% vimage_subsetD
tff(fact_3131_vimage__fst,axiom,
! [B: $tType,A: $tType,A4: set(A)] : aa(set(A),set(product_prod(A,B)),vimage(product_prod(A,B),A,product_fst(A,B)),A4) = product_Sigma(A,B,A4,aTP_Lamp_jn(A,set(B))) ).
% vimage_fst
tff(fact_3132_vimage__snd,axiom,
! [A: $tType,B: $tType,A4: set(B)] : aa(set(B),set(product_prod(A,B)),vimage(product_prod(A,B),B,product_snd(A,B)),A4) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_jm(set(B),fun(A,set(B)),A4)) ).
% vimage_snd
tff(fact_3133_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A4: set(A)] :
( ~ finite_finite2(A,A4)
=> ( lattic7752659483105999362nf_fin(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).
% Inf_fin.infinite
tff(fact_3134_rtrancl__last__visit__node,axiom,
! [A: $tType,S: A,S3: A,R4: set(product_prod(A,A)),Sh: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S),S3),transitive_rtrancl(A,R4)))
=> ( ( ( S != Sh )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S),S3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_na(A,fun(A,set(A)),Sh)))))) )
| ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),S),Sh),transitive_rtrancl(A,R4)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Sh),S3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_na(A,fun(A,set(A)),Sh)))))) ) ) ) ).
% rtrancl_last_visit_node
tff(fact_3135_floor__rat__def,axiom,
! [X2: rat] : archim6421214686448440834_floor(rat,X2) = the(int,aTP_Lamp_nb(rat,fun(int,bool),X2)) ).
% floor_rat_def
tff(fact_3136_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
& pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),one_one(int))))) ) ).
% rat_floor_lemma
tff(fact_3137_num__of__integer_Otransfer,axiom,
pp(aa(fun(code_integer,num),bool,aa(fun(int,num),fun(fun(code_integer,num),bool),bNF_rel_fun(int,code_integer,num,num,code_pcr_integer,fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),code_num_of_integer)) ).
% num_of_integer.transfer
tff(fact_3138_prod__decode__triangle__add,axiom,
! [K: nat,M: nat] : nat_prod_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M)) = nat_prod_decode_aux(K,M) ).
% prod_decode_triangle_add
tff(fact_3139_wf__insert,axiom,
! [A: $tType,Y: A,X2: A,R: set(product_prod(A,A))] :
( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2)),R))
<=> ( wf(A,R)
& ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R))) ) ) ).
% wf_insert
tff(fact_3140_add__rat,axiom,
! [B2: int,D3: int,A3: int,C2: int] :
( ( B2 != zero_zero(int) )
=> ( ( D3 != zero_zero(int) )
=> ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D3)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D3)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D3)) ) ) ) ).
% add_rat
tff(fact_3141_r__le__rtrancl,axiom,
! [A: $tType,S2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S2),transitive_rtrancl(A,S2))) ).
% r_le_rtrancl
tff(fact_3142_rtrancl__mono__mp,axiom,
! [A: $tType,U3: set(product_prod(A,A)),V5: set(product_prod(A,A)),X2: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U3),V5))
=> ( pp(member(product_prod(A,A),X2,transitive_rtrancl(A,U3)))
=> pp(member(product_prod(A,A),X2,transitive_rtrancl(A,V5))) ) ) ).
% rtrancl_mono_mp
tff(fact_3143_rtrancl__mono__rightI,axiom,
! [A: $tType,S2: set(product_prod(A,A)),S5: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S2),S5))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S2),transitive_rtrancl(A,S5))) ) ).
% rtrancl_mono_rightI
tff(fact_3144_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,R4)))
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),Vh),R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Vh),V),transitive_rtrancl(A,R4))) ) ) ) ) ).
% converse_rtranclE'
tff(fact_3145_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R)))
=> ( ( A22 != A1 )
=> ~ ! [B3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B3),transitive_rtrancl(A,R)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22),R)) ) ) ) ).
% rtrancl.cases
tff(fact_3146_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_rtrancl(A,R)))
<=> ( ? [A7: A] :
( ( A1 = A7 )
& ( A22 = A7 ) )
| ? [A7: A,B7: A,C4: A] :
( ( A1 = A7 )
& ( A22 = C4 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),transitive_rtrancl(A,R)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4),R)) ) ) ) ).
% rtrancl.simps
tff(fact_3147_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3),transitive_rtrancl(A,R))) ).
% rtrancl.rtrancl_refl
tff(fact_3148_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_rtrancl(A,R))) ) ) ).
% rtrancl.rtrancl_into_rtrancl
tff(fact_3149_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R)))
=> ( ( A3 != B2 )
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y4),transitive_rtrancl(A,R)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),B2),R)) ) ) ) ).
% rtranclE
tff(fact_3150_rtrancl__trans,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),Z2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_rtrancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),transitive_rtrancl(A,R))) ) ) ).
% rtrancl_trans
tff(fact_3151_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R)))
=> ( pp(aa(A,bool,P,A3))
=> ( ! [Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y4),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> ( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,P,Z3)) ) ) )
=> pp(aa(A,bool,P,B2)) ) ) ) ).
% rtrancl_induct
tff(fact_3152_converse__rtranclE,axiom,
! [A: $tType,X2: A,Z2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),transitive_rtrancl(A,R)))
=> ( ( X2 != Z2 )
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),R))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),transitive_rtrancl(A,R))) ) ) ) ).
% converse_rtranclE
tff(fact_3153_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R)))
=> ( pp(aa(A,bool,P,B2))
=> ( ! [Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_rtrancl(A,R)))
=> ( pp(aa(A,bool,P,Z3))
=> pp(aa(A,bool,P,Y4)) ) ) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% converse_rtrancl_induct
tff(fact_3154_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_rtrancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_rtrancl(A,R))) ) ) ).
% converse_rtrancl_into_rtrancl
tff(fact_3155_in__rtrancl__insert,axiom,
! [A: $tType,X2: product_prod(A,A),R4: set(product_prod(A,A)),R: product_prod(A,A)] :
( pp(member(product_prod(A,A),X2,transitive_rtrancl(A,R4)))
=> pp(member(product_prod(A,A),X2,transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),R),R4)))) ) ).
% in_rtrancl_insert
tff(fact_3156_integer_Orep__transfer,axiom,
pp(aa(fun(code_integer,int),bool,aa(fun(int,int),fun(fun(code_integer,int),bool),bNF_rel_fun(int,code_integer,int,int,code_pcr_integer,fequal(int)),aTP_Lamp_ck(int,int)),code_int_of_integer)) ).
% integer.rep_transfer
tff(fact_3157_integer_Oid__abs__transfer,axiom,
pp(aa(fun(int,code_integer),bool,aa(fun(int,int),fun(fun(int,code_integer),bool),bNF_rel_fun(int,int,int,code_integer,fequal(int),code_pcr_integer),aTP_Lamp_ck(int,int)),code_integer_of_int)) ).
% integer.id_abs_transfer
tff(fact_3158_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q))))
=> ( ! [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2),transitive_rtrancl(A,P)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),Q))
=> ( Y4 = X3 ) ) )
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,P))) ) ) ).
% rtrancl_Un_separator_converseE
tff(fact_3159_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set(product_prod(A,A)),Q: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P),Q))))
=> ( ! [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3),transitive_rtrancl(A,P)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),Q))
=> ( X3 = Y4 ) ) )
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,P))) ) ) ).
% rtrancl_Un_separatorE
tff(fact_3160_zero__integer_Otransfer,axiom,
pp(aa(code_integer,bool,aa(int,fun(code_integer,bool),code_pcr_integer,zero_zero(int)),zero_zero(code_integer))) ).
% zero_integer.transfer
tff(fact_3161_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By))
=> ( ! [A5: A,B3: B,Aa2: A,Ba: B] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R))
=> ( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay)) ) ) ) ).
% converse_rtrancl_induct2
tff(fact_3162_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)),transitive_rtrancl(product_prod(A,B),R)))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
=> ~ ! [A5: A,B3: B] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),R))
=> ~ pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)),transitive_rtrancl(product_prod(A,B),R))) ) ) ) ).
% converse_rtranclE2
tff(fact_3163_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay))
=> ( ! [A5: A,B3: B,Aa2: A,Ba: B] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).
% rtrancl_induct2
tff(fact_3164_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R4: set(product_prod(A,A)),X2: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R4)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),X2),R4)))) ).
% rtrancl_sub_insert_rtrancl
tff(fact_3165_plus__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),plus_plus(int)),plus_plus(code_integer))) ).
% plus_integer.transfer
tff(fact_3166_quotient__of__eq,axiom,
! [A3: int,B2: int,P3: int,Q3: int] :
( ( quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,P3),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).
% quotient_of_eq
tff(fact_3167_less__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(code_integer,fun(code_integer,bool)),bool),bNF_rel_fun(int,code_integer,fun(int,bool),fun(code_integer,bool),code_pcr_integer,bNF_rel_fun(int,code_integer,bool,bool,code_pcr_integer,fequal(bool))),ord_less(int)),ord_less(code_integer))) ).
% less_integer.transfer
tff(fact_3168_one__integer_Otransfer,axiom,
pp(aa(code_integer,bool,aa(int,fun(code_integer,bool),code_pcr_integer,one_one(int)),one_one(code_integer))) ).
% one_integer.transfer
tff(fact_3169_less__eq__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,bool)),bool,aa(fun(int,fun(int,bool)),fun(fun(code_integer,fun(code_integer,bool)),bool),bNF_rel_fun(int,code_integer,fun(int,bool),fun(code_integer,bool),code_pcr_integer,bNF_rel_fun(int,code_integer,bool,bool,code_pcr_integer,fequal(bool))),ord_less_eq(int)),ord_less_eq(code_integer))) ).
% less_eq_integer.transfer
tff(fact_3170_uminus__integer_Otransfer,axiom,
pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),uminus_uminus(int)),uminus_uminus(code_integer))) ).
% uminus_integer.transfer
tff(fact_3171_times__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),times_times(int)),times_times(code_integer))) ).
% times_integer.transfer
tff(fact_3172_minus__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),minus_minus(int)),minus_minus(code_integer))) ).
% minus_integer.transfer
tff(fact_3173_abs__integer_Otransfer,axiom,
pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),abs_abs(int)),abs_abs(code_integer))) ).
% abs_integer.transfer
tff(fact_3174_nat__of__integer_Otransfer,axiom,
pp(aa(fun(code_integer,nat),bool,aa(fun(int,nat),fun(fun(code_integer,nat),bool),bNF_rel_fun(int,code_integer,nat,nat,code_pcr_integer,fequal(nat)),nat2),code_nat_of_integer)) ).
% nat_of_integer.transfer
tff(fact_3175_normalize__eq,axiom,
! [A3: int,B2: int,P3: int,Q3: int] :
( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,P3),Q3) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).
% normalize_eq
tff(fact_3176_integer__of__nat_Otransfer,axiom,
pp(aa(fun(nat,code_integer),bool,aa(fun(nat,int),fun(fun(nat,code_integer),bool),bNF_rel_fun(nat,nat,int,code_integer,fequal(nat),code_pcr_integer),semiring_1_of_nat(int)),code_integer_of_nat)) ).
% integer_of_nat.transfer
tff(fact_3177_divide__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),divide_divide(int)),divide_divide(code_integer))) ).
% divide_integer.transfer
tff(fact_3178_modulo__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),modulo_modulo(int)),modulo_modulo(code_integer))) ).
% modulo_integer.transfer
tff(fact_3179_sgn__integer_Otransfer,axiom,
pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),sgn_sgn(int)),sgn_sgn(code_integer))) ).
% sgn_integer.transfer
tff(fact_3180_dup_Otransfer,axiom,
pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),aTP_Lamp_hf(int,int)),code_dup)) ).
% dup.transfer
tff(fact_3181_take__bit__integer_Otransfer,axiom,
pp(aa(fun(nat,fun(code_integer,code_integer)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se2584673776208193580ke_bit(int)),bit_se2584673776208193580ke_bit(code_integer))) ).
% take_bit_integer.transfer
tff(fact_3182_not__integer_Otransfer,axiom,
pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),bit_ri4277139882892585799ns_not(int)),bit_ri4277139882892585799ns_not(code_integer))) ).
% not_integer.transfer
tff(fact_3183_and__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se5824344872417868541ns_and(int)),bit_se5824344872417868541ns_and(code_integer))) ).
% and_integer.transfer
tff(fact_3184_bit__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(nat,bool)),bool,aa(fun(int,fun(nat,bool)),fun(fun(code_integer,fun(nat,bool)),bool),bNF_rel_fun(int,code_integer,fun(nat,bool),fun(nat,bool),code_pcr_integer,fequal(fun(nat,bool))),bit_se5641148757651400278ts_bit(int)),bit_se5641148757651400278ts_bit(code_integer))) ).
% bit_integer.transfer
tff(fact_3185_or__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se1065995026697491101ons_or(int)),bit_se1065995026697491101ons_or(code_integer))) ).
% or_integer.transfer
tff(fact_3186_xor__integer_Otransfer,axiom,
pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se5824344971392196577ns_xor(int)),bit_se5824344971392196577ns_xor(code_integer))) ).
% xor_integer.transfer
tff(fact_3187_drop__bit__integer_Otransfer,axiom,
pp(aa(fun(nat,fun(code_integer,code_integer)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se4197421643247451524op_bit(int)),bit_se4197421643247451524op_bit(code_integer))) ).
% drop_bit_integer.transfer
tff(fact_3188_push__bit__integer_Otransfer,axiom,
pp(aa(fun(nat,fun(code_integer,code_integer)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se4730199178511100633sh_bit(int)),bit_se4730199178511100633sh_bit(code_integer))) ).
% push_bit_integer.transfer
tff(fact_3189_mask__integer_Otransfer,axiom,
pp(aa(fun(nat,code_integer),bool,aa(fun(nat,int),fun(fun(nat,code_integer),bool),bNF_rel_fun(nat,nat,int,code_integer,fequal(nat),code_pcr_integer),bit_se2239418461657761734s_mask(int)),bit_se2239418461657761734s_mask(code_integer))) ).
% mask_integer.transfer
tff(fact_3190_unset__bit__integer_Otransfer,axiom,
pp(aa(fun(nat,fun(code_integer,code_integer)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(code_integer))) ).
% unset_bit_integer.transfer
tff(fact_3191_set__bit__integer_Otransfer,axiom,
pp(aa(fun(nat,fun(code_integer,code_integer)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(code_integer))) ).
% set_bit_integer.transfer
tff(fact_3192_division__segment__integer_Otransfer,axiom,
pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),euclid7384307370059645450egment(int)),euclid7384307370059645450egment(code_integer))) ).
% division_segment_integer.transfer
tff(fact_3193_flip__bit__integer_Otransfer,axiom,
pp(aa(fun(nat,fun(code_integer,code_integer)),bool,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(code_integer,code_integer)),bool),bNF_rel_fun(nat,nat,fun(int,int),fun(code_integer,code_integer),fequal(nat),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(code_integer))) ).
% flip_bit_integer.transfer
tff(fact_3194_euclidean__size__integer_Otransfer,axiom,
pp(aa(fun(code_integer,nat),bool,aa(fun(int,nat),fun(fun(code_integer,nat),bool),bNF_rel_fun(int,code_integer,nat,nat,code_pcr_integer,fequal(nat)),euclid6346220572633701492n_size(int)),euclid6346220572633701492n_size(code_integer))) ).
% euclidean_size_integer.transfer
tff(fact_3195_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),A4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4))))
=> ( ( A3 = B2 )
| pp(member(A,A3,A4)) ) ) ) ).
% trancl_subset_Sigma_aux
tff(fact_3196_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W: A,E4: set(product_prod(A,A)),U3: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),E4),product_Sigma(A,A,U3,aTP_Lamp_ma(set(A),fun(A,set(A)),U3))))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_rtrancl(A,E4))) ) ).
% Restr_rtrancl_mono
tff(fact_3197_quotient__of__Fract,axiom,
! [A3: int,B2: int] : quotient_of(aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) ).
% quotient_of_Fract
tff(fact_3198_sub_Otransfer,axiom,
pp(aa(fun(num,fun(num,code_integer)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,code_integer)),bool),bNF_rel_fun(num,num,fun(num,int),fun(num,code_integer),fequal(num),bNF_rel_fun(num,num,int,code_integer,fequal(num),code_pcr_integer)),aTP_Lamp_hn(num,fun(num,int))),code_sub)) ).
% sub.transfer
tff(fact_3199_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E4: set(product_prod(A,A)),F: fun(A,B)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,E4)))
=> pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,A3)),aa(A,B,F,B2)),transitive_rtrancl(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),image2(product_prod(A,A),product_prod(B,B),pairself(A,B,F)),E4)))) ) ).
% rtrancl_mapI
tff(fact_3200_Fract_Oabs__eq,axiom,
! [Xa: int,X2: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa),X2) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),X2),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),X2))) ).
% Fract.abs_eq
tff(fact_3201_rtrancl__last__touch,axiom,
! [A: $tType,Q3: A,Q4: A,R4: set(product_prod(A,A)),S2: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4),transitive_rtrancl(A,R4)))
=> ( pp(member(A,Q3,S2))
=> ~ ! [Qt: A] :
( pp(member(A,Qt,S2))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_rtrancl(A,R4)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_ma(set(A),fun(A,set(A)),S2)))))) ) ) ) ) ).
% rtrancl_last_touch
tff(fact_3202_rtrancl__last__visit_H,axiom,
! [A: $tType,Q3: A,Q4: A,R4: set(product_prod(A,A)),S2: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4),transitive_rtrancl(A,R4)))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_ma(set(A),fun(A,set(A)),S2))))))
=> ~ ! [Qt: A] :
( pp(member(A,Qt,S2))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_rtrancl(A,R4)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_ma(set(A),fun(A,set(A)),S2)))))) ) ) ) ) ).
% rtrancl_last_visit'
tff(fact_3203_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_nc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).
% rtrancl_insert
tff(fact_3204_Fract_Otransfer,axiom,
pp(aa(fun(int,fun(int,rat)),bool,aa(fun(int,fun(int,product_prod(int,int))),fun(fun(int,fun(int,rat)),bool),bNF_rel_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),fequal(int),bNF_rel_fun(int,int,product_prod(int,int),rat,fequal(int),pcr_rat)),aTP_Lamp_kx(int,fun(int,product_prod(int,int)))),fract)) ).
% Fract.transfer
tff(fact_3205_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N != zero_zero(int) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)),N) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(int,rat,aa(int,fun(int,rat),fract,M),N)),one_one(rat)) ) ) ).
% Fract_add_one
tff(fact_3206_Fract__def,axiom,
fract = aa(fun(int,fun(int,product_prod(int,int))),fun(int,fun(int,rat)),map_fun(int,int,fun(int,product_prod(int,int)),fun(int,rat),id(int),map_fun(int,int,product_prod(int,int),rat,id(int),abs_Rat)),aTP_Lamp_kx(int,fun(int,product_prod(int,int)))) ).
% Fract_def
tff(fact_3207_old_Orec__unit__def,axiom,
! [T: $tType,X: T,Xa3: product_unit] : product_rec_unit(T,X,Xa3) = the(T,product_rec_set_unit(T,X,Xa3)) ).
% old.rec_unit_def
tff(fact_3208_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType,X: fun(A,fun(B,T)),Xa3: product_prod(A,B)] : product_rec_prod(A,B,T,X,Xa3) = the(T,product_rec_set_prod(A,B,T,X,Xa3)) ).
% old.rec_prod_def
tff(fact_3209_list__decode_Opinduct,axiom,
! [A0: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),A0))
=> ( ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat)))
=> pp(aa(nat,bool,P,zero_zero(nat))) )
=> ( ! [N3: nat] :
( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N3)))
=> ( ! [X: nat,Y5: nat] :
( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y5) = nat_prod_decode(N3) )
=> pp(aa(nat,bool,P,Y5)) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) )
=> pp(aa(nat,bool,P,A0)) ) ) ) ).
% list_decode.pinduct
tff(fact_3210_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),E4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_ma(set(A),fun(A,set(A)),R4))))))
=> ( ~ pp(member(A,U,R4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,rel_restrict(A,E4,R4)))) ) ) ).
% rtrancl_restrictI
tff(fact_3211_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),A3: B] :
( inj_on(A,B,F,A4)
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),inf_inf(set(A),aa(set(B),set(A),vimage(A,B,F),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),A4)),aa(set(A),set(A),insert2(A,the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_nd(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F),A4),A3))),bot_bot(set(A))))) ) ).
% inj_on_vimage_singleton
tff(fact_3212_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : inj_on(A,B,F,bot_bot(set(A))) ).
% inj_on_empty
tff(fact_3213_inj__uminus,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A4: set(A)] : inj_on(A,A,uminus_uminus(A),A4) ) ).
% inj_uminus
tff(fact_3214_rel__restrict__empty,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : rel_restrict(A,R4,bot_bot(set(A))) = R4 ).
% rel_restrict_empty
tff(fact_3215_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ne(A,fun(B,fun(A,fun(B,bool))),X2),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y) ).
% The_split_eq
tff(fact_3216_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] :
( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),top_top(set(product_prod(A,B))))
<=> inj_on(A,C,F,top_top(set(A))) ) ).
% inj_apfst
tff(fact_3217_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] :
( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F),top_top(set(product_prod(A,B))))
<=> inj_on(B,C,F,top_top(set(B))) ) ).
% inj_apsnd
tff(fact_3218_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C),A4: set(A)] :
( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),product_Sigma(A,B,A4,aTP_Lamp_jn(A,set(B))))
<=> inj_on(A,C,F,A4) ) ).
% inj_on_apfst
tff(fact_3219_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C),A4: set(B)] :
( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_jm(set(B),fun(A,set(B)),A4)))
<=> inj_on(B,C,F,A4) ) ).
% inj_on_apsnd
tff(fact_3220_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,A4: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),insert2(A,A3),A4))
<=> ( inj_on(A,B,F,A4)
& ~ pp(member(B,aa(A,B,F,A3),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))))) ) ) ).
% inj_on_insert
tff(fact_3221_inj__on__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( ( inj_on(A,B,F,A4)
| inj_on(A,B,F,B5) )
=> inj_on(A,B,F,aa(set(A),set(A),inf_inf(set(A),A4),B5)) ) ).
% inj_on_Int
tff(fact_3222_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),top_top(set(fun(D,A)))) ) ).
% fun.inj_map
tff(fact_3223_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F2: fun(C,B),F: fun(A,C),A4: set(A)] :
( inj_on(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F2),F),A4)
=> inj_on(A,C,F,A4) ) ).
% inj_on_imageI2
tff(fact_3224_inj__on__of__nat,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N5: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N5) ) ).
% inj_on_of_nat
tff(fact_3225_inj__on__add,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,A4: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),A4) ) ).
% inj_on_add
tff(fact_3226_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A4: set(A),G: fun(A,B),F: fun(A,A)] :
( ! [X3: A] :
( pp(member(A,X3,A4))
=> ! [Xa4: A] :
( pp(member(A,Xa4,A4))
=> ( ( aa(A,B,G,aa(A,A,F,X3)) = aa(A,B,G,aa(A,A,F,Xa4)) )
<=> ( aa(A,B,G,X3) = aa(A,B,G,Xa4) ) ) ) )
=> ( inj_on(A,A,F,A4)
=> ( inj_on(A,B,G,aa(set(A),set(A),image2(A,A,F),A4))
<=> inj_on(A,B,G,A4) ) ) ) ).
% inj_on_image_iff
tff(fact_3227_inj__on__id2,axiom,
! [A: $tType,A4: set(A)] : inj_on(A,A,aTP_Lamp_ae(A,A),A4) ).
% inj_on_id2
tff(fact_3228_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A4: set(A),G: fun(B,A),F: fun(A,B)] :
( ! [X3: A] :
( pp(member(A,X3,A4))
=> ( aa(B,A,G,aa(A,B,F,X3)) = X3 ) )
=> inj_on(A,B,F,A4) ) ).
% inj_on_inverseI
tff(fact_3229_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: A,Y: A] :
( inj_on(A,B,F,A4)
=> ( ( X2 != Y )
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( aa(A,B,F,X2) != aa(A,B,F,Y) ) ) ) ) ) ).
% inj_on_contraD
tff(fact_3230_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: A,Y: A] :
( inj_on(A,B,F,A4)
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( ( aa(A,B,F,X2) = aa(A,B,F,Y) )
<=> ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
tff(fact_3231_inj__on__cong,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B),G: fun(A,B)] :
( ! [A5: A] :
( pp(member(A,A5,A4))
=> ( aa(A,B,F,A5) = aa(A,B,G,A5) ) )
=> ( inj_on(A,B,F,A4)
<=> inj_on(A,B,G,A4) ) ) ).
% inj_on_cong
tff(fact_3232_inj__on__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,A4)
<=> ! [X4: A] :
( pp(member(A,X4,A4))
=> ! [Xa2: A] :
( pp(member(A,Xa2,A4))
=> ( ( aa(A,B,F,X4) = aa(A,B,F,Xa2) )
=> ( X4 = Xa2 ) ) ) ) ) ).
% inj_on_def
tff(fact_3233_inj__onI,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,A4))
=> ( pp(member(A,Y4,A4))
=> ( ( aa(A,B,F,X3) = aa(A,B,F,Y4) )
=> ( X3 = Y4 ) ) ) )
=> inj_on(A,B,F,A4) ) ).
% inj_onI
tff(fact_3234_inj__onD,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: A,Y: A] :
( inj_on(A,B,F,A4)
=> ( ( aa(A,B,F,X2) = aa(A,B,F,Y) )
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
tff(fact_3235_inj__on__diff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,A4)
=> inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) ) ).
% inj_on_diff
tff(fact_3236_inj__on__id,axiom,
! [A: $tType,A4: set(A)] : inj_on(A,A,id(A),A4) ).
% inj_on_id
tff(fact_3237_inj__Some,axiom,
! [A: $tType,A4: set(A)] : inj_on(A,option(A),some(A),A4) ).
% inj_Some
tff(fact_3238_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A4))
=> inj_on(A,B,F,B5) ) ) ).
% inj_on_subset
tff(fact_3239_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B5: set(A),A4: set(A)] :
( inj_on(A,B,F,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
=> inj_on(A,B,F,A4) ) ) ).
% subset_inj_on
tff(fact_3240_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(A,fun(C,B),aTP_Lamp_nf(fun(A,B),fun(A,fun(C,B)),F),top_top(set(A))) ) ).
% inj_fun
tff(fact_3241_inj__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
<=> ! [X4: A,Y3: A] :
( ( aa(A,B,F,X4) = aa(A,B,F,Y3) )
=> ( X4 = Y3 ) ) ) ).
% inj_def
tff(fact_3242_inj__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X2: A,Y: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(A,B,F,X2) = aa(A,B,F,Y) )
<=> ( X2 = Y ) ) ) ).
% inj_eq
tff(fact_3243_injI,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( ! [X3: A,Y4: A] :
( ( aa(A,B,F,X3) = aa(A,B,F,Y4) )
=> ( X3 = Y4 ) )
=> inj_on(A,B,F,top_top(set(A))) ) ).
% injI
tff(fact_3244_injD,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X2: A,Y: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(A,B,F,X2) = aa(A,B,F,Y) )
=> ( X2 = Y ) ) ) ).
% injD
tff(fact_3245_inj__swap,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_swap(A,B),A4) ).
% inj_swap
tff(fact_3246_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X7: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_ng(fun(A,B),fun(A,product_prod(A,B)),F),X7) ).
% inj_on_convol_ident
tff(fact_3247_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: fun(A,B),S2: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_ng(fun(A,B),fun(A,product_prod(A,B)),C2),S2) ).
% inj_Pair(1)
tff(fact_3248_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: fun(A,B),S2: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_nh(fun(A,B),fun(A,product_prod(B,A)),C2),S2) ).
% inj_Pair(2)
tff(fact_3249_inj__on__add_H,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,A4: set(A)] : inj_on(A,A,aTP_Lamp_ni(A,fun(A,A),A3),A4) ) ).
% inj_on_add'
tff(fact_3250_rel__restrict__lift,axiom,
! [A: $tType,X2: A,Y: A,E4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),rel_restrict(A,E4,R4)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),E4)) ) ).
% rel_restrict_lift
tff(fact_3251_rel__restrictI,axiom,
! [A: $tType,X2: A,R4: set(A),Y: A,E4: set(product_prod(A,A))] :
( ~ pp(member(A,X2,R4))
=> ( ~ pp(member(A,Y,R4))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),E4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),rel_restrict(A,E4,R4))) ) ) ) ).
% rel_restrictI
tff(fact_3252_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X2: A,Y: A,A4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),rel_restrict(A,A4,R4)))
=> ~ pp(member(A,X2,R4)) ) ).
% rel_restrict_notR(1)
tff(fact_3253_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X2: A,Y: A,A4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),rel_restrict(A,A4,R4)))
=> ~ pp(member(A,Y,R4)) ) ).
% rel_restrict_notR(2)
tff(fact_3254_rel__restrict__union,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A),B5: set(A)] : rel_restrict(A,R4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = rel_restrict(A,rel_restrict(A,R4,A4),B5) ).
% rel_restrict_union
tff(fact_3255_rel__restrict__sub,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R4,A4)),R4)) ).
% rel_restrict_sub
tff(fact_3256_rel__restrict__mono,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B5: set(product_prod(A,A)),R4: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A4),B5))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A4,R4)),rel_restrict(A,B5,R4))) ) ).
% rel_restrict_mono
tff(fact_3257_finite__rel__restrict,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] :
( finite_finite2(product_prod(A,A),R4)
=> finite_finite2(product_prod(A,A),rel_restrict(A,R4,A4)) ) ).
% finite_rel_restrict
tff(fact_3258_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),top_top(set(option(A)))) ) ).
% option.inj_map
tff(fact_3259_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( order(A)
=> ! [A4: set(A),F: fun(A,B)] :
( ! [X3: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
=> ( pp(member(A,X3,A4))
=> ( pp(member(A,Y4,A4))
=> ( aa(A,B,F,X3) != aa(A,B,F,Y4) ) ) ) )
=> ( ! [X3: A,Y4: A] :
( pp(member(A,X3,A4))
=> ( pp(member(A,Y4,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) ) ) )
=> inj_on(A,B,F,A4) ) ) ) ).
% linorder_inj_onI
tff(fact_3260_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [F: fun(A,B)] :
( ! [X3: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
=> ( aa(A,B,F,X3) != aa(A,B,F,Y4) ) )
=> inj_on(A,B,F,top_top(set(A))) ) ) ).
% linorder_injI
tff(fact_3261_inj__add__left,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) ) ).
% inj_add_left
tff(fact_3262_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C5))
=> ( ( aa(set(A),set(B),image2(A,B,F),A4) = aa(set(A),set(B),image2(A,B,F),B5) )
<=> ( A4 = B5 ) ) ) ) ) ).
% inj_on_image_eq_iff
tff(fact_3263_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B5: set(A),A3: A,A4: set(A)] :
( inj_on(A,B,F,B5)
=> ( pp(member(A,A3,B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
=> ( pp(member(B,aa(A,B,F,A3),aa(set(A),set(B),image2(A,B,F),A4)))
<=> pp(member(A,A3,A4)) ) ) ) ) ).
% inj_on_image_mem_iff
tff(fact_3264_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: B,B5: set(B)] :
( inj_on(A,B,F,A4)
=> ( ~ pp(member(B,X2,B5))
=> ( ( aa(set(B),set(B),insert2(B,X2),B5) = aa(set(A),set(B),image2(A,B,F),A4) )
=> ~ ! [X5: A,A9: set(A)] :
( ~ pp(member(A,X5,A9))
=> ( ( A4 = aa(set(A),set(A),insert2(A,X5),A9) )
=> ( ( X2 = aa(A,B,F,X5) )
=> ( B5 != aa(set(A),set(B),image2(A,B,F),A9) ) ) ) ) ) ) ) ).
% inj_img_insertE
tff(fact_3265_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B2: B] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(member(B,B2,aa(set(A),set(B),image2(A,B,F),top_top(set(A)))))
<=> ? [X4: A] :
( ( B2 = aa(A,B,F,X4) )
& ! [Y3: A] :
( ( B2 = aa(A,B,F,Y3) )
=> ( Y3 = X4 ) ) ) ) ) ).
% range_ex1_eq
tff(fact_3266_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(set(A),set(B),image2(A,B,F),A4) = aa(set(A),set(B),image2(A,B,F),B5) )
<=> ( A4 = B5 ) ) ) ).
% inj_image_eq_iff
tff(fact_3267_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,A4: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(member(B,aa(A,B,F,A3),aa(set(A),set(B),image2(A,B,F),A4)))
<=> pp(member(A,A3,A4)) ) ) ).
% inj_image_mem_iff
tff(fact_3268_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),A4: set(A),G: fun(B,C)] :
( inj_on(A,B,F,A4)
=> ( inj_on(B,C,G,aa(set(A),set(B),image2(A,B,F),A4))
=> inj_on(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),A4) ) ) ).
% comp_inj_on
tff(fact_3269_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(C,B),F: fun(A,C),A4: set(A)] :
( inj_on(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),F),A4)
=> inj_on(C,B,G,aa(set(A),set(C),image2(A,C,F),A4)) ) ).
% inj_on_imageI
tff(fact_3270_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),A4: set(A),F2: fun(B,C)] :
( inj_on(A,B,F,A4)
=> ( inj_on(B,C,F2,aa(set(A),set(B),image2(A,B,F),A4))
<=> inj_on(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F),A4) ) ) ).
% comp_inj_on_iff
tff(fact_3271_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))
=> ( ( aa(set(A),set(B),image2(A,B,F),A4) = aa(set(A),set(B),image2(A,B,F),B5) )
<=> ( A4 = B5 ) ) ) ).
% inj_on_Un_image_eq_iff
tff(fact_3272_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( inj_on(C,A,G,top_top(set(C)))
=> inj_on(C,B,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G),top_top(set(C))) ) ) ).
% inj_compose
tff(fact_3273_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B5: set(A),A4: set(A)] :
( inj_on(A,B,F,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B5))) ) ) ).
% inj_on_strict_subset
tff(fact_3274_inj__on__fun__updI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),Y: B,X2: A] :
( inj_on(A,B,F,A4)
=> ( ~ pp(member(B,Y,aa(set(A),set(B),image2(A,B,F),A4)))
=> inj_on(A,B,fun_upd(A,B,F,X2,Y),A4) ) ) ).
% inj_on_fun_updI
tff(fact_3275_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M)) ).
% inj_on_fst_map_to_set
tff(fact_3276_inj__diff__right,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : inj_on(A,A,aTP_Lamp_go(A,fun(A,A),A3),top_top(set(A))) ) ).
% inj_diff_right
tff(fact_3277_inj__singleton,axiom,
! [A: $tType,A4: set(A)] : inj_on(A,set(A),aTP_Lamp_nj(A,set(A)),A4) ).
% inj_singleton
tff(fact_3278_finite__Collect,axiom,
! [A: $tType,B: $tType,S2: set(A),F: fun(B,A)] :
( finite_finite2(A,S2)
=> ( inj_on(B,A,F,top_top(set(B)))
=> finite_finite2(B,aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_nk(set(A),fun(fun(B,A),fun(B,bool)),S2),F))) ) ) ).
% finite_Collect
tff(fact_3279_swap__inj__on,axiom,
! [B: $tType,A: $tType,A4: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_it(A,fun(B,product_prod(B,A)))),A4) ).
% swap_inj_on
tff(fact_3280_rel__restrict__mono2,axiom,
! [A: $tType,R4: set(A),S2: set(A),A4: set(product_prod(A,A))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R4),S2))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,A4,S2)),rel_restrict(A,A4,R4))) ) ).
% rel_restrict_mono2
tff(fact_3281_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B5)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5)) ) ) ).
% inj_image_subset_iff
tff(fact_3282_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C5))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),inf_inf(set(A),A4),B5)) = aa(set(B),set(B),inf_inf(set(B),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B5)) ) ) ) ) ).
% inj_on_image_Int
tff(fact_3283_image__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),inf_inf(set(A),A4),B5)) = aa(set(B),set(B),inf_inf(set(B),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B5)) ) ) ).
% image_Int
tff(fact_3284_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C5))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B5)) ) ) ) ) ).
% inj_on_image_set_diff
tff(fact_3285_sum_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [H: fun(B,C),A4: set(B),G: fun(C,A)] :
( inj_on(B,C,H,A4)
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A4) ) ) ) ).
% sum.reindex
tff(fact_3286_image__set__diff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,F),B5)) ) ) ).
% image_set_diff
tff(fact_3287_prod_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [H: fun(B,C),A4: set(B),G: fun(C,A)] :
( inj_on(B,C,H,A4)
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A4) ) ) ) ).
% prod.reindex
tff(fact_3288_inj__vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(B),set(A),vimage(A,B,F),aa(set(A),set(B),image2(A,B,F),A4)) = A4 ) ) ).
% inj_vimage_image_eq
tff(fact_3289_rel__restrict__def,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : rel_restrict(A,R4,A4) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_nl(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),R4),A4))) ).
% rel_restrict_def
tff(fact_3290_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A4: set(A)] :
( finite_finite2(A,A4)
=> ~ ! [F3: fun(A,nat)] :
( ? [N3: nat] : aa(set(A),set(nat),image2(A,nat,F3),A4) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_cw(nat,fun(nat,bool),N3))
=> ~ inj_on(A,nat,F3,A4) ) ) ).
% finite_imp_inj_to_nat_seg'
tff(fact_3291_set__to__map__inverse,axiom,
! [B: $tType,A: $tType,S2: set(product_prod(A,B))] :
( inj_on(product_prod(A,B),A,product_fst(A,B),S2)
=> ( map_to_set(A,B,set_to_map(A,B,S2)) = S2 ) ) ).
% set_to_map_inverse
tff(fact_3292_rel__restrict__Int__empty,axiom,
! [A: $tType,A4: set(A),R4: set(product_prod(A,A))] :
( ( aa(set(A),set(A),inf_inf(set(A),A4),field2(A,R4)) = bot_bot(set(A)) )
=> ( rel_restrict(A,R4,A4) = R4 ) ) ).
% rel_restrict_Int_empty
tff(fact_3293_Field__rel__restrict,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),field2(A,rel_restrict(A,R4,A4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),field2(A,R4)),A4))) ).
% Field_rel_restrict
tff(fact_3294_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B5: set(B),A4: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F),A4)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),vimage(A,B,F),B5)),A4)) ) ) ).
% vimage_subsetI
tff(fact_3295_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),uminus_uminus(set(A)),A4))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F),A4)))) ) ).
% inj_image_Compl_subset
tff(fact_3296_The__case__prod,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P)) = the(product_prod(A,B),aTP_Lamp_nm(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P)) ).
% The_case_prod
tff(fact_3297_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),G: fun(A,B),B5: set(A)] :
( inj_on(A,B,F,A4)
=> ( inj_on(A,B,G,B5)
=> ( ( aa(set(B),set(B),inf_inf(set(B),aa(set(A),set(B),image2(A,B,F),A4)),aa(set(A),set(B),image2(A,B,G),B5)) = bot_bot(set(B)) )
=> inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_nn(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) ) ) ) ).
% inj_on_disjoint_Un
tff(fact_3298_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S2: set(product_prod(A,B)),K: A,V: B] :
( inj_on(product_prod(A,B),A,product_fst(A,B),S2)
=> ( ( aa(A,option(B),set_to_map(A,B,S2),K) = aa(B,option(B),some(B),V) )
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V),S2)) ) ) ).
% set_to_map_simp
tff(fact_3299_rel__restrict__compl,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),rel_restrict(A,R4,A4)),rel_restrict(A,R4,aa(set(A),set(A),uminus_uminus(set(A)),A4))) = bot_bot(set(product_prod(A,A))) ).
% rel_restrict_compl
tff(fact_3300_homo__rel__restrict__mono,axiom,
! [A: $tType,R4: set(product_prod(A,A)),B5: set(A),A4: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),product_Sigma(A,A,B5,aTP_Lamp_ma(set(A),fun(A,set(A)),B5))))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,R4,A4)),product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4),aa(set(A),fun(A,set(A)),aTP_Lamp_no(set(A),fun(set(A),fun(A,set(A))),B5),A4)))) ) ).
% homo_rel_restrict_mono
tff(fact_3301_rel__restrict__alt__def,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : rel_restrict(A,R4,A4) = aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R4),product_Sigma(A,A,aa(set(A),set(A),uminus_uminus(set(A)),A4),aTP_Lamp_np(set(A),fun(A,set(A)),A4))) ).
% rel_restrict_alt_def
tff(fact_3302_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))
<=> ( inj_on(A,B,F,A4)
& inj_on(A,B,F,B5)
& ( aa(set(B),set(B),inf_inf(set(B),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),B5))),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A4))) = bot_bot(set(B)) ) ) ) ).
% inj_on_Un
tff(fact_3303_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: B] :
( inj_on(A,B,F,top_top(set(A)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),vimage(A,B,F),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),aa(set(A),set(A),insert2(A,the(A,aa(B,fun(A,bool),aTP_Lamp_nq(fun(A,B),fun(B,fun(A,bool)),F),A3))),bot_bot(set(A))))) ) ).
% inj_vimage_singleton
tff(fact_3304_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C5: set(A),G: fun(A,B),X2: A] :
( pp(member(A,I,C5))
=> ( inj_on(A,B,G,C5)
=> ( aa(A,A,aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_nr(set(A),fun(fun(A,B),fun(A,fun(B,A))),C5),G),X2)),G),I) = aa(A,A,id(A),I) ) ) ) ).
% If_the_inv_into_f_f
tff(fact_3305_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),X2: B,Y: A,Z2: A] :
( ( aa(B,option(A),M,X2) = aa(A,option(A),some(A),Y) )
=> ( inj_on(B,option(A),M,dom(B,A,M))
=> ( ~ pp(member(A,Z2,ran(B,A,M)))
=> ( ran(B,A,fun_upd(B,option(A),M,X2,aa(A,option(A),some(A),Z2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),ran(B,A,M)),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))),aa(set(A),set(A),insert2(A,Z2),bot_bot(set(A)))) ) ) ) ) ).
% ran_map_upd_Some
tff(fact_3306_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),X7: set(A),M: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),product_Sigma(A,A,X7,aTP_Lamp_ns(A,fun(A,set(A)),M))))))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ~ ! [X3: A] :
( pp(member(A,X3,X7))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3),transitive_rtrancl(A,R)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),B2),transitive_rtrancl(A,R))) ) ) ) ) ).
% trancl_multi_insert
tff(fact_3307_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),M: A,X7: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(A),set(A),insert2(A,M),bot_bot(set(A))),aTP_Lamp_ma(set(A),fun(A,set(A)),X7))))))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ~ ! [X3: A] :
( pp(member(A,X3,X7))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),M),transitive_rtrancl(A,R)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),B2),transitive_rtrancl(A,R))) ) ) ) ) ).
% trancl_multi_insert2
tff(fact_3308_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Z2: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(A,A,aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),the_inv_into(A,B,top_top(set(A)),F)),F),Z2) = aa(A,A,id(A),Z2) ) ) ).
% the_inv_f_o_f_id
tff(fact_3309_restrict__map__self,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : restrict_map(A,B,M,dom(A,B,M)) = M ).
% restrict_map_self
tff(fact_3310_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),R4: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P,R4)) = same_fst(A,B,P,aTP_Lamp_nt(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R4)) ).
% same_fst_trancl
tff(fact_3311_dom__eq__empty__conv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B))] :
( ( dom(A,B,F) = bot_bot(set(A)) )
<=> ! [X4: A] : aa(A,option(B),F,X4) = none(B) ) ).
% dom_eq_empty_conv
tff(fact_3312_fun__upd__None__if__notin__dom,axiom,
! [B: $tType,A: $tType,K: A,M: fun(A,option(B))] :
( ~ pp(member(A,K,dom(A,B,M)))
=> ( fun_upd(A,option(B),M,K,none(B)) = M ) ) ).
% fun_upd_None_if_notin_dom
tff(fact_3313_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,A4)
=> ( aa(set(B),set(A),image2(B,A,the_inv_into(A,B,A4,F)),aa(set(A),set(B),image2(A,B,F),A4)) = A4 ) ) ).
% the_inv_into_onto
tff(fact_3314_dom__empty,axiom,
! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_hj(A,option(B))) = bot_bot(set(A)) ).
% dom_empty
tff(fact_3315_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_nu(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),K),V),M)) = aa(set(A),set(A),insert2(A,K),dom(A,B,M)) ).
% map_update_eta_repair(1)
tff(fact_3316_dom__const_H,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : dom(A,B,aTP_Lamp_nv(fun(A,B),fun(A,option(B)),F)) = top_top(set(A)) ).
% dom_const'
tff(fact_3317_dom__mmupd,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),K2: set(A),V: B] : dom(A,B,map_mmupd(A,B,M,K2,V)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,M)),K2) ).
% dom_mmupd
tff(fact_3318_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),bot_bot(set(product_prod(A,A)))) ).
% trancl_single
tff(fact_3319_restrict__map__inv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] : aa(A,option(B),restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),dom(A,B,F))),X) = none(B) ).
% restrict_map_inv
tff(fact_3320_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option(B),F: fun(A,option(B)),X2: A] :
( ( ( Y = none(B) )
=> ( dom(A,B,fun_upd(A,option(B),F,X2,Y)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F)),aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))) ) )
& ( ( Y != none(B) )
=> ( dom(A,B,fun_upd(A,option(B),F,X2,Y)) = aa(set(A),set(A),insert2(A,X2),dom(A,B,F)) ) ) ) ).
% dom_fun_upd
tff(fact_3321_domI,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B2: A] :
( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B2) )
=> pp(member(B,A3,dom(B,A,M))) ) ).
% domI
tff(fact_3322_domD,axiom,
! [A: $tType,B: $tType,A3: A,M: fun(A,option(B))] :
( pp(member(A,A3,dom(A,B,M)))
=> ? [B3: B] : aa(A,option(B),M,A3) = aa(B,option(B),some(B),B3) ) ).
% domD
tff(fact_3323_domIff,axiom,
! [A: $tType,B: $tType,A3: A,M: fun(A,option(B))] :
( pp(member(A,A3,dom(A,B,M)))
<=> ( aa(A,option(B),M,A3) != none(B) ) ) ).
% domIff
tff(fact_3324_nempty__dom,axiom,
! [B: $tType,A: $tType,E2: fun(A,option(B))] :
( ~ ! [X: A] : aa(A,option(B),E2,X) = none(B)
=> ~ ! [M4: A] : ~ pp(member(A,M4,dom(A,B,E2))) ) ).
% nempty_dom
tff(fact_3325_trancl__sub,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),transitive_trancl(A,R4))) ).
% trancl_sub
tff(fact_3326_trancl__mono__mp,axiom,
! [A: $tType,U3: set(product_prod(A,A)),V5: set(product_prod(A,A)),X2: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),U3),V5))
=> ( pp(member(product_prod(A,A),X2,transitive_trancl(A,U3)))
=> pp(member(product_prod(A,A),X2,transitive_trancl(A,V5))) ) ) ).
% trancl_mono_mp
tff(fact_3327_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R)))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),R))
=> ~ ! [B3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B3),transitive_trancl(A,R)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22),R)) ) ) ) ).
% trancl.cases
tff(fact_3328_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22),transitive_trancl(A,R)))
<=> ( ? [A7: A,B7: A] :
( ( A1 = A7 )
& ( A22 = B7 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),R)) )
| ? [A7: A,B7: A,C4: A] :
( ( A1 = A7 )
& ( A22 = C4 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),transitive_trancl(A,R)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4),R)) ) ) ) ).
% trancl.simps
tff(fact_3329_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R))) ) ).
% trancl.r_into_trancl
tff(fact_3330_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ~ ! [C3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3),transitive_trancl(A,R)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C3),B2),R)) ) ) ) ).
% tranclE
tff(fact_3331_trancl__trans,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),Z2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),transitive_trancl(A,R))) ) ) ).
% trancl_trans
tff(fact_3332_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ( ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y4),R))
=> pp(aa(A,bool,P,Y4)) )
=> ( ! [Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y4),transitive_trancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> ( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,P,Z3)) ) ) )
=> pp(aa(A,bool,P,B2)) ) ) ) ).
% trancl_induct
tff(fact_3333_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R4: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R4))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R4))) ) ) ).
% r_r_into_trancl
tff(fact_3334_converse__tranclE,axiom,
! [A: $tType,X2: A,Z2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),transitive_trancl(A,R)))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),R))
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),R))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),transitive_trancl(A,R))) ) ) ) ).
% converse_tranclE
tff(fact_3335_irrefl__trancl__rD,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y: A] :
( ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( X2 != Y ) ) ) ).
% irrefl_trancl_rD
tff(fact_3336_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R))) ) ) ).
% Transitive_Closure.trancl_into_trancl
tff(fact_3337_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_trancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R))) ) ) ).
% trancl_into_trancl2
tff(fact_3338_trancl__trans__induct,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),P: fun(A,fun(A,bool))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,R)))
=> ( ! [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y4)) )
=> ( ! [X3: A,Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),transitive_trancl(A,R)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y4))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),transitive_trancl(A,R)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Z3)) ) ) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X2),Y)) ) ) ) ).
% trancl_trans_induct
tff(fact_3339_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ( ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),B2),R))
=> pp(aa(A,bool,P,Y4)) )
=> ( ! [Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_trancl(A,R)))
=> ( pp(aa(A,bool,P,Z3))
=> pp(aa(A,bool,P,Y4)) ) ) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% converse_trancl_induct
tff(fact_3340_dom__def,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_nw(fun(A,option(B)),fun(A,bool),M)) ).
% dom_def
tff(fact_3341_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: A] :
( inj_on(A,B,F,A4)
=> ( pp(member(A,X2,A4))
=> ( aa(B,A,the_inv_into(A,B,A4,F),aa(A,B,F,X2)) = X2 ) ) ) ).
% the_inv_into_f_f
tff(fact_3342_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: A,Y: B] :
( inj_on(A,B,F,A4)
=> ( ( aa(A,B,F,X2) = Y )
=> ( pp(member(A,X2,A4))
=> ( aa(B,A,the_inv_into(A,B,A4,F),Y) = X2 ) ) ) ) ).
% the_inv_into_f_eq
tff(fact_3343_insert__dom,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),X2: B,Y: A] :
( ( aa(B,option(A),F,X2) = aa(A,option(A),some(A),Y) )
=> ( aa(set(B),set(B),insert2(B,X2),dom(B,A,F)) = dom(B,A,F) ) ) ).
% insert_dom
tff(fact_3344_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_rtrancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R))) ) ) ).
% trancl_rtrancl_trancl
tff(fact_3345_rtrancl__trancl__trancl,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),Z2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_trancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),transitive_trancl(A,R))) ) ) ).
% rtrancl_trancl_trancl
tff(fact_3346_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),transitive_rtrancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R))) ) ) ).
% rtrancl_into_trancl2
tff(fact_3347_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C2),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2),transitive_trancl(A,R))) ) ) ).
% rtrancl_into_trancl1
tff(fact_3348_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X2: A,Y: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R4)))
<=> ( ( X2 = Y )
| ( ( X2 != Y )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,R4))) ) ) ) ).
% rtrancl_eq_or_trancl
tff(fact_3349_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R))) ) ).
% trancl_into_rtrancl
tff(fact_3350_tranclD2,axiom,
! [A: $tType,X2: A,Y: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,R4)))
=> ? [Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3),transitive_rtrancl(A,R4)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),R4)) ) ) ).
% tranclD2
tff(fact_3351_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_rtrancl(A,R4)))
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,R4))) ) ) ) ).
% rtranclD
tff(fact_3352_tranclD,axiom,
! [A: $tType,X2: A,Y: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,R4)))
=> ? [Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3),R4))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y),transitive_rtrancl(A,R4))) ) ) ).
% tranclD
tff(fact_3353_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B))] :
( finite_finite2(A,dom(A,B,M2))
=> finite_finite2(B,ran(A,B,M2)) ) ).
% map_dom_ran_finite
tff(fact_3354_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W: A,E4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,rel_restrict(A,E4,R4))))
=> ~ pp(member(A,W,R4)) ) ).
% rel_restrict_trancl_notR(2)
tff(fact_3355_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W: A,E4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,rel_restrict(A,E4,R4))))
=> ~ pp(member(A,V,R4)) ) ).
% rel_restrict_trancl_notR(1)
tff(fact_3356_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,A4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,rel_restrict(A,A4,R4))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),rel_restrict(A,transitive_trancl(A,A4),R4))) ) ).
% rel_restrict_trancl_mem
tff(fact_3357_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),P: fun(A,fun(B,bool))] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R)))
=> ( ! [A5: A,B3: B] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),R))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3)) )
=> ( ! [A5: A,B3: B,Aa2: A,Ba: B] :
( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),transitive_trancl(product_prod(A,B),R)))
=> ( pp(member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).
% trancl_induct2
tff(fact_3358_card__map__to__set,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : finite_card(product_prod(A,B),map_to_set(A,B,M)) = finite_card(A,dom(A,B,M)) ).
% card_map_to_set
tff(fact_3359_trancl__sub__insert__trancl,axiom,
! [A: $tType,R4: set(product_prod(A,A)),X2: product_prod(A,A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R4)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),X2),R4)))) ).
% trancl_sub_insert_trancl
tff(fact_3360_rel__restrict__trancl__sub,axiom,
! [A: $tType,A4: set(product_prod(A,A)),R4: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,rel_restrict(A,A4,R4))),rel_restrict(A,transitive_trancl(A,A4),R4))) ).
% rel_restrict_trancl_sub
tff(fact_3361_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M: fun(A,option(B)),M7: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M7))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),dom(A,B,M)),dom(A,B,M7))) ) ) ).
% le_map_dom_mono
tff(fact_3362_the__inv__into__def,axiom,
! [A: $tType,B: $tType,A4: set(A),F: fun(A,B),X: B] : aa(B,A,the_inv_into(A,B,A4,F),X) = the(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_nx(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A4),F),X)) ).
% the_inv_into_def
tff(fact_3363_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B))] :
( finite_finite2(A,dom(A,B,F))
=> ( ~ finite_finite2(A,top_top(set(A)))
=> ? [X3: A] : aa(A,option(B),F,X3) = none(B) ) ) ).
% finite_map_freshness
tff(fact_3364_dom__minus,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),X2: B,A4: set(B)] :
( ( aa(B,option(A),F,X2) = none(A) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F)),aa(set(B),set(B),insert2(B,X2),A4)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F)),A4) ) ) ).
% dom_minus
tff(fact_3365_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M: fun(B,option(A)),A4: set(B),K: B] :
( ( aa(B,option(A),restrict_map(B,A,M,A4),K) = none(A) )
<=> ~ pp(member(B,K,aa(set(B),set(B),inf_inf(set(B),dom(B,A,M)),A4))) ) ).
% restrict_map_eq(1)
tff(fact_3366_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M2: fun(A,option(B)),X2: A,Y: A] :
( finite_finite2(A,dom(A,B,M2))
=> ( ( finite_card(A,dom(A,B,M2)) = finite_card(B,ran(A,B,M2)) )
=> ( pp(member(A,X2,dom(A,B,M2)))
=> ( ( aa(A,option(B),M2,X2) = aa(A,option(B),M2,Y) )
<=> ( X2 = Y ) ) ) ) ) ).
% map_card_eq_iff
tff(fact_3367_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),P: fun(B,bool),K: B,M: fun(B,A)] :
( wf(A,R)
=> ( ! [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),transitive_trancl(A,R)))
<=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),transitive_rtrancl(A,R))) )
=> ( pp(aa(B,bool,P,K))
=> ? [X3: B] :
( pp(aa(B,bool,P,X3))
& ! [Y5: B] :
( pp(aa(B,bool,P,Y5))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M,X3)),aa(B,A,M,Y5)),transitive_rtrancl(A,R))) ) ) ) ) ) ).
% wf_linord_ex_has_least
tff(fact_3368_trancl__union__outside,axiom,
! [A: $tType,V: A,W: A,E4: set(product_prod(A,A)),U3: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E4),U3))))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,E4)))
=> ? [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),X3),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E4),U3))))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),U3))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),W),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),E4),U3)))) ) ) ) ).
% trancl_union_outside
tff(fact_3369_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W: A,V1: A,V22: A,E4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V1),V22)),E4))))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),W),transitive_trancl(A,E4)))
=> ~ ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V1),transitive_rtrancl(A,E4)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V22),W),transitive_rtrancl(A,E4))) ) ) ) ).
% trancl_over_edgeE
tff(fact_3370_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] :
( finite_finite2(product_prod(A,B),map_to_set(A,B,M))
<=> finite_finite2(A,dom(A,B,M)) ) ).
% finite_map_to_set
tff(fact_3371_map__to__set__dom,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),map_to_set(A,B,M)) ).
% map_to_set_dom
tff(fact_3372_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,A4)
=> inj_on(B,A,the_inv_into(A,B,A4,F),aa(set(A),set(B),image2(A,B,F),A4)) ) ).
% inj_on_the_inv_into
tff(fact_3373_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),Y: B] :
( inj_on(A,B,F,A4)
=> ( pp(member(B,Y,aa(set(A),set(B),image2(A,B,F),A4)))
=> ( aa(A,B,F,aa(B,A,the_inv_into(A,B,A4,F),Y)) = Y ) ) ) ).
% f_the_inv_into_f
tff(fact_3374_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X2: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(B,A,the_inv_into(A,B,top_top(set(A)),F),aa(A,B,F,X2)) = X2 ) ) ).
% the_inv_f_f
tff(fact_3375_set__to__map__dom,axiom,
! [B: $tType,A: $tType,S2: set(product_prod(A,B))] : dom(A,B,set_to_map(A,B,S2)) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S2) ).
% set_to_map_dom
tff(fact_3376_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W: A,E4: set(product_prod(A,A)),U3: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),E4),product_Sigma(A,A,U3,aTP_Lamp_ma(set(A),fun(A,set(A)),U3))))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V),W),transitive_trancl(A,E4))) ) ).
% Restr_trancl_mono
tff(fact_3377_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D4: set(A),M: fun(A,option(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D4),dom(A,B,M)))
=> ( inj_on(A,option(B),M,D4)
=> inj_on(A,B,aa(fun(A,option(B)),fun(A,B),aa(fun(option(B),B),fun(fun(A,option(B)),fun(A,B)),comp(option(B),B,A),the2(B)),M),D4) ) ) ).
% inj_on_map_the
tff(fact_3378_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(fun(A,option(B)),bool)] :
( finite_finite2(A,dom(A,B,M))
=> ( pp(aa(fun(A,option(B)),bool,P,aTP_Lamp_hj(A,option(B))))
=> ( ! [K3: A,V4: B,M4: fun(A,option(B))] :
( finite_finite2(A,dom(A,B,M4))
=> ( ~ pp(member(A,K3,dom(A,B,M4)))
=> ( pp(aa(fun(A,option(B)),bool,P,M4))
=> pp(aa(fun(A,option(B)),bool,P,fun_upd(A,option(B),M4,K3,aa(B,option(B),some(B),V4)))) ) ) )
=> pp(aa(fun(A,option(B)),bool,P,M)) ) ) ) ).
% finite_Map_induct
tff(fact_3379_ran__is__image,axiom,
! [A: $tType,B: $tType,M2: fun(B,option(A))] : ran(B,A,M2) = aa(set(B),set(A),image2(B,A,aa(fun(B,option(A)),fun(B,A),aa(fun(option(A),A),fun(fun(B,option(A)),fun(B,A)),comp(option(A),A,B),the2(A)),M2)),dom(B,A,M2)) ).
% ran_is_image
tff(fact_3380_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(set(A),set(product_prod(A,B)),image2(A,product_prod(A,B),aTP_Lamp_ny(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).
% graph_eq_to_snd_dom
tff(fact_3381_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_nz(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).
% trancl_insert2
tff(fact_3382_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A4: set(A),R4: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),rel_restrict(A,transitive_trancl(A,product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4))),R4)),transitive_trancl(A,product_Sigma(A,A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),R4),aa(set(A),fun(A,set(A)),aTP_Lamp_no(set(A),fun(set(A),fun(A,set(A))),A4),R4))))) ).
% rel_restrict_Sigma_sub
tff(fact_3383_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: B,B5: set(A)] :
( inj_on(A,B,F,A4)
=> ( pp(member(B,X2,aa(set(A),set(B),image2(A,B,F),A4)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
=> pp(member(A,aa(B,A,the_inv_into(A,B,A4,F),X2),B5)) ) ) ) ).
% the_inv_into_into
tff(fact_3384_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A4: set(C),X2: B] :
( inj_on(A,B,F,aa(set(C),set(A),image2(C,A,G),A4))
=> ( inj_on(C,A,G,A4)
=> ( pp(member(B,X2,aa(set(A),set(B),image2(A,B,F),aa(set(C),set(A),image2(C,A,G),A4))))
=> ( aa(B,C,the_inv_into(C,B,A4,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)),X2) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),the_inv_into(C,A,A4,G)),the_inv_into(A,B,aa(set(C),set(A),image2(C,A,G),A4),F)),X2) ) ) ) ) ).
% the_inv_into_comp
tff(fact_3385_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X2: A] :
( ( dom(A,B,F) = aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))) )
<=> ? [V3: B] : F = fun_upd(A,option(B),aTP_Lamp_hj(A,option(B)),X2,aa(B,option(B),some(B),V3)) ) ).
% dom_eq_singleton_conv
tff(fact_3386_rtrancl__last__visit,axiom,
! [A: $tType,Q3: A,Q4: A,R4: set(product_prod(A,A)),S2: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4),transitive_rtrancl(A,R4)))
=> ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Q4),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_ma(set(A),fun(A,set(A)),S2))))))
=> ~ ! [Qt: A] :
( pp(member(A,Qt,S2))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),Qt),transitive_trancl(A,R4)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Qt),Q4),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R4),product_Sigma(A,A,top_top(set(A)),aTP_Lamp_ma(set(A),fun(A,set(A)),S2)))))) ) ) ) ) ).
% rtrancl_last_visit
tff(fact_3387_trancl__insert,axiom,
! [A: $tType,Y: A,X2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_nc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y),X2),R)))) ).
% trancl_insert
tff(fact_3388_finite__trancl__ntranl,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( finite_finite2(product_prod(A,A),R4)
=> ( transitive_trancl(A,R4) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(product_prod(A,A),R4)),one_one(nat)),R4) ) ) ).
% finite_trancl_ntranl
tff(fact_3389_ran__add,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
( ( aa(set(A),set(A),inf_inf(set(A),dom(A,B,F)),dom(A,B,G)) = bot_bot(set(A)) )
=> ( ran(A,B,map_add(A,B,F,G)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,F)),ran(A,B,G)) ) ) ).
% ran_add
tff(fact_3390_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(fun(A,option(B)),fun(A,option(B))),A4: fun(A,option(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))))
=> ( ( A4 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4),aa(set(A),set(A),uminus_uminus(set(A)),S)) )
<=> ( map_le(A,B,A4,aa(fun(A,option(B)),fun(A,option(B)),F,A4))
& ( S = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))),dom(A,B,A4)) ) ) ) ) ).
% eq_f_restr_ss_eq
tff(fact_3391_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S: set(A),F: fun(fun(A,option(B)),fun(A,option(B))),A4: fun(A,option(B))] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))))
& ( A4 = restrict_map(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4),aa(set(A),set(A),uminus_uminus(set(A)),S)) ) )
<=> ( map_le(A,B,A4,aa(fun(A,option(B)),fun(A,option(B)),F,A4))
& ( S = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),F,A4))),dom(A,B,A4)) ) ) ) ).
% eq_f_restr_conv
tff(fact_3392_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F4: $tType,Rep: fun(A,B),Abs: fun(B,A),S2: fun(B,set(D)),M2: fun(C,B),F: fun(E,D),S5: fun(C,set(E)),G: fun(F4,C)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( aa(fun(C,B),fun(C,set(D)),aa(fun(B,set(D)),fun(fun(C,B),fun(C,set(D))),comp(B,set(D),C),S2),M2) = aa(fun(C,set(E)),fun(C,set(D)),aa(fun(set(E),set(D)),fun(fun(C,set(E)),fun(C,set(D))),comp(set(E),set(D),C),image2(E,D,F)),S5) )
=> ( aa(fun(F4,A),fun(F4,set(D)),aa(fun(A,set(D)),fun(fun(F4,A),fun(F4,set(D))),comp(A,set(D),F4),aa(fun(A,B),fun(A,set(D)),aa(fun(B,set(D)),fun(fun(A,B),fun(A,set(D))),comp(B,set(D),A),S2),Rep)),aa(fun(F4,C),fun(F4,A),aa(fun(C,A),fun(fun(F4,C),fun(F4,A)),comp(C,A,F4),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Abs),M2)),G)) = aa(fun(F4,set(E)),fun(F4,set(D)),aa(fun(set(E),set(D)),fun(fun(F4,set(E)),fun(F4,set(D))),comp(set(E),set(D),F4),image2(E,D,F)),aa(fun(F4,C),fun(F4,set(E)),aa(fun(C,set(E)),fun(fun(F4,C),fun(F4,set(E))),comp(C,set(E),F4),S5),G)) ) ) ) ).
% type_copy_set_map0
tff(fact_3393_map__add__find__right,axiom,
! [B: $tType,A: $tType,N: fun(B,option(A)),K: B,Xx: A,M: fun(B,option(A))] :
( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),Xx) )
=> ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),Xx) ) ) ).
% map_add_find_right
tff(fact_3394_map__add__eq__empty__iff,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
( ! [X4: A] : aa(A,option(B),map_add(A,B,F,G),X4) = none(B)
<=> ( ! [X4: A] : aa(A,option(B),F,X4) = none(B)
& ! [X4: A] : aa(A,option(B),G,X4) = none(B) ) ) ).
% map_add_eq_empty_iff
tff(fact_3395_map__add__None,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B] :
( ( aa(B,option(A),map_add(B,A,M,N),K) = none(A) )
<=> ( ( aa(B,option(A),N,K) = none(A) )
& ( aa(B,option(A),M,K) = none(A) ) ) ) ).
% map_add_None
tff(fact_3396_empty__map__add,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,aTP_Lamp_hj(A,option(B)),M) = M ).
% empty_map_add
tff(fact_3397_map__add__empty,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,M,aTP_Lamp_hj(A,option(B))) = M ).
% map_add_empty
tff(fact_3398_empty__eq__map__add__iff,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
( ( aTP_Lamp_hj(A,option(B)) = map_add(A,B,F,G) )
<=> ( ! [X4: A] : aa(A,option(B),F,X4) = none(B)
& ! [X4: A] : aa(A,option(B),G,X4) = none(B) ) ) ).
% empty_eq_map_add_iff
tff(fact_3399_map__add__upd,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B)),X2: A,Y: B] : map_add(A,B,F,fun_upd(A,option(B),G,X2,aa(B,option(B),some(B),Y))) = fun_upd(A,option(B),map_add(A,B,F,G),X2,aa(B,option(B),some(B),Y)) ).
% map_add_upd
tff(fact_3400_map__add__upd__left,axiom,
! [A: $tType,B: $tType,M: A,E22: fun(A,option(B)),E1: fun(A,option(B)),U1: B] :
( ~ pp(member(A,M,dom(A,B,E22)))
=> ( map_add(A,B,fun_upd(A,option(B),E1,M,aa(B,option(B),some(B),U1)),E22) = fun_upd(A,option(B),map_add(A,B,E1,E22),M,aa(B,option(B),some(B),U1)) ) ) ).
% map_add_upd_left
tff(fact_3401_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),K2: set(A),V: B] : map_le(A,B,M,map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K2),dom(A,B,M)),V)) ).
% le_map_mmupd_not_dom
tff(fact_3402_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: fun(B,option(A)),K: B,F: fun(B,option(A))] :
( ( aa(B,option(A),G,K) = none(A) )
=> ( aa(B,option(A),map_add(B,A,F,G),K) = aa(B,option(A),F,K) ) ) ).
% map_add_find_left
tff(fact_3403_map__add__left__None,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),K: B,G: fun(B,option(A))] :
( ( aa(B,option(A),F,K) = none(A) )
=> ( aa(B,option(A),map_add(B,A,F,G),K) = aa(B,option(A),G,K) ) ) ).
% map_add_left_None
tff(fact_3404_map__leI,axiom,
! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
( ! [X3: A,V4: B] :
( ( aa(A,option(B),M1,X3) = aa(B,option(B),some(B),V4) )
=> ( aa(A,option(B),M22,X3) = aa(B,option(B),some(B),V4) ) )
=> map_le(A,B,M1,M22) ) ).
% map_leI
tff(fact_3405_map__leD,axiom,
! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),K: A,V: B] :
( map_le(A,B,M1,M22)
=> ( ( aa(A,option(B),M1,K) = aa(B,option(B),some(B),V) )
=> ( aa(A,option(B),M22,K) = aa(B,option(B),some(B),V) ) ) ) ).
% map_leD
tff(fact_3406_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [M: fun(A,option(B)),M7: fun(A,option(B)),N: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M7))
=> pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N)),map_add(A,B,M7,N))) ) ) ).
% map_add_first_le
tff(fact_3407_map__le__empty,axiom,
! [B: $tType,A: $tType,G: fun(A,option(B))] : map_le(A,B,aTP_Lamp_hj(A,option(B)),G) ).
% map_le_empty
tff(fact_3408_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B,X2: A] :
( ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),X2) )
<=> ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),X2) )
| ( ( aa(B,option(A),N,K) = none(A) )
& ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X2) ) ) ) ) ).
% map_add_Some_iff
tff(fact_3409_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B,X2: A] :
( ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),X2) )
=> ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),X2) )
| ( ( aa(B,option(A),N,K) = none(A) )
& ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X2) ) ) ) ) ).
% map_add_SomeD
tff(fact_3410_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F4: $tType,Rep: fun(A,B),Abs: fun(B,A),Rep2: fun(C,D),Abs2: fun(D,C),Rep3: fun(E,F4),Abs3: fun(F4,E),M2: fun(F4,D),M13: fun(B,D),M24: fun(F4,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( type_definition(C,D,Rep2,Abs2,top_top(set(D)))
=> ( type_definition(E,F4,Rep3,Abs3,top_top(set(F4)))
=> ( ( aa(fun(E,F4),fun(E,C),aa(fun(F4,C),fun(fun(E,F4),fun(E,C)),comp(F4,C,E),aa(fun(F4,D),fun(F4,C),aa(fun(D,C),fun(fun(F4,D),fun(F4,C)),comp(D,C,F4),Abs2),M2)),Rep3) = aa(fun(E,A),fun(E,C),aa(fun(A,C),fun(fun(E,A),fun(E,C)),comp(A,C,E),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),Abs2),M13)),Rep)),aa(fun(E,F4),fun(E,A),aa(fun(F4,A),fun(fun(E,F4),fun(E,A)),comp(F4,A,E),aa(fun(F4,B),fun(F4,A),aa(fun(B,A),fun(fun(F4,B),fun(F4,A)),comp(B,A,F4),Abs),M24)),Rep3)) )
=> ( aa(fun(F4,B),fun(F4,D),aa(fun(B,D),fun(fun(F4,B),fun(F4,D)),comp(B,D,F4),M13),M24) = M2 ) ) ) ) ) ).
% type_copy_map_comp0_undo
tff(fact_3411_type__copy__map__comp0,axiom,
! [F4: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: fun(A,B),Abs: fun(B,A),M2: fun(C,D),M13: fun(B,D),M24: fun(C,B),F: fun(D,F4),G: fun(E,C)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( M2 = aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),M13),M24) )
=> ( aa(fun(E,C),fun(E,F4),aa(fun(C,F4),fun(fun(E,C),fun(E,F4)),comp(C,F4,E),aa(fun(C,D),fun(C,F4),aa(fun(D,F4),fun(fun(C,D),fun(C,F4)),comp(D,F4,C),F),M2)),G) = aa(fun(E,A),fun(E,F4),aa(fun(A,F4),fun(fun(E,A),fun(E,F4)),comp(A,F4,E),aa(fun(A,B),fun(A,F4),aa(fun(B,F4),fun(fun(A,B),fun(A,F4)),comp(B,F4,A),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),F),M13)),Rep)),aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Abs),M24)),G)) ) ) ) ).
% type_copy_map_comp0
tff(fact_3412_upd__None__map__le,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),X2: A] : map_le(A,B,fun_upd(A,option(B),F,X2,none(B)),F) ).
% upd_None_map_le
tff(fact_3413_map__add__def,axiom,
! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X: A] : aa(A,option(B),map_add(A,B,M1,M22),X) = aa(option(B),option(B),aa(fun(B,option(B)),fun(option(B),option(B)),aa(option(B),fun(fun(B,option(B)),fun(option(B),option(B))),case_option(option(B),B),aa(A,option(B),M1,X)),some(B)),aa(A,option(B),M22,X)) ).
% map_add_def
tff(fact_3414_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A4: fun(A,option(B)),B5: fun(A,option(B)),C5: fun(A,option(B))] :
( ( aa(set(A),set(A),inf_inf(set(A),dom(A,B,A4)),dom(A,B,B5)) = bot_bot(set(A)) )
=> ( map_add(A,B,A4,map_add(A,B,B5,C5)) = map_add(A,B,B5,map_add(A,B,A4,C5)) ) ) ).
% map_add_left_comm
tff(fact_3415_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),M2: fun(B,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( M2 = id(B) )
=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),aa(fun(B,B),fun(B,A),aa(fun(B,A),fun(fun(B,B),fun(B,A)),comp(B,A,B),Abs),M2)),Rep) = id(A) ) ) ) ).
% type_copy_map_id0
tff(fact_3416_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),Abs),Rep) = id(A) ) ) ).
% type_copy_Abs_o_Rep
tff(fact_3417_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),Rep),Abs) = id(B) ) ) ).
% type_copy_Rep_o_Abs
tff(fact_3418_type__definition__integer,axiom,
type_definition(code_integer,int,code_int_of_integer,code_integer_of_int,top_top(set(int))) ).
% type_definition_integer
tff(fact_3419_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M1: fun(A,option(B)),M22: fun(A,option(B)),X2: A,Y: B] :
( map_le(A,B,M1,M22)
=> map_le(A,B,fun_upd(A,option(B),M1,X2,none(B)),fun_upd(A,option(B),M22,X2,aa(B,option(B),some(B),Y))) ) ).
% map_le_imp_upd_le
tff(fact_3420_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( preorder(B)
=> ! [M: fun(A,option(B)),M7: fun(A,option(B)),N: fun(A,option(B)),N2: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),M),M7))
=> ( pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),N),N2))
=> ( ( aa(set(A),set(A),inf_inf(set(A),dom(A,B,M7)),dom(A,B,N2)) = bot_bot(set(A)) )
=> pp(aa(fun(A,option(B)),bool,aa(fun(A,option(B)),fun(fun(A,option(B)),bool),ord_less_eq(fun(A,option(B))),map_add(A,B,M,N)),map_add(A,B,M7,N2))) ) ) ) ) ).
% map_add_distinct_le
tff(fact_3421_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K2: set(A),K8: set(A),M: fun(A,option(B)),V: B] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K2),K8))
=> map_le(A,B,map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K2),dom(A,B,M)),V),map_mmupd(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),K8),dom(A,B,M)),V)) ) ).
% map_mmupd_update_less
tff(fact_3422_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(product_prod(A,A),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ).
% trancl_set_ntrancl
tff(fact_3423_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( pp(member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N),transitive_rtrancl(nat,pred_nat)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% pred_nat_trancl_eq_le
tff(fact_3424_less__eq,axiom,
! [M: nat,N: nat] :
( pp(member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),N),transitive_trancl(nat,pred_nat)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% less_eq
tff(fact_3425_Gcd__fin__def,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).
% Gcd_fin_def
tff(fact_3426_relImage__def,axiom,
! [A: $tType,B: $tType,R4: set(product_prod(B,B)),F: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R4,F) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_oa(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R4),F)) ).
% relImage_def
tff(fact_3427_set__map__filter,axiom,
! [A: $tType,B: $tType,G: fun(B,option(A)),Xs: list(B)] : aa(list(A),set(A),set2(A),map_filter(B,A,G,Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(B),fun(A,bool),aTP_Lamp_ob(fun(B,option(A)),fun(list(B),fun(A,bool)),G),Xs)) ).
% set_map_filter
tff(fact_3428_ran__map__of,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xs))),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)))) ).
% ran_map_of
tff(fact_3429_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A)),S2: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_trancl(A,E4)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E4,S2)),S2))
=> ( pp(member(A,U,S2))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),E4),product_Sigma(A,A,S2,aTP_Lamp_ma(set(A),fun(A,set(A)),S2)))))) ) ) ) ).
% trancl_restrict_reachable
tff(fact_3430_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E4: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),U)),E4) = image(B,A,E4,aa(set(B),set(B),insert2(B,U),bot_bot(set(B)))) ).
% pair_vimage_is_Image
tff(fact_3431_set__bind__option,axiom,
! [A: $tType,B: $tType,X2: option(B),F: fun(B,option(A))] : aa(option(A),set(A),set_option(A),aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),X2),F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,option(A)),fun(B,set(A)),aa(fun(option(A),set(A)),fun(fun(B,option(A)),fun(B,set(A))),comp(option(A),set(A),B),set_option(A)),F)),aa(option(B),set(B),set_option(B),X2))) ).
% set_bind_option
tff(fact_3432_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B)),A4: set(A)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R))
=> ( pp(member(A,A3,A4))
=> pp(member(B,B2,image(A,B,R,A4))) ) ) ).
% ImageI
tff(fact_3433_elem__set,axiom,
! [A: $tType,X2: A,Xo: option(A)] :
( pp(member(A,X2,aa(option(A),set(A),set_option(A),Xo)))
<=> ( Xo = aa(A,option(A),some(A),X2) ) ) ).
% elem_set
tff(fact_3434_set__empty__eq,axiom,
! [A: $tType,Xo: option(A)] :
( ( aa(option(A),set(A),set_option(A),Xo) = bot_bot(set(A)) )
<=> ( Xo = none(A) ) ) ).
% set_empty_eq
tff(fact_3435_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A3: B] :
( pp(member(A,B2,image(B,A,R,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))))
<=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2),R)) ) ).
% Image_singleton_iff
tff(fact_3436_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set(A),B2: B,R: set(product_prod(A,B))] :
( pp(member(A,A3,A4))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R))
=> pp(member(B,B2,image(A,B,R,A4))) ) ) ).
% rev_ImageI
tff(fact_3437_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A4: set(B)] :
( pp(member(A,B2,image(B,A,R,A4)))
<=> ? [X4: B] :
( pp(member(B,X4,A4))
& pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),B2),R)) ) ) ).
% Image_iff
tff(fact_3438_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A4: set(B)] :
( pp(member(A,B2,image(B,A,R,A4)))
=> ~ ! [X3: B] :
( pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),B2),R))
=> ~ pp(member(B,X3,A4)) ) ) ).
% ImageE
tff(fact_3439_ospec,axiom,
! [A: $tType,A4: option(A),P: fun(A,bool),X2: A] :
( ! [X3: A] :
( pp(member(A,X3,aa(option(A),set(A),set_option(A),A4)))
=> pp(aa(A,bool,P,X3)) )
=> ( ( A4 = aa(A,option(A),some(A),X2) )
=> pp(aa(A,bool,P,X2)) ) ) ).
% ospec
tff(fact_3440_option_Oset__intros,axiom,
! [A: $tType,X22: A] : pp(member(A,X22,aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X22)))) ).
% option.set_intros
tff(fact_3441_option_Oset__cases,axiom,
! [A: $tType,E2: A,A3: option(A)] :
( pp(member(A,E2,aa(option(A),set(A),set_option(A),A3)))
=> ( A3 = aa(A,option(A),some(A),E2) ) ) ).
% option.set_cases
tff(fact_3442_map__option__idI,axiom,
! [A: $tType,X2: option(A),F: fun(A,A)] :
( ! [Y4: A] :
( pp(member(A,Y4,aa(option(A),set(A),set_option(A),X2)))
=> ( aa(A,A,F,Y4) = Y4 ) )
=> ( aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),F),X2) = X2 ) ) ).
% map_option_idI
tff(fact_3443_option_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: option(A),Xa: option(A),F: fun(A,B),Fa: fun(A,B)] :
( ! [Z3: A,Za: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),X2)))
=> ( pp(member(A,Za,aa(option(A),set(A),set_option(A),Xa)))
=> ( ( aa(A,B,F,Z3) = aa(A,B,Fa,Za) )
=> ( Z3 = Za ) ) ) )
=> ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Fa),Xa) )
=> ( X2 = Xa ) ) ) ).
% option.inj_map_strong
tff(fact_3444_option_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: option(A),F: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),X2)))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),X2) ) ) ).
% option.map_cong0
tff(fact_3445_option_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: option(A),Ya: option(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),Ya)))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Ya) ) ) ) ).
% option.map_cong
tff(fact_3446_bind__option__cong,axiom,
! [B: $tType,A: $tType,X2: option(A),Y: option(A),F: fun(A,option(B)),G: fun(A,option(B))] :
( ( X2 = Y )
=> ( ! [Z3: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),Y)))
=> ( aa(A,option(B),F,Z3) = aa(A,option(B),G,Z3) ) )
=> ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),X2),F) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind(A,B),Y),G) ) ) ) ).
% bind_option_cong
tff(fact_3447_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q3: A,R4: set(product_prod(A,A)),Q0: set(A),X2: A] :
( pp(member(A,Q3,image(A,A,transitive_rtrancl(A,R4),Q0)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X2),transitive_rtrancl(A,R4)))
=> pp(member(A,X2,image(A,A,transitive_rtrancl(A,R4),Q0))) ) ) ).
% rtrancl_image_advance_rtrancl
tff(fact_3448_rtrancl__image__advance,axiom,
! [A: $tType,Q3: A,R4: set(product_prod(A,A)),Q0: set(A),X2: A] :
( pp(member(A,Q3,image(A,A,transitive_rtrancl(A,R4),Q0)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),X2),R4))
=> pp(member(A,X2,image(A,A,transitive_rtrancl(A,R4),Q0))) ) ) ).
% rtrancl_image_advance
tff(fact_3449_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B5: set(B),C5: set(product_prod(B,A))] :
( finite_finite2(A,image(B,A,A4,B5))
=> ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),C5),A4))
=> finite_finite2(A,image(B,A,C5,B5)) ) ) ).
% finite_Image_subset
tff(fact_3450_rtrancl__image__unfold__right,axiom,
! [A: $tType,E4: set(product_prod(A,A)),V5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E4,image(A,A,transitive_rtrancl(A,E4),V5))),image(A,A,transitive_rtrancl(A,E4),V5))) ).
% rtrancl_image_unfold_right
tff(fact_3451_rtrancl__reachable__induct,axiom,
! [A: $tType,I4: set(A),INV: set(A),E4: set(product_prod(A,A))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),I4),INV))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E4,INV)),INV))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,E4),I4)),INV)) ) ) ).
% rtrancl_reachable_induct
tff(fact_3452_trancl__Image__unfold__right,axiom,
! [A: $tType,E4: set(product_prod(A,A)),S2: set(A)] : image(A,A,transitive_trancl(A,E4),S2) = image(A,A,E4,image(A,A,transitive_rtrancl(A,E4),S2)) ).
% trancl_Image_unfold_right
tff(fact_3453_trancl__Image__unfold__left,axiom,
! [A: $tType,E4: set(product_prod(A,A)),S2: set(A)] : image(A,A,transitive_trancl(A,E4),S2) = image(A,A,transitive_rtrancl(A,E4),image(A,A,E4,S2)) ).
% trancl_Image_unfold_left
tff(fact_3454_option_Osimps_I14_J,axiom,
! [A: $tType] : aa(option(A),set(A),set_option(A),none(A)) = bot_bot(set(A)) ).
% option.simps(14)
tff(fact_3455_option_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: option(A)] : aa(option(B),set(B),set_option(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),V)) = aa(set(A),set(B),image2(A,B,F),aa(option(A),set(A),set_option(A),V)) ).
% option.set_map
tff(fact_3456_option_Oset__sel,axiom,
! [A: $tType,A3: option(A)] :
( ( A3 != none(A) )
=> pp(member(A,aa(option(A),A,the2(A),A3),aa(option(A),set(A),set_option(A),A3))) ) ).
% option.set_sel
tff(fact_3457_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R4: set(product_prod(A,A)),V: A] :
( ( image(A,A,R4,aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( image(A,A,transitive_rtrancl(A,R4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = aa(set(A),set(A),insert2(A,V),bot_bot(set(A))) ) ) ).
% Image_empty_rtrancl_Image_id
tff(fact_3458_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R4: set(product_prod(A,A)),V: A] :
( ( image(A,A,R4,aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( image(A,A,transitive_trancl(A,R4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ).
% Image_empty_trancl_Image_empty
tff(fact_3459_reachable__mono,axiom,
! [A: $tType,R4: set(product_prod(A,A)),R6: set(product_prod(A,A)),X7: set(A),X8: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),R6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),X8))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,R4),X7)),image(A,A,transitive_rtrancl(A,R6),X8))) ) ) ).
% reachable_mono
tff(fact_3460_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B5: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(B,A,A4,B5)),aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),A4))) ).
% Image_subset_snd_image
tff(fact_3461_trancl__image__by__rtrancl,axiom,
! [A: $tType,E4: set(product_prod(A,A)),Vi: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),image(A,A,transitive_trancl(A,E4),Vi)),Vi) = image(A,A,transitive_rtrancl(A,E4),Vi) ).
% trancl_image_by_rtrancl
tff(fact_3462_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X2: B,L: list(product_prod(A,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L)))
=> ? [X3: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X3) ) ).
% weak_map_of_SomeI
tff(fact_3463_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ) ).
% map_of_SomeD
tff(fact_3464_Image__singleton,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),A3: B] : image(B,A,R,aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_au(set(product_prod(B,A)),fun(B,fun(A,bool)),R),A3)) ).
% Image_singleton
tff(fact_3465_option_Osimps_I15_J,axiom,
! [A: $tType,X22: A] : aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X22)) = aa(set(A),set(A),insert2(A,X22),bot_bot(set(A))) ).
% option.simps(15)
tff(fact_3466_rtrancl__apply__insert,axiom,
! [A: $tType,R4: set(product_prod(A,A)),X2: A,S2: set(A)] : image(A,A,transitive_rtrancl(A,R4),aa(set(A),set(A),insert2(A,X2),S2)) = aa(set(A),set(A),insert2(A,X2),image(A,A,transitive_rtrancl(A,R4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),image(A,A,R4,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))))))) ).
% rtrancl_apply_insert
tff(fact_3467_rtrancl__Image__in__Field,axiom,
! [A: $tType,R4: set(product_prod(A,A)),V5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,R4),V5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),field2(A,R4)),V5))) ).
% rtrancl_Image_in_Field
tff(fact_3468_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,E4)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E4,R4)),R4))
=> ( ~ pp(member(A,V,R4))
=> ~ ( ~ pp(member(A,U,R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),transitive_rtrancl(A,rel_restrict(A,E4,R4)))) ) ) ) ) ).
% E_closed_restr_reach_cases
tff(fact_3469_rel__restrict__tranclI,axiom,
! [A: $tType,X2: A,Y: A,E4: set(product_prod(A,A)),R4: set(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,E4)))
=> ( ~ pp(member(A,X2,R4))
=> ( ~ pp(member(A,Y,R4))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,E4,R4)),R4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,rel_restrict(A,E4,R4)))) ) ) ) ) ).
% rel_restrict_tranclI
tff(fact_3470_refl__on__reflcl__Image,axiom,
! [A: $tType,B5: set(A),A4: set(product_prod(A,A)),C5: set(A)] :
( refl_on(A,B5,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),B5))
=> ( image(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A4),id2(A)),C5) = image(A,A,A4,C5) ) ) ) ).
% refl_on_reflcl_Image
tff(fact_3471_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),X2: B] :
( ( aa(B,option(A),map_of(B,A,Xys),X2) = none(A) )
<=> ~ pp(member(B,X2,aa(set(product_prod(B,A)),set(B),image2(product_prod(B,A),B,product_fst(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xys)))) ) ).
% map_of_eq_None_iff
tff(fact_3472_finite__reachable__advance,axiom,
! [A: $tType,E4: set(product_prod(A,A)),V0: A,V: A] :
( finite_finite2(A,image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,V0),bot_bot(set(A)))))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),V0),V),transitive_rtrancl(A,E4)))
=> finite_finite2(A,image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A))))) ) ) ).
% finite_reachable_advance
tff(fact_3473_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A))))),image(A,A,transitive_rtrancl(A,E4),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))))) ) ).
% rtrancl_Image_advance_ss
tff(fact_3474_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),U),V),E4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,E4),aa(set(A),set(A),insert2(A,V),bot_bot(set(A))))),image(A,A,transitive_trancl(A,E4),aa(set(A),set(A),insert2(A,U),bot_bot(set(A)))))) ) ).
% trancl_Image_advance_ss
tff(fact_3475_subset__Image1__Image1__iff,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( order_preorder_on(A,field2(A,R),R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R)) ) ) ) ) ).
% subset_Image1_Image1_iff
tff(fact_3476_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(product_prod(A,B)),Y: B] :
( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
=> ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_hj(A,option(B)),X2,aa(B,option(B),some(B),Y)),map_of(A,B,Xs)) = fun_upd(A,option(B),map_of(A,B,Xs),X2,aa(B,option(B),some(B),Y)) ) ) ).
% map_of_distinct_upd
tff(fact_3477_subset__Image__Image__iff,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B5: set(A)] :
( order_preorder_on(A,field2(A,R),R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),field2(A,R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,A4)),image(A,A,R,B5)))
<=> ! [X4: A] :
( pp(member(A,X4,A4))
=> ? [Xa2: A] :
( pp(member(A,Xa2,B5))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R)) ) ) ) ) ) ) ).
% subset_Image_Image_iff
tff(fact_3478_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_ng(fun(A,B),fun(A,product_prod(A,B)),F)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),aa(fun(B,option(B)),fun(fun(A,B),fun(A,option(B))),comp(B,option(B),A),some(B)),F),aa(list(A),set(A),set2(A),Ks)) ).
% map_of_map_restrict
tff(fact_3479_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),T2: list(product_prod(A,C)),K: A,X2: C] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(A,option(C),map_of(A,C,T2),K) = aa(C,option(C),some(C),X2) )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_oc(fun(A,B),fun(A,fun(C,product_prod(B,C))),F))),T2)),aa(A,B,F,K)) = aa(C,option(C),some(C),X2) ) ) ) ).
% map_of_mapk_SomeI
tff(fact_3480_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),aa(fun(list(C),list(B)),fun(fun(list(A),list(C)),fun(list(A),list(B))),comp(list(C),list(B),list(A)),map(C,B,F)),map(A,C,G)) = map(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G)) ).
% map_comp_map
tff(fact_3481_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F),aa(list(C),list(B),map(C,B,G),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G)),Xs) ).
% map_map
tff(fact_3482_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),List: list(A)] : aa(list(B),list(C),map(B,C,F),aa(list(A),list(B),map(A,B,G),List)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),List) ).
% List.map.compositionality
tff(fact_3483_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: list(A)] : aa(list(B),list(C),map(B,C,G),aa(list(A),list(B),map(A,B,F),V)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% list.map_comp
tff(fact_3484_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),list(C)),aa(fun(list(B),list(C)),fun(fun(list(A),list(B)),fun(list(A),list(C))),comp(list(B),list(C),list(A)),map(B,C,F)),map(A,B,G)) = map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).
% List.map.comp
tff(fact_3485_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))),K)),L)) = L ).
% map_fst_mk_snd
tff(fact_3486_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K)),L)) = L ).
% map_snd_mk_fst
tff(fact_3487_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X: list(A)] : aa(list(A),set(B),aa(fun(list(A),list(B)),fun(list(A),set(B)),aa(fun(list(B),set(B)),fun(fun(list(A),list(B)),fun(list(A),set(B))),comp(list(B),set(B),list(A)),set2(B)),map(A,B,F)),X) = aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),X)) ).
% set_oo_map_alt
tff(fact_3488_Id__on__set,axiom,
! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set2(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_je(A,product_prod(A,A))),Xs)) ).
% Id_on_set
tff(fact_3489_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
=> ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_ny(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).
% map_of_map_keys
tff(fact_3490_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_od(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),map_of(A,C,Xs)) ).
% map_of_map
tff(fact_3491_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list(A),A4: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A4))
=> ( inj_on(A,B,F,A4)
=> ( aa(list(B),list(A),map(B,A,inv_on(A,B,F,A4)),aa(list(A),list(B),map(A,B,F),L)) = L ) ) ) ).
% inj_on_map_inv_f
tff(fact_3492_Misc_Oran__distinct,axiom,
! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
=> ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).
% Misc.ran_distinct
tff(fact_3493_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X2: A,Y: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( ( aa(A,option(B),map_of(A,B,Xys),X2) = aa(B,option(B),some(B),Y) )
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).
% map_of_eq_Some_iff
tff(fact_3494_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,X2: A] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),X2) )
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).
% Some_eq_map_of_iff
tff(fact_3495_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X2: A,Y: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)))
=> ( aa(A,option(B),map_of(A,B,Xys),X2) = aa(B,option(B),some(B),Y) ) ) ) ).
% map_of_is_SomeI
tff(fact_3496_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),X2: A] :
( inj_on(A,B,F,A4)
=> ( pp(member(A,X2,A4))
=> ( aa(B,A,inv_on(A,B,F,A4),aa(A,B,F,X2)) = X2 ) ) ) ).
% inv_on_f_f
tff(fact_3497_distinct__mapI,axiom,
! [A: $tType,B: $tType,F: fun(B,A),L: list(B)] :
( distinct(A,aa(list(B),list(A),map(B,A,F),L))
=> distinct(B,L) ) ).
% distinct_mapI
tff(fact_3498_distinct__finite__set,axiom,
! [A: $tType,X2: set(A)] : finite_finite2(list(A),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_oe(set(A),fun(list(A),bool),X2))) ).
% distinct_finite_set
tff(fact_3499_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),L: list(B),X2: B,Y: B] :
( distinct(A,aa(list(B),list(A),map(B,A,F),L))
=> ( ( aa(B,A,F,X2) = aa(B,A,F,Y) )
=> ( pp(member(B,X2,aa(list(B),set(B),set2(B),L)))
=> ( pp(member(B,Y,aa(list(B),set(B),set2(B),L)))
=> ( X2 = Y ) ) ) ) ) ).
% distinct_map_eq
tff(fact_3500_finite__set__image,axiom,
! [A: $tType,A4: set(list(A))] :
( finite_finite2(set(A),aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),A4))
=> ( ! [Xs2: list(A)] :
( pp(member(list(A),Xs2,A4))
=> distinct(A,Xs2) )
=> finite_finite2(list(A),A4) ) ) ).
% finite_set_image
tff(fact_3501_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F: fun(B,A),A4: set(B)] :
( pp(member(A,Y,aa(set(B),set(A),image2(B,A,F),A4)))
=> pp(member(B,aa(A,B,inv_on(B,A,F,A4),Y),A4)) ) ).
% inv_on_f_range
tff(fact_3502_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F: fun(B,A),A4: set(B)] :
( pp(member(A,Y,aa(set(B),set(A),image2(B,A,F),A4)))
=> ( aa(B,A,F,aa(A,B,inv_on(B,A,F,A4),Y)) = Y ) ) ).
% f_inv_on_f
tff(fact_3503_distinct__finite__subset,axiom,
! [A: $tType,X2: set(A)] :
( finite_finite2(A,X2)
=> finite_finite2(list(A),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_of(set(A),fun(list(A),bool),X2))) ) ).
% distinct_finite_subset
tff(fact_3504_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),X2: A,Y: B,Z2: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
tff(fact_3505_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),K: A,V1: B,V22: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs)))
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
tff(fact_3506_map__to__set__map__of,axiom,
! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
=> ( map_to_set(A,B,map_of(A,B,L)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) ) ) ).
% map_to_set_map_of
tff(fact_3507_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list(product_prod(A,B)),M: fun(A,option(B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
=> ( ( map_of(A,B,L) = M )
<=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M) ) ) ) ).
% map_of_map_to_set
tff(fact_3508_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_og(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).
% set_map_of_compr
tff(fact_3509_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set(A),K: nat] :
( finite_finite2(A,A4)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),finite_card(A,A4)))
=> ( finite_card(list(A),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_oh(set(A),fun(nat,fun(list(A),bool)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bx(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,A4)),K)),one_one(nat)),finite_card(A,A4))) ) ) ) ).
% card_lists_distinct_length_eq
tff(fact_3510_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),finite_card(A,A4)))
=> ( finite_card(list(A),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),bool),aTP_Lamp_oi(nat,fun(set(A),fun(list(A),bool)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_bx(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),finite_card(A,A4)),K)),one_one(nat)),finite_card(A,A4))) ) ) ).
% card_lists_distinct_length_eq'
tff(fact_3511_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S2: list(A),I4: fun(set(A),fun(B,bool)),Sigma_0: B,F: fun(B,fun(A,B))] :
( distinct(A,S2)
=> ( pp(aa(B,bool,aa(set(A),fun(B,bool),I4,aa(list(A),set(A),set2(A),S2)),Sigma_0))
=> ( ! [X3: A,It: set(A),Sigma: B] :
( pp(member(A,X3,It))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S2)))
=> ( pp(aa(B,bool,aa(set(A),fun(B,bool),I4,It),Sigma))
=> pp(aa(B,bool,aa(set(A),fun(B,bool),I4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),It),aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F,Sigma),X3))) ) ) )
=> pp(aa(B,bool,aa(set(A),fun(B,bool),I4,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F),Sigma_0),S2))) ) ) ) ).
% distinct_foldl_invar
tff(fact_3512_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A,P: fun(product_prod(B,A),bool)] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ( ~ pp(aa(product_prod(B,A),bool,P,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)))
=> ( distinct(B,aa(list(product_prod(B,A)),list(B),map(product_prod(B,A),B,product_fst(B,A)),Xs))
=> ( aa(B,option(A),map_of(B,A,filter(product_prod(B,A),P,Xs)),K) = none(A) ) ) ) ) ).
% map_of_Some_filter_not_in
tff(fact_3513_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M,map_of(A,B,Ps)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(product_prod(A,B),fun(A,option(B)),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_oj(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps),M) ).
% map_add_map_of_foldr
tff(fact_3514_foldl__length,axiom,
! [A: $tType,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_ok(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ).
% foldl_length
tff(fact_3515_foldl__length__aux,axiom,
! [A: $tType,A3: nat,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_ok(nat,fun(A,nat))),A3),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).
% foldl_length_aux
tff(fact_3516_foldl__A1__eq,axiom,
! [A: $tType,F: fun(A,fun(A,A)),N: A,I: A,Ww: list(A)] :
( ! [E3: A] : aa(A,A,aa(A,fun(A,A),F,N),E3) = E3
=> ( ! [E3: A] : aa(A,A,aa(A,fun(A,A),F,E3),N) = E3
=> ( ! [A5: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B3),C3)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B3)),C3)
=> ( aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F),I),Ww) = aa(A,A,aa(A,fun(A,A),F,I),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F),N),Ww)) ) ) ) ) ).
% foldl_A1_eq
tff(fact_3517_sum__length__filter__compl,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter(A,aTP_Lamp_ol(fun(A,bool),fun(A,bool),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).
% sum_length_filter_compl
tff(fact_3518_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Xs: list(A),A3: B] :
( finite6289374366891150609ommute(A,B,F)
=> ( aa(B,B,foldr(A,B,F,Xs),A3) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_om(fun(A,fun(B,B)),fun(B,fun(A,B)),F)),A3),Xs) ) ) ).
% comp_fun_commute.foldr_conv_foldl
tff(fact_3519_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),A3: A,B2: B,Xs: list(A)] :
( finite6289374366891150609ommute(A,B,F)
=> ( aa(B,B,aa(A,fun(B,B),F,A3),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_om(fun(A,fun(B,B)),fun(B,fun(A,B)),F)),B2),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_om(fun(A,fun(B,B)),fun(B,fun(A,B)),F)),aa(B,B,aa(A,fun(B,B),F,A3),B2)),Xs) ) ) ).
% comp_fun_commute.foldl_f_commute
tff(fact_3520_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),X2: B,P: fun(product_prod(B,A),bool)] :
( ( aa(B,option(A),map_of(B,A,Xs),X2) = none(A) )
=> ( aa(B,option(A),map_of(B,A,filter(product_prod(B,A),P,Xs)),X2) = none(A) ) ) ).
% map_of_None_filterD
tff(fact_3521_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(C,A)),G: fun(A,fun(B,fun(C,B))),A3: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(list(C),product_prod(A,B),aa(product_prod(A,B),fun(list(C),product_prod(A,B)),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_on(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F),G))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,F),A3),Xs) ).
% fst_foldl
tff(fact_3522_foldl__absorb1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X2: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),X2),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),one_one(A)),Zs)) = aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),X2),Zs) ) ).
% foldl_absorb1
tff(fact_3523_foldl__un__empty__eq,axiom,
! [A: $tType,I: set(A),Ww: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),I),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),Ww)) ).
% foldl_un_empty_eq
tff(fact_3524_distinct__map__fst__filterI,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),P: fun(product_prod(A,B),bool)] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),filter(product_prod(A,B),P,Xs))) ) ).
% distinct_map_fst_filterI
tff(fact_3525_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,Z2: A,P: fun(B,fun(A,bool))] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Z2) )
=> ( pp(aa(A,bool,aa(B,fun(A,bool),P,K),Z2))
=> ( aa(B,option(A),map_of(B,A,filter(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P),Xs)),K) = aa(A,option(A),some(A),Z2) ) ) ) ).
% map_of_filter_in
tff(fact_3526_distinct__length__le,axiom,
! [A: $tType,Ys: list(A),Xs: list(A)] :
( distinct(A,Ys)
=> ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% distinct_length_le
tff(fact_3527_foldl__set,axiom,
! [A: $tType,L: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_oo(list(set(A)),fun(set(A),bool),L))) ).
% foldl_set
tff(fact_3528_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),map(B,A,F),filter(B,P,Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_op(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F),P),Xs) ).
% map_filter_map_filter
tff(fact_3529_map__filter__def,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F,Xs) = aa(list(A),list(B),map(A,B,aa(fun(A,option(B)),fun(A,B),aa(fun(option(B),B),fun(fun(A,option(B)),fun(A,B)),comp(option(B),B,A),the2(B)),F)),filter(A,aTP_Lamp_nw(fun(A,option(B)),fun(A,bool),F),Xs)) ).
% map_filter_def
tff(fact_3530_execute__of__list,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_of_list(A,Xs)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_oq(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Xs)),array_alloc(A,Xs,H))) ) ).
% execute_of_list
tff(fact_3531_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list(A),Xs22: list(A),Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs1)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
& ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs22)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).
% mergesort_by_rel_split_length
tff(fact_3532_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( aa(set(A),set(A),inf_inf(set(A),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
=> ( finite_card(list(A),shuffles(A,Xs,Ys)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).
% card_disjoint_shuffles
tff(fact_3533_of__list__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_or(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).
% of_list_def
tff(fact_3534_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),From),To))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From) ) ) ) ).
% slice_len
tff(fact_3535_slice__complete,axiom,
! [A: $tType,Xs: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs),Xs) = Xs ).
% slice_complete
tff(fact_3536_foldr__length,axiom,
! [A: $tType,L: list(A)] : aa(nat,nat,foldr(A,nat,aTP_Lamp_os(A,fun(nat,nat)),L),zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).
% foldr_length
tff(fact_3537_length__shuffles,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
( pp(member(list(A),Zs,shuffles(A,Xs,Ys)))
=> ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).
% length_shuffles
tff(fact_3538_foldr__length__aux,axiom,
! [A: $tType,L: list(A),A3: nat] : aa(nat,nat,foldr(A,nat,aTP_Lamp_os(A,fun(nat,nat)),L),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).
% foldr_length_aux
tff(fact_3539_effect__of__listI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H3: heap_ext(product_unit),Xs: list(A),H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H3) = array_alloc(A,Xs,H) )
=> heap_Time_effect(array(A),array_of_list(A,Xs),H,H3,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% effect_of_listI
tff(fact_3540_execute__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_make(A,N,F)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ot(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N)),H))) ) ).
% execute_make
tff(fact_3541_execute__new,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ot(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,replicate(A,N,X2),H))) ) ).
% execute_new
tff(fact_3542_Array__Time_Omake__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A)] : array_make(A,N,F) = heap_Time_heap(array(A),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ou(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),F)) ) ).
% Array_Time.make_def
tff(fact_3543_new__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A] : aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2) = heap_Time_heap(array(A),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ov(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),X2)) ) ).
% new_def
tff(fact_3544_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( upt(zero_zero(nat),J) = nil(nat) )
<=> ( J = zero_zero(nat) ) ) ).
% upt_0_eq_Nil_conv
tff(fact_3545_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
( ! [X4: A] : aa(A,option(B),map_of(A,B,Xys),X4) = none(B)
<=> ( Xys = nil(product_prod(A,B)) ) ) ).
% map_of_eq_empty_iff
tff(fact_3546_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).
% slice_Nil
tff(fact_3547_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs: list(A)] : slice(A,I,I,Xs) = nil(A) ).
% slice_eq_bounds_empty
tff(fact_3548_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
( ( aTP_Lamp_hj(A,option(B)) = map_of(A,B,Xys) )
<=> ( Xys = nil(product_prod(A,B)) ) ) ).
% empty_eq_map_of_iff
tff(fact_3549_length__ge__1__conv,axiom,
! [A: $tType,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L)))
<=> ( L != nil(A) ) ) ).
% length_ge_1_conv
tff(fact_3550_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).
% map_fst_mk_fst
tff(fact_3551_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_it(A,fun(B,product_prod(B,A))),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).
% map_snd_mk_snd
tff(fact_3552_map__add__upt_H,axiom,
! [Ofs: nat,A3: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ow(nat,fun(nat,nat),Ofs)),upt(A3,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).
% map_add_upt'
tff(fact_3553_len__greater__imp__nonempty,axiom,
! [A: $tType,X2: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),L)))
=> ( L != nil(A) ) ) ).
% len_greater_imp_nonempty
tff(fact_3554_map__add__upt,axiom,
! [N: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ow(nat,fun(nat,nat),N)),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).
% map_add_upt
tff(fact_3555_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K) = none(A) ).
% map_of_Cons_code(1)
tff(fact_3556_map__of_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,X: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X) = none(B) ).
% map_of.simps(1)
tff(fact_3557_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list(A),Xs22: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) ).
% mergesort_by_rel_split.simps(1)
tff(fact_3558_upt__filter__extend,axiom,
! [U: nat,U4: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),U4))
=> ( ! [I2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),U),I2))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),U4)) )
=> ~ pp(aa(nat,bool,P,I2)) )
=> ( filter(nat,P,upt(zero_zero(nat),U)) = filter(nat,P,upt(zero_zero(nat),U4)) ) ) ) ).
% upt_filter_extend
tff(fact_3559_effect__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H3: heap_ext(product_unit),N: nat,X2: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H3) = array_alloc(A,replicate(A,N,X2),H) )
=> heap_Time_effect(array(A),aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2),H,H3,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_newI
tff(fact_3560_effect__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H3: heap_ext(product_unit),F: fun(nat,A),N: nat,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H3) = array_alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N)),H) )
=> heap_Time_effect(array(A),array_make(A,N,F),H,H3,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_makeI
tff(fact_3561_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),P: fun(fun(A,fun(A,bool)),fun(list(A),bool))] :
( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),A0),A1)))
=> ( ! [R7: fun(A,fun(A,bool)),Xs2: list(A)] :
( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R7),Xs2)))
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R7),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)))) )
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R7),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)))) )
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,R7),Xs2)) ) ) )
=> pp(aa(list(A),bool,aa(fun(A,fun(A,bool)),fun(list(A),bool),P,A0),A1)) ) ) ).
% mergesort_by_rel.pinduct
tff(fact_3562_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),mergesort_by_rel(A,X2),Xa) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( Y = Xa ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( Y = merges9089515139780605204_merge(A,X2,aa(list(A),list(A),mergesort_by_rel(A,X2),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),aa(list(A),list(A),mergesort_by_rel(A,X2),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa)))) ) ) ) ) ).
% mergesort_by_rel.elims
tff(fact_3563_mergesort__by__rel_Osimps,axiom,
! [A: $tType,Xs: list(A),R4: fun(A,fun(A,bool))] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R4),Xs) = Xs ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R4),Xs) = merges9089515139780605204_merge(A,R4,aa(list(A),list(A),mergesort_by_rel(A,R4),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R4),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)))) ) ) ) ).
% mergesort_by_rel.simps
tff(fact_3564_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M: nat,A3: A] : enumerate(A,N,replicate(A,M,A3)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_ox(A,fun(nat,product_prod(nat,A)),A3)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).
% enumerate_replicate_eq
tff(fact_3565_nth__step__trancl,axiom,
! [A: $tType,Xs: list(A),R4: set(product_prod(A,A)),N: nat,M: nat] :
( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N3))),aa(nat,A,nth(A,Xs),N3)),R4)) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N)),aa(nat,A,nth(A,Xs),M)),transitive_trancl(A,R4))) ) ) ) ).
% nth_step_trancl
tff(fact_3566_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool))] : aa(list(A),list(A),mergesort_by_rel(A,R4),nil(A)) = nil(A) ).
% mergesort_by_rel_simps(1)
tff(fact_3567_set__mergesort__by__rel,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R4),Xs)) = aa(list(A),set(A),set2(A),Xs) ).
% set_mergesort_by_rel
tff(fact_3568_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Ys: list(A)] : merges9089515139780605204_merge(A,R4,nil(A),Ys) = Ys ).
% mergesort_by_rel_merge_simps(3)
tff(fact_3569_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J))
=> ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).
% nth_upt
tff(fact_3570_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R4,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).
% set_mergesort_by_rel_merge
tff(fact_3571_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).
% map_fst_enumerate
tff(fact_3572_nth__image__indices,axiom,
! [A: $tType,L: list(A)] : aa(set(nat),set(A),image2(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ).
% nth_image_indices
tff(fact_3573_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_oy(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).
% ran_nth_set_encoding_conv
tff(fact_3574_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(A,fun(A,bool)),list(A))] :
~ ! [R7: fun(A,fun(A,bool)),Xs2: list(A)] : X2 != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R7),Xs2) ).
% mergesort_by_rel.cases
tff(fact_3575_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),L: list(B),L4: list(B),I: nat] :
( ( aa(list(B),list(A),map(B,A,F),L) = aa(list(B),list(A),map(B,A,F),L4) )
=> ( aa(B,A,F,aa(nat,B,nth(B,L),I)) = aa(B,A,F,aa(nat,B,nth(B,L4),I)) ) ) ).
% map_eq_nth_eq
tff(fact_3576_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).
% nth_enumerate_eq
tff(fact_3577_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A)] : merges9089515139780605204_merge(A,R4,Xs,nil(A)) = Xs ).
% mergesort_by_rel_merge_simps(2)
tff(fact_3578_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: fun(A,fun(nat,bool))] :
( ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
=> ? [Li: A] : pp(aa(nat,bool,aa(A,fun(nat,bool),P,Li),I2)) )
=> ~ ! [L5: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L5) = N )
=> ~ ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
=> pp(aa(nat,bool,aa(A,fun(nat,bool),P,aa(nat,A,nth(A,L5),I5)),I5)) ) ) ) ).
% obtain_list_from_elements
tff(fact_3579_in__set__enumerate__eq,axiom,
! [A: $tType,P3: product_prod(nat,A),N: nat,Xs: list(A)] :
( pp(member(product_prod(nat,A),P3,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
& ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).
% in_set_enumerate_eq
tff(fact_3580_all__set__conv__nth,axiom,
! [A: $tType,L: list(A),P: fun(A,bool)] :
( ! [X4: A] :
( pp(member(A,X4,aa(list(A),set(A),set2(A),L)))
=> pp(aa(A,bool,P,X4)) )
<=> ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,L),I3))) ) ) ).
% all_set_conv_nth
tff(fact_3581_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list(A),L4: list(A),F: fun(A,B)] :
( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L4) )
=> ( ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(A,B,F,aa(nat,A,nth(A,L),I2)) = aa(A,B,F,aa(nat,A,nth(A,L4),I2)) ) )
=> ( aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image2(A,B,F),aa(list(A),set(A),set2(A),L4)) ) ) ) ).
% set_image_eq_pointwiseI
tff(fact_3582_in__set__image__conv__nth,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B,L: list(B)] :
( pp(member(A,aa(B,A,F,X2),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),L))))
<=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),L)))
& ( aa(B,A,F,aa(nat,B,nth(B,L),I3)) = aa(B,A,F,X2) ) ) ) ).
% in_set_image_conv_nth
tff(fact_3583_distinct__idx,axiom,
! [B: $tType,A: $tType,F: fun(B,A),L: list(B),I: nat,J: nat] :
( distinct(A,aa(list(B),list(A),map(B,A,F),L))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L)))
=> ( ( aa(B,A,F,aa(nat,B,nth(B,L),I)) = aa(B,A,F,aa(nat,B,nth(B,L),J)) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
tff(fact_3584_array__of__list__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : array_of_list(A,Xs) = array_make(A,aa(list(A),nat,size_size(list(A)),Xs),nth(A,Xs)) ) ).
% array_of_list_make
tff(fact_3585_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M: nat,F: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
=> ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F),upt(M,N))),I) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).
% nth_map_upt
tff(fact_3586_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A)] :
( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),R4),Xs)))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R4),Xs) = Xs ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( aa(list(A),list(A),mergesort_by_rel(A,R4),Xs) = merges9089515139780605204_merge(A,R4,aa(list(A),list(A),mergesort_by_rel(A,R4),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R4),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)))) ) ) ) ) ).
% mergesort_by_rel.psimps
tff(fact_3587_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Y: list(A)] :
( ( aa(list(A),list(A),mergesort_by_rel(A,X2),Xa) = Y )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( Y = Xa ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( Y = merges9089515139780605204_merge(A,X2,aa(list(A),list(A),mergesort_by_rel(A,X2),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa))),aa(list(A),list(A),mergesort_by_rel(A,X2),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xa)))) ) ) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa))) ) ) ) ).
% mergesort_by_rel.pelims
tff(fact_3588_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F: fun(nat,A),M: nat] : enumerate(A,N,aa(list(nat),list(A),map(nat,A,F),upt(N,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_oz(fun(nat,A),fun(nat,product_prod(nat,A)),F)),upt(N,M)) ).
% enumerate_map_upt
tff(fact_3589_map__upt__eqI,axiom,
! [A: $tType,Xs: list(A),N: nat,M: nat,F: fun(nat,A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) )
=> ( ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
=> ( aa(list(nat),list(A),map(nat,A,F),upt(M,N)) = Xs ) ) ) ).
% map_upt_eqI
tff(fact_3590_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From)))
=> ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).
% slice_nth
tff(fact_3591_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).
% product_nth
tff(fact_3592_mergesort__def,axiom,
! [A: $tType] :
( ord(A)
=> ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).
% mergesort_def
tff(fact_3593_sorted__in__between,axiom,
! [A: $tType] :
( linorder(A)
=> ! [I: nat,J: nat,L: list(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(nat,A,nth(A,L),J)))
=> ~ ! [K3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),J))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),K3)),X2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K3),one_one(nat))))) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
tff(fact_3594_Un__set__drop__extend,axiom,
! [A: $tType,J: nat,L: list(set(A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(set(A)),nat,size_size(list(set(A))),L)))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),J,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).
% Un_set_drop_extend
tff(fact_3595_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),X1: A,X22: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aTP_Lamp_pa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),R4)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),nil(A))),Xs)) ).
% mergesort_by_rel_simps(3)
tff(fact_3596_drop__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list(A)] : drop(A,N,drop(A,M,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs) ).
% drop_drop
tff(fact_3597_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( ! [X3: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R4,X3),Y4))
| pp(aa(A,bool,aa(A,fun(A,bool),R4,Y4),X3)) )
=> ( ! [X3: A,Y4: A,Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R4,X3),Y4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,Y4),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),R4,X3),Z3)) ) )
=> ( sorted_wrt(A,R4,merges9089515139780605204_merge(A,R4,Xs,Ys))
<=> ( sorted_wrt(A,R4,Xs)
& sorted_wrt(A,R4,Ys) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
tff(fact_3598_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),X2: A] : aa(list(A),list(A),mergesort_by_rel(A,R4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) ).
% mergesort_by_rel_simps(2)
tff(fact_3599_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X2: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),N),X2)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).
% enumerate_simps(2)
tff(fact_3600_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [L: list(product_prod(A,B))] :
( sorted_wrt(product_prod(A,B),aTP_Lamp_pb(product_prod(A,B),fun(product_prod(A,B),bool)),L)
<=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).
% sorted_wrt_map_linord
tff(fact_3601_nth__Cons__numeral,axiom,
! [A: $tType,X2: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).
% nth_Cons_numeral
tff(fact_3602_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X2: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).
% drop_Cons_numeral
tff(fact_3603_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,drop(A,N,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ) ) ).
% nth_drop
tff(fact_3604_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X2: A,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).
% nth_Cons_pos
tff(fact_3605_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),list(A),foldr(A,list(A),cons(A),Xs),nil(A)) = Xs ).
% Misc.foldr_Cons
tff(fact_3606_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list(A),F: fun(B,A),Ww: list(B),A3: B] :
( ( W = aa(list(B),list(A),map(B,A,F),Ww) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,A3)),W) = aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww)) ) ) ).
% map_consI(1)
tff(fact_3607_map__eq__consE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fa: A,Fl: list(A)] :
( ( aa(list(B),list(A),map(B,A,F),Ls) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
=> ~ ! [A5: B,L5: list(B)] :
( ( Ls = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A5),L5) )
=> ( ( aa(B,A,F,A5) = Fa )
=> ( aa(list(B),list(A),map(B,A,F),L5) != Fl ) ) ) ) ).
% map_eq_consE
tff(fact_3608_neq__NilE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [X3: A,Xs2: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) ) ).
% neq_NilE
tff(fact_3609_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),W1: list(A),W2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
=> ( ! [E3: A,W12: list(A),W22: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W12),W22))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),W12)),W22)) )
=> ( ! [E3: B,W13: list(A),W23: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W13),W23))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E3),W23))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,W1),W2)) ) ) ) ).
% list_2pre_induct
tff(fact_3610_list__induct__first2,axiom,
! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [X3: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))
=> ( ! [X12: A,X23: A,Xs2: list(A)] :
( pp(aa(list(A),bool,P,Xs2))
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)))) )
=> pp(aa(list(A),bool,P,Xs)) ) ) ) ).
% list_induct_first2
tff(fact_3611_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),R4: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( ! [Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),nil(B)))
=> ( ! [Ys2: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),Ys2))
=> ( ! [X3: A,Xs2: list(A),Y4: B,Ys2: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,X3),Y4))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys2)))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys2))) ) )
=> ( ! [X3: A,Xs2: list(A),Y4: B,Ys2: list(B)] :
( ~ pp(aa(B,bool,aa(A,fun(B,bool),R4,X3),Y4))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys2))) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% mergesort_by_rel_merge_induct
tff(fact_3612_sorted__wrt__mergesort__by__rel,axiom,
! [X9: $tType,R4: fun(X9,fun(X9,bool)),Xs: list(X9)] :
( ! [X3: X9,Y4: X9] :
( pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,X3),Y4))
| pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,Y4),X3)) )
=> ( ! [X3: X9,Y4: X9,Z3: X9] :
( pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,X3),Y4))
=> ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,Y4),Z3))
=> pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,X3),Z3)) ) )
=> sorted_wrt(X9,R4,aa(list(X9),list(X9),mergesort_by_rel(X9,R4),Xs)) ) ) ).
% sorted_wrt_mergesort_by_rel
tff(fact_3613_list__tail__coinc,axiom,
! [A: $tType,N12: A,R1: list(A),N23: A,R22: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N12),R1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N23),R22) )
=> ( ( N12 = N23 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
tff(fact_3614_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list(A),X2: A,Xs3: list(A)] :
( ( drop(A,N,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs3) )
=> ( drop(A,aa(nat,nat,suc,N),Xs) = Xs3 ) ) ).
% drop_eq_ConsD
tff(fact_3615_drop__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) )
& ( ( N != zero_zero(nat) )
=> ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ) ).
% drop_Cons'
tff(fact_3616_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),X2: A,Y: A,Xs: list(A),Ys: list(A)] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X2),Y))
=> ( merges9089515139780605204_merge(A,R4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merges9089515139780605204_merge(A,R4,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),R4,X2),Y))
=> ( merges9089515139780605204_merge(A,R4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R4,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys)) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
tff(fact_3617_sorted__filter_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),L)
=> sorted_wrt(A,ord_less_eq(A),filter(A,P,L)) ) ) ).
% sorted_filter'
tff(fact_3618_sorted__mergesort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_by_rel(A,ord_less_eq(A)),Xs)) ) ).
% sorted_mergesort_by_rel
tff(fact_3619_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),X2: B,Xs: list(B)] : map_filter(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs)) = aa(option(A),list(A),aa(fun(A,list(A)),fun(option(A),list(A)),aa(list(A),fun(fun(A,list(A)),fun(option(A),list(A))),case_option(list(A),A),map_filter(B,A,F,Xs)),aa(list(B),fun(A,list(A)),aTP_Lamp_pc(fun(B,option(A)),fun(list(B),fun(A,list(A))),F),Xs)),aa(B,option(A),F,X2)) ).
% map_filter_simps(1)
tff(fact_3620_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X2: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
( ! [F3: fun(A,fun(B,C))] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
=> ( ! [F3: fun(A,fun(B,C)),A5: A,As: list(A),B3: B,Bs: list(B)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))
=> ( ! [A5: fun(A,fun(B,C)),V4: A,Va: list(A)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(B)))
=> ~ ! [A5: fun(A,fun(B,C)),V4: B,Va: list(B)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A5),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va))) ) ) ) ).
% zipf.cases
tff(fact_3621_merge_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: product_prod(list(A),list(A))] :
( ! [L22: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L22)
=> ( ! [V4: A,Va: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(A))
=> ~ ! [X12: A,L12: list(A),X23: A,L22: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22)) ) ) ) ).
% merge.cases
tff(fact_3622_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))] :
( ! [P4: fun(A,fun(B,bool))] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
=> ( ! [P4: fun(A,fun(B,bool)),A5: A,As: list(A),B3: B,Bs: list(B)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))
=> ( ! [P4: fun(A,fun(B,bool)),V4: A,Va: list(A)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(B)))
=> ~ ! [P4: fun(A,fun(B,bool)),V4: B,Va: list(B)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va))) ) ) ) ).
% list_all_zip.cases
tff(fact_3623_partition__rev_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))] :
( ! [P4: fun(A,bool),Yes: list(A),No: list(A)] : X2 != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P4),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
=> ~ ! [P4: fun(A,bool),Yes: list(A),No: list(A),X3: A,Xs2: list(A)] : X2 != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),P4),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ).
% partition_rev.cases
tff(fact_3624_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
( ! [R7: fun(A,fun(A,bool)),Sl: list(A)] : X2 != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
=> ~ ! [R7: fun(A,fun(A,bool)),Sl: list(A),X3: A,Xs2: list(A)] : X2 != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ).
% quicksort_by_rel.cases
tff(fact_3625_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))] :
( ! [R7: fun(A,fun(A,bool)),X3: A,Xs2: list(A),Y4: A,Ys2: list(A)] : X2 != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2)))
=> ( ! [R7: fun(A,fun(A,bool)),Xs2: list(A)] : X2 != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
=> ~ ! [R7: fun(A,fun(A,bool)),V4: A,Va: list(A)] : X2 != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va))) ) ) ).
% mergesort_by_rel_merge.cases
tff(fact_3626_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X2: product_prod(product_prod(list(A),list(A)),list(A))] :
( ! [Xs12: list(A),Xs23: list(A)] : X2 != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),nil(A))
=> ( ! [Xs12: list(A),Xs23: list(A),X3: A] : X2 != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
=> ~ ! [Xs12: list(A),Xs23: list(A),X12: A,X23: A,Xs2: list(A)] : X2 != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))) ) ) ).
% mergesort_by_rel_split.cases
tff(fact_3627_shuffles_Ocases,axiom,
! [A: $tType,X2: product_prod(list(A),list(A))] :
( ! [Ys2: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)
=> ( ! [Xs2: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A))
=> ~ ! [X3: A,Xs2: list(A),Y4: A,Ys2: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2)) ) ) ).
% shuffles.cases
tff(fact_3628_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
( ! [F3: fun(A,B),Bs: list(B)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs))
=> ~ ! [F3: fun(A,B),A5: A,As: list(A),Bs: list(B)] : X2 != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),Bs)) ) ).
% map_tailrec_rev.cases
tff(fact_3629_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A,Xs: list(A)] : foldr(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),foldr(A,B,F,Xs)) ).
% foldr.foldr_Cons
tff(fact_3630_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X2: product_prod(fun(A,B),list(A))] :
( ! [F3: fun(A,B),X3: A] : X2 != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
=> ( ! [F3: fun(A,B),X3: A,Y4: A,Zs2: list(A)] : X2 != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))
=> ~ ! [A5: fun(A,B)] : X2 != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A5),nil(A)) ) ) ) ).
% arg_min_list.cases
tff(fact_3631_successively_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(A,fun(A,bool)),list(A))] :
( ! [P4: fun(A,fun(A,bool))] : X2 != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P4),nil(A))
=> ( ! [P4: fun(A,fun(A,bool)),X3: A] : X2 != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
=> ~ ! [P4: fun(A,fun(A,bool)),X3: A,Y4: A,Xs2: list(A)] : X2 != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2))) ) ) ).
% successively.cases
tff(fact_3632_sorted__wrt_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(A,fun(A,bool)),list(A))] :
( ! [P4: fun(A,fun(A,bool))] : X2 != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P4),nil(A))
=> ~ ! [P4: fun(A,fun(A,bool)),X3: A,Ys2: list(A)] : X2 != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2)) ) ).
% sorted_wrt.cases
tff(fact_3633_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X2,Xa,Xb) = Y )
=> ( ! [X3: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ! [Y4: A,Ys2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2) )
=> ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X2,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),merges9089515139780605204_merge(A,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys2)) ) ) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V4: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
tff(fact_3634_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),V: A,Va2: list(A)] : merges9089515139780605204_merge(A,R4,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2) ).
% mergesort_by_rel_merge.simps(3)
tff(fact_3635_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list(A),Xs22: list(A),X1: A,X22: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),Xs22)),Xs) ).
% mergesort_by_rel_split.simps(3)
tff(fact_3636_filter__conv__foldr,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter(A,P,Xs) = aa(list(A),list(A),foldr(A,list(A),aTP_Lamp_pd(fun(A,bool),fun(A,fun(list(A),list(A))),P),Xs),nil(A)) ).
% filter_conv_foldr
tff(fact_3637_sorted__wrt01,axiom,
! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> sorted_wrt(A,P,Xs) ) ).
% sorted_wrt01
tff(fact_3638_length__compl__induct,axiom,
! [A: $tType,P: fun(list(A),bool),L: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [E3: A,L5: list(A)] :
( ! [Ll: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L5)))
=> pp(aa(list(A),bool,P,Ll)) )
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),L5))) )
=> pp(aa(list(A),bool,P,L)) ) ) ).
% length_compl_induct
tff(fact_3639_list__decomp__1,axiom,
! [A: $tType,L: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
=> ? [A5: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),nil(A)) ) ).
% list_decomp_1
tff(fact_3640_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V: C,Ps: list(product_prod(B,C))] :
( ( ( L = K )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V)),Ps)),K) = aa(C,option(C),some(C),V) ) )
& ( ( L != K )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V)),Ps)),K) = aa(B,option(C),map_of(B,C,Ps),K) ) ) ) ).
% map_of_Cons_code(2)
tff(fact_3641_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list(A),Xs22: list(A),X2: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs1)),Xs22) ).
% mergesort_by_rel_split.simps(2)
tff(fact_3642_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X2: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
( ( merges295452479951948502_split(A,X2,Xa) = Y )
=> ( ! [Xs12: list(A),Xs23: list(A)] :
( ( X2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) ) ) )
=> ( ! [Xs12: list(A),Xs23: list(A)] :
( ( X2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs12)),Xs23) ) ) )
=> ~ ! [Xs12: list(A),Xs23: list(A)] :
( ( X2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ! [X12: A,X23: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)) )
=> ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs23)),Xs2) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
tff(fact_3643_sorted01,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).
% sorted01
tff(fact_3644_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).
% list.size(4)
tff(fact_3645_nth__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),N) = X2 ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).
% nth_Cons'
tff(fact_3646_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs: list(nat)] :
( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
& ( I = X2 )
& ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_3647_in__set__drop__conv__nth,axiom,
! [A: $tType,X2: A,N: nat,L: list(A)] :
( pp(member(A,X2,aa(list(A),set(A),set2(A),drop(A,N,L))))
<=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L)))
& ( X2 = aa(nat,A,nth(A,L),I3) ) ) ) ).
% in_set_drop_conv_nth
tff(fact_3648_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list(A)] : aa(list(A),set(A),set2(A),drop(A,N,L)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_pe(nat,fun(list(A),fun(A,bool)),N),L)) ).
% set_drop_conv
tff(fact_3649_list__decomp__2,axiom,
! [A: $tType,L: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
=> ? [A5: A,B3: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),nil(A))) ) ).
% list_decomp_2
tff(fact_3650_nth__non__equal__first__eq,axiom,
! [A: $tType,X2: A,Y: A,Xs: list(A),N: nat] :
( ( X2 != Y )
=> ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),N) = Y )
<=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).
% nth_non_equal_first_eq
tff(fact_3651_Cons__replicate__eq,axiom,
! [A: $tType,X2: A,Xs: list(A),N: nat,Y: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) = replicate(A,N,Y) )
<=> ( ( X2 = Y )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
& ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X2) ) ) ) ).
% Cons_replicate_eq
tff(fact_3652_distinct__sorted__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),L)
=> ( distinct(A,L)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))) ) ) ) ) ) ).
% distinct_sorted_mono
tff(fact_3653_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I: nat,J: nat] :
( distinct(A,L)
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J)) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
tff(fact_3654_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A),I: nat,J: nat] :
( distinct(A,L)
=> ( sorted_wrt(A,ord_less_eq(A),L)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J)) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
tff(fact_3655_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P3: product_prod(A,B),Ps: list(product_prod(A,B))] : map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P3),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P3),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P3))) ).
% map_of.simps(2)
tff(fact_3656_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X2: A,Xs: list(A)] :
( ( ( ( Begin = zero_zero(nat) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
=> ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)) ) )
& ( ~ ( ( Begin = zero_zero(nat) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),End)) )
=> ( slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Begin),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs) ) ) ) ).
% slice_Cons
tff(fact_3657_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X2: product_prod(list(A),list(A)),Xa: list(A),Y: product_prod(list(A),list(A))] :
( ( merges295452479951948502_split(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),X2),Xa)))
=> ( ! [Xs12: list(A),Xs23: list(A)] :
( ( X2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),nil(A)))) ) ) )
=> ( ! [Xs12: list(A),Xs23: list(A)] :
( ( X2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs12)),Xs23) )
=> ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) ) )
=> ~ ! [Xs12: list(A),Xs23: list(A)] :
( ( X2 = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) )
=> ! [X12: A,X23: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)) )
=> ( ( Y = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs23)),Xs2) )
=> ~ pp(aa(product_prod(product_prod(list(A),list(A)),list(A)),bool,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
tff(fact_3658_list__decode_Opelims,axiom,
! [X2: nat,Y: list(nat)] :
( ( nat_list_decode(X2) = Y )
=> ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),X2))
=> ( ( ( X2 = zero_zero(nat) )
=> ( ( Y = nil(nat) )
=> ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat))) ) )
=> ~ ! [N3: nat] :
( ( X2 = aa(nat,nat,suc,N3) )
=> ( ( Y = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_pf(nat,fun(nat,list(nat)))),nat_prod_decode(N3)) )
=> ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N3))) ) ) ) ) ) ).
% list_decode.pelims
tff(fact_3659_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( merges9089515139780605204_merge(A,X2,Xa,Xb) = Y )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb))))
=> ( ! [X3: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ! [Y4: A,Ys2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2) )
=> ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),merges9089515139780605204_merge(A,X2,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),merges9089515139780605204_merge(A,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys2)) ) ) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2))))) ) ) )
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A))))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V4: A,Va: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va))))) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
tff(fact_3660_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X2: product_prod(list(A),list(A))] :
( ! [Ys2: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)
=> ~ ! [X3: A,Xs2: list(A),Ys2: list(A)] : X2 != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys2) ) ).
% subset_eq_mset_impl.cases
tff(fact_3661_drop__upt,axiom,
! [M: nat,I: nat,J: nat] : drop(nat,M,upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M),J) ).
% drop_upt
tff(fact_3662_natural__zero__minus__one,axiom,
aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),zero_zero(code_natural)),one_one(code_natural)) = zero_zero(code_natural) ).
% natural_zero_minus_one
tff(fact_3663_list__decode_Opsimps_I1_J,axiom,
( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat)))
=> ( nat_list_decode(zero_zero(nat)) = nil(nat) ) ) ).
% list_decode.psimps(1)
tff(fact_3664_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N)))
=> ( nat_list_decode(aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_pf(nat,fun(nat,list(nat)))),nat_prod_decode(N)) ) ) ).
% list_decode.psimps(2)
tff(fact_3665_list__encode_Oelims,axiom,
! [X2: list(nat),Y: nat] :
( ( nat_list_encode(X2) = Y )
=> ( ( ( X2 = nil(nat) )
=> ( Y != zero_zero(nat) ) )
=> ~ ! [X3: nat,Xs2: list(nat)] :
( ( X2 = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) )
=> ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),nat_list_encode(Xs2)))) ) ) ) ) ).
% list_encode.elims
tff(fact_3666_mergesort__remdups__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( distinct(A,aa(list(A),list(A),mergesort_remdups(A),L))
& sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_remdups(A),L))
& ( aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_remdups(A),L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).
% mergesort_remdups_correct
tff(fact_3667_filter__upt__last,axiom,
! [A: $tType,P: fun(A,bool),L: list(A),Js: list(nat),J: nat,I: nat] :
( ( filter(nat,aa(list(A),fun(nat,bool),aTP_Lamp_pg(fun(A,bool),fun(list(A),fun(nat,bool)),P),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I))) ) ) ) ).
% filter_upt_last
tff(fact_3668_empty__append__eq__id,axiom,
! [A: $tType,X: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X) = X ).
% empty_append_eq_id
tff(fact_3669_op__conc__empty__img__id,axiom,
! [A: $tType,L6: set(list(A))] : aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A))),L6) = L6 ).
% op_conc_empty_img_id
tff(fact_3670_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list(A),E5: A,L2: list(A),E2: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) )
<=> ( ( L1 = nil(A) )
& ( E5 = E2 )
& ( L2 = nil(A) ) ) ) ).
% list_e_eq_lel(2)
tff(fact_3671_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E2: A,L1: list(A),E5: A,L2: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L2)) )
<=> ( ( L1 = nil(A) )
& ( E5 = E2 )
& ( L2 = nil(A) ) ) ) ).
% list_e_eq_lel(1)
tff(fact_3672_list__se__match_I4_J,axiom,
! [A: $tType,L2: list(A),A3: A,L1: list(A)] :
( ( L2 != nil(A) )
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L2) )
<=> ( ( L1 = nil(A) )
& ( L2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).
% list_se_match(4)
tff(fact_3673_list__se__match_I3_J,axiom,
! [A: $tType,L1: list(A),A3: A,L2: list(A)] :
( ( L1 != nil(A) )
=> ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L2) )
<=> ( ( L1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
& ( L2 = nil(A) ) ) ) ) ).
% list_se_match(3)
tff(fact_3674_list__se__match_I2_J,axiom,
! [A: $tType,L2: list(A),L1: list(A),A3: A] :
( ( L2 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
<=> ( ( L1 = nil(A) )
& ( L2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) ) ) ) ) ).
% list_se_match(2)
tff(fact_3675_list__se__match_I1_J,axiom,
! [A: $tType,L1: list(A),L2: list(A),A3: A] :
( ( L1 != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),L2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
<=> ( ( L1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),nil(A)) )
& ( L2 = nil(A) ) ) ) ) ).
% list_se_match(1)
tff(fact_3676_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list(A),E12: A,E23: A,L2: list(A),E1: A,E22: A] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L2))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) )
<=> ( ( L1 = nil(A) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L2 = nil(A) ) ) ) ).
% list_ee_eq_leel(2)
tff(fact_3677_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L1: list(A),E12: A,E23: A,L2: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L2))) )
<=> ( ( L1 = nil(A) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L2 = nil(A) ) ) ) ).
% list_ee_eq_leel(1)
tff(fact_3678_length__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).
% length_append
tff(fact_3679_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K)) )
=> ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,K)) = upt(I,K) ) ) ).
% upt_merge
tff(fact_3680_nth__append__first,axiom,
! [A: $tType,I: nat,L: list(A),L4: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L4)),I) = aa(nat,A,nth(A,L),I) ) ) ).
% nth_append_first
tff(fact_3681_nth__append__length__plus,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).
% nth_append_length_plus
tff(fact_3682_list__match__lel__lel,axiom,
! [A: $tType,C1: list(A),Qs: A,C22: list(A),C12: list(A),Qs2: A,C23: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) )
=> ( ! [C21: list(A)] :
( ( C1 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
=> ( C23 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list(A)] :
( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
=> ( C22 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) ) ) ) ) ) ).
% list_match_lel_lel
tff(fact_3683_foldl__conc__empty__eq,axiom,
! [A: $tType,I: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),I),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),nil(A)),Ww)) ).
% foldl_conc_empty_eq
tff(fact_3684_Misc_Omap__eq__append__conv,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
( ( aa(list(B),list(A),map(B,A,F),Ls) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
<=> ? [L3: list(B),L7: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L7) )
& ( aa(list(B),list(A),map(B,A,F),L3) = Fl )
& ( aa(list(B),list(A),map(B,A,F),L7) = Fl2 ) ) ) ).
% Misc.map_eq_append_conv
tff(fact_3685_Misc_Oappend__eq__map__conv,axiom,
! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F: fun(B,A),Ls: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F),Ls) )
<=> ? [L3: list(B),L7: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L3),L7) )
& ( aa(list(B),list(A),map(B,A,F),L3) = Fl )
& ( aa(list(B),list(A),map(B,A,F),L7) = Fl2 ) ) ) ).
% Misc.append_eq_map_conv
tff(fact_3686_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
( ( aa(list(B),list(A),map(B,A,F),Ls) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
=> ~ ! [L5: list(B),L8: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L5),L8) )
=> ( ( aa(list(B),list(A),map(B,A,F),L5) = Fl )
=> ( aa(list(B),list(A),map(B,A,F),L8) != Fl2 ) ) ) ) ).
% map_eq_appendE
tff(fact_3687_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F: fun(B,A),Ls: list(B)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F),Ls) )
=> ~ ! [L5: list(B),L8: list(B)] :
( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L5),L8) )
=> ( ( aa(list(B),list(A),map(B,A,F),L5) = Fl )
=> ( aa(list(B),list(A),map(B,A,F),L8) != Fl2 ) ) ) ) ).
% append_eq_mapE
tff(fact_3688_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).
% enumerate_append_eq
tff(fact_3689_neq__Nil__revE,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ~ ! [Ll2: list(A),E3: A] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A))) ) ).
% neq_Nil_revE
tff(fact_3690_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
=> ( ! [X3: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),nil(B)))
=> ( ! [Y4: B,Ys2: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),nil(B)))))
=> ( ! [X3: A,Xs2: list(A),Y4: B,Ys2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys2))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),nil(B))))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).
% rev_induct2'
tff(fact_3691_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
<=> ? [Xs4: list(A),X4: A] : L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))) ) ).
% neq_Nil_rev_conv
tff(fact_3692_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),bool))] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ! [X3: A,Y4: B] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),nil(B))))
=> ( ! [X3: A,Xs2: list(A),Y4: B] :
( ( Xs2 != nil(A) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),nil(B)))) )
=> ( ! [X3: A,Y4: B,Ys2: list(B)] :
( ( Ys2 != nil(B) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),nil(B))))) )
=> ( ! [X3: A,Xs2: list(A),Y4: B,Ys2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys2))
=> ( ( Xs2 != nil(A) )
=> ( ( Ys2 != nil(B) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),nil(B))))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
tff(fact_3693_list__Cons__eq__append__cases,axiom,
! [A: $tType,X2: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
=> ( ( ( Ys = nil(A) )
=> ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) )
=> ~ ! [Ys3: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) != Xs ) ) ) ) ).
% list_Cons_eq_append_cases
tff(fact_3694_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list(A),Zs: list(A),X2: A,Xs: list(A)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) )
=> ( ( ( Ys = nil(A) )
=> ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) )
=> ~ ! [Ys3: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys3) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),Zs) != Xs ) ) ) ) ).
% list_append_eq_Cons_cases
tff(fact_3695_in__set__list__format,axiom,
! [A: $tType,E2: A,L: list(A)] :
( pp(member(A,E2,aa(list(A),set(A),set2(A),L)))
=> ~ ! [L12: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L22)) ) ).
% in_set_list_format
tff(fact_3696_xy__in__set__cases,axiom,
! [A: $tType,X2: A,L: list(A),Y: A] :
( pp(member(A,X2,aa(list(A),set(A),set2(A),L)))
=> ( pp(member(A,Y,aa(list(A),set(A),set2(A),L)))
=> ( ( ( X2 = Y )
=> ! [L12: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L22)) )
=> ( ( ( X2 != Y )
=> ! [L12: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) )
=> ~ ( ( X2 != Y )
=> ! [L12: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),L32)))) ) ) ) ) ) ).
% xy_in_set_cases
tff(fact_3697_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list(A),L: list(A),F: fun(B,A),Ww: list(B),A3: B] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F),Ww)),L) )
=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,A3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),L)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),Ww))),L) ) ) ).
% map_consI(2)
tff(fact_3698_list__rest__coinc,axiom,
! [A: $tType,S22: list(A),S1: list(A),R1: list(A),R22: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1)))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S1),R1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S22),R22) )
=> ? [R1p: list(A)] : R22 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),R1p),R1) ) ) ).
% list_rest_coinc
tff(fact_3699_set__union__code,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).
% set_union_code
tff(fact_3700_distinct__match,axiom,
! [A: $tType,Al: list(A),E2: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl)))
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl2)) )
<=> ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
tff(fact_3701_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( ( upt(I,J) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys) )
<=> ? [K4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K4))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K4),J))
& ( upt(I,K4) = Xs )
& ( upt(K4,J) = Ys ) ) ) ) ).
% upt_eq_append_conv
tff(fact_3702_replicate__add,axiom,
! [A: $tType,N: nat,M: nat,X2: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),X2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X2)),replicate(A,M,X2)) ).
% replicate_add
tff(fact_3703_upt__append,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I)),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).
% upt_append
tff(fact_3704_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).
% upt_add_eq_append
tff(fact_3705_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I4: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2))] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,Sigma_0),nil(A)),L0))
=> ( ! [L12: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,Sigma),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),L22)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F),Sigma_0),L0)),L0),nil(A))) ) ) ).
% foldl_rule
tff(fact_3706_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I4: fun(Sigma2,fun(list(A),fun(list(A),bool))),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2)),P: fun(Sigma2,bool)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,Sigma_0),nil(A)),L0))
=> ( ! [L12: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,Sigma),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))),L22)) ) )
=> ( ! [Sigma: Sigma2] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(Sigma2,fun(list(A),fun(list(A),bool)),I4,Sigma),L0),nil(A)))
=> pp(aa(Sigma2,bool,P,Sigma)) )
=> pp(aa(Sigma2,bool,P,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F),Sigma_0),L0))) ) ) ) ).
% foldl_rule_P
tff(fact_3707_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I4: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2))] :
( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,Sigma_0),L0))
=> ( ! [L12: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
=> ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
=> pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X3)),L22)) ) )
=> pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F),Sigma_0),L0)),nil(A))) ) ) ).
% foldl_rule_aux
tff(fact_3708_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I4: fun(Sigma2,fun(list(A),bool)),Sigma_0: Sigma2,L0: list(A),F: fun(Sigma2,fun(A,Sigma2)),P: fun(Sigma2,bool)] :
( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,Sigma_0),L0))
=> ( ! [L12: list(A),L22: list(A),X3: A,Sigma: Sigma2] :
( ( L0 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)) )
=> ( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,Sigma),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),L22)))
=> pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,aa(A,Sigma2,aa(Sigma2,fun(A,Sigma2),F,Sigma),X3)),L22)) ) )
=> ( ! [Sigma: Sigma2] :
( pp(aa(list(A),bool,aa(Sigma2,fun(list(A),bool),I4,Sigma),nil(A)))
=> pp(aa(Sigma2,bool,P,Sigma)) )
=> pp(aa(Sigma2,bool,P,aa(list(A),Sigma2,aa(Sigma2,fun(list(A),Sigma2),foldl(Sigma2,A,F),Sigma_0),L0))) ) ) ) ).
% foldl_rule_aux_P
tff(fact_3709_map__by__foldl,axiom,
! [B: $tType,A: $tType,F: fun(A,B),L: list(A)] : aa(list(A),list(B),aa(list(B),fun(list(A),list(B)),foldl(list(B),A,aTP_Lamp_ph(fun(A,B),fun(list(B),fun(A,list(B))),F)),nil(B)),L) = aa(list(A),list(B),map(A,B,F),L) ).
% map_by_foldl
tff(fact_3710_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [Ys4: list(A),Y3: A] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
& ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).
% length_Suc_rev_conv
tff(fact_3711_length__compl__rev__induct,axiom,
! [A: $tType,P: fun(list(A),bool),L: list(A)] :
( pp(aa(list(A),bool,P,nil(A)))
=> ( ! [L5: list(A),E3: A] :
( ! [Ll: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L5)))
=> pp(aa(list(A),bool,P,Ll)) )
=> pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A))))) )
=> pp(aa(list(A),bool,P,L)) ) ) ).
% length_compl_rev_induct
tff(fact_3712_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list(A)] :
( ~ distinct(A,Xs)
=> ~ ! [Y4: A,Ys2: list(A)] :
( distinct(A,Ys2)
=> ( pp(member(A,Y4,aa(list(A),set(A),set2(A),Ys2)))
=> ! [Zs2: list(A)] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))),Zs2)) ) ) ) ).
% not_distinct_split_distinct
tff(fact_3713_filter__eq__snocD,axiom,
! [A: $tType,P: fun(A,bool),L: list(A),L4: list(A),X2: A] :
( ( filter(A,P,L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) )
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),L)))
& pp(aa(A,bool,P,X2)) ) ) ).
% filter_eq_snocD
tff(fact_3714_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X2: A] : replicate(A,aa(nat,nat,suc,N),X2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) ).
% replicate_Suc_conv_snoc
tff(fact_3715_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs)) ) ) ) ).
% slice_prepend
tff(fact_3716_sorted__append__bigger,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Y: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ! [X3: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y)) )
=> sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ) ) ).
% sorted_append_bigger
tff(fact_3717_upt__eq__lel__conv,axiom,
! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
( ( upt(L,H) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),Is2)) )
<=> ( ( Is1 = upt(L,I) )
& ( Is2 = upt(aa(nat,nat,suc,I),H) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),I))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),H)) ) ) ).
% upt_eq_lel_conv
tff(fact_3718_list__encode_Osimps_I2_J,axiom,
! [X2: nat,Xs: list(nat)] : nat_list_encode(aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X2),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),nat_list_encode(Xs)))) ).
% list_encode.simps(2)
tff(fact_3719_list__encode_Opelims,axiom,
! [X2: list(nat),Y: nat] :
( ( nat_list_encode(X2) = Y )
=> ( pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),X2))
=> ( ( ( X2 = nil(nat) )
=> ( ( Y = zero_zero(nat) )
=> ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),nil(nat))) ) )
=> ~ ! [X3: nat,Xs2: list(nat)] :
( ( X2 = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2) )
=> ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X3),nat_list_encode(Xs2)))) )
=> ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2))) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_3720_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( map_upds(A,B,M,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys),X2,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).
% map_upds_append1
tff(fact_3721_lexn__conv,axiom,
! [A: $tType,R: set(product_prod(A,A)),N: nat] : lexn(A,R,N) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_pi(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R),N))) ).
% lexn_conv
tff(fact_3722_drop__last__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ( drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,L)),nil(A)) ) ) ).
% drop_last_conv
tff(fact_3723_Misc_Olast__in__set,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> pp(member(A,last(A,L),aa(list(A),set(A),set2(A),L))) ) ).
% Misc.last_in_set
tff(fact_3724_last__upt,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).
% last_upt
tff(fact_3725_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,As2: list(A),B2: B,Bs2: list(B)] : map_upds(A,B,M,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2)) = map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As2,Bs2) ).
% map_upds_Cons
tff(fact_3726_map__upds__twist,axiom,
! [A: $tType,B: $tType,A3: A,As2: list(A),M: fun(A,option(B)),B2: B,Bs2: list(B)] :
( ~ pp(member(A,A3,aa(list(A),set(A),set2(A),As2)))
=> ( map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As2,Bs2) = fun_upd(A,option(B),map_upds(A,B,M,As2,Bs2),A3,aa(B,option(B),some(B),B2)) ) ) ).
% map_upds_twist
tff(fact_3727_last__filter,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ( Xs != nil(A) )
=> ( pp(aa(A,bool,P,last(A,Xs)))
=> ( last(A,filter(A,P,Xs)) = last(A,Xs) ) ) ) ).
% last_filter
tff(fact_3728_lexn__length,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),N: nat] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexn(A,R,N)))
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
& ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).
% lexn_length
tff(fact_3729_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2)),Ys)),product(A,B,Xs,Ys)) ).
% product.simps(2)
tff(fact_3730_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V) )
=> ? [Ys2: list(product_prod(B,A)),Zs2: list(product_prod(B,A))] :
( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys2),aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)),Zs2)) )
& ( aa(B,option(A),map_of(B,A,Ys2),K) = none(A) ) ) ) ).
% map_of_Some_split
tff(fact_3731_last__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).
% last_conv_nth
tff(fact_3732_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
=> ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))))
=> ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),Ys))),X2) = aa(B,option(B),some(B),Y) ) ) ) ).
% map_of_distinct_lookup
tff(fact_3733_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B,Y6: B] :
( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
=> ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y6)),Ys))),X2,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_of_distinct_upd3
tff(fact_3734_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
=> ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)),X2,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_of_distinct_upd2
tff(fact_3735_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
=> ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))))
=> ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),Ys))),X2,none(B)) ) ) ) ).
% map_of_distinct_upd4
tff(fact_3736_lex__conv,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pj(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lex_conv
tff(fact_3737_shuffles_Opelims,axiom,
! [A: $tType,X2: list(A),Xa: list(A),Y: set(list(A))] :
( ( shuffles(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Xa)))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = aa(set(list(A)),set(list(A)),insert2(list(A),Xa),bot_bot(set(list(A)))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa))) ) )
=> ( ( ( Xa = nil(A) )
=> ( ( Y = aa(set(list(A)),set(list(A)),insert2(list(A),X2),bot_bot(set(list(A)))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),nil(A)))) ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ! [Y4: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2) )
=> ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys2))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2)))) ) ) ) ) ) ) ) ).
% shuffles.pelims
tff(fact_3738_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list(product_prod(A,C)),Yzs: list(product_prod(C,B))] : relcomp(A,C,B,aa(list(product_prod(A,C)),set(product_prod(A,C)),set2(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set2(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_pl(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).
% set_relcomp
tff(fact_3739_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: fun(code_natural,T)] : aa(code_natural,T,code_case_natural(T,F1,F22),zero_zero(code_natural)) = F1 ).
% natural.simps(4)
tff(fact_3740_Cons__in__lex,axiom,
! [A: $tType,X2: A,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),lex(A,R)))
<=> ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
& ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
| ( ( X2 = Y )
& pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R))) ) ) ) ).
% Cons_in_lex
tff(fact_3741_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list(list(A))] : concat(A,filter(list(A),aTP_Lamp_pm(list(A),bool),Xs)) = concat(A,Xs) ).
% concat_filter_neq_Nil
tff(fact_3742_Nil__notin__lex,axiom,
! [A: $tType,Ys: list(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys),lex(A,R))) ).
% Nil_notin_lex
tff(fact_3743_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)),lex(A,R))) ).
% Nil2_notin_lex
tff(fact_3744_lex__append__leftI,axiom,
! [A: $tType,Ys: list(A),Zs: list(A),R: set(product_prod(A,A)),Xs: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R))) ) ).
% lex_append_leftI
tff(fact_3745_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,A)),A3: A,Xs: list(list(B))] : aa(list(list(B)),A,aa(A,fun(list(list(B)),A),foldl(A,list(B),foldl(A,B,F)),A3),Xs) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F),A3),concat(B,Xs)) ).
% foldl_foldl_conv_concat
tff(fact_3746_product__concat__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_pn(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).
% product_concat_map
tff(fact_3747_lex__append__left__iff,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R)))
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R))) ) ) ).
% lex_append_left_iff
tff(fact_3748_lex__append__leftD,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R))) ) ) ).
% lex_append_leftD
tff(fact_3749_lex__append__rightI,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R)))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),lex(A,R))) ) ) ).
% lex_append_rightI
tff(fact_3750_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list(list(product_prod(A,B)))] : map_of(A,B,concat(product_prod(A,B),Xss)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(list(product_prod(A,B)),fun(A,option(B)),aTP_Lamp_po(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Xss),aTP_Lamp_hj(A,option(B))) ).
% map_of_concat
tff(fact_3751_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(A) )
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
=> ? [N3: nat,Zs2: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N3))
& ( concat(A,replicate(list(A),N3,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).
% comm_append_is_replicate
tff(fact_3752_shuffles_Opinduct,axiom,
! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1)))
=> ( ! [Ys2: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys2)) )
=> ( ! [Xs2: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A))))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),nil(A))) )
=> ( ! [X3: A,Xs2: list(A),Y4: A,Ys2: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2))))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2)))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys2))) ) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).
% shuffles.pinduct
tff(fact_3753_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys)))
=> ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),insert2(list(A),Ys),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(1)
tff(fact_3754_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A))))
=> ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),insert2(list(A),Xs),bot_bot(set(list(A)))) ) ) ).
% shuffles.psimps(2)
tff(fact_3755_distinct__concat_H,axiom,
! [A: $tType,Xs: list(list(A))] :
( distinct(list(A),filter(list(A),aTP_Lamp_pm(list(A),bool),Xs))
=> ( ! [Ys2: list(A)] :
( pp(member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> distinct(A,Ys2) )
=> ( ! [Ys2: list(A),Zs2: list(A)] :
( pp(member(list(A),Ys2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( pp(member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
=> ( ( Ys2 != Zs2 )
=> ( aa(set(A),set(A),inf_inf(set(A),aa(list(A),set(A),set2(A),Ys2)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
=> distinct(A,concat(A,Xs)) ) ) ) ).
% distinct_concat'
tff(fact_3756_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list(A),Y: A,Ys: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))))
=> ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys))) ) ) ).
% shuffles.psimps(3)
tff(fact_3757_product__code,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_pn(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).
% product_code
tff(fact_3758_lenlex__conv,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pp(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lenlex_conv
tff(fact_3759_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I))))
=> ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_3760_splice_Opinduct,axiom,
! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1)))
=> ( ! [Ys2: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys2)))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys2)) )
=> ( ! [X3: A,Xs2: list(A),Ys2: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys2)))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys2),Xs2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys2)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).
% splice.pinduct
tff(fact_3761_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ns),lenlex(A,R)))
<=> ( Ns != nil(A) ) ) ).
% Nil_lenlex_iff1
tff(fact_3762_lenlex__irreflexive,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
( ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R))
=> ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lenlex(A,R))) ) ).
% lenlex_irreflexive
tff(fact_3763_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ns),nil(A)),lenlex(A,R))) ).
% Nil_lenlex_iff2
tff(fact_3764_lenlex__length,axiom,
! [A: $tType,Ms: list(A),Ns: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).
% lenlex_length
tff(fact_3765_lenlex__append1,axiom,
! [A: $tType,Us: list(A),Xs: list(A),R4: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs),lenlex(A,R4)))
=> ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),lenlex(A,R4))) ) ) ).
% lenlex_append1
tff(fact_3766_member__product,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),A4: set(A),B5: set(B)] :
( pp(member(product_prod(A,B),X2,product_product(A,B,A4,B5)))
<=> pp(member(product_prod(A,B),X2,product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))) ) ).
% member_product
tff(fact_3767_Product__Type_Oproduct__def,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] : product_product(A,B,A4,B5) = product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)) ).
% Product_Type.product_def
tff(fact_3768_lenlex__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R)),aTP_Lamp_pq(list(A),product_prod(nat,list(A)))) ).
% lenlex_def
tff(fact_3769_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns)),lenlex(A,R)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
| ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N),R)) )
| ( ( M = N )
& pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R))) ) ) ) ).
% Cons_lenlex_iff
tff(fact_3770_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)))
=> ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_3771_splice_Opelims,axiom,
! [A: $tType,X2: list(A),Xa: list(A),Y: list(A)] :
( ( splice(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Xa)))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = Xa )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa))) ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xa,Xs2)) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xa))) ) ) ) ) ) ).
% splice.pelims
tff(fact_3772_sorted__find__Min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ? [X: A] :
( pp(member(A,X,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X)) )
=> ( find(A,P,Xs) = aa(A,option(A),some(A),lattic643756798350308766er_Min(A,aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_pr(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).
% sorted_find_Min
tff(fact_3773_part__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Pivot: A,Xs: list(B)] : linorder_part(B,A,F,Pivot,Xs) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),filter(B,aa(A,fun(B,bool),aTP_Lamp_ps(fun(B,A),fun(A,fun(B,bool)),F),Pivot),Xs)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),filter(B,aa(A,fun(B,bool),aTP_Lamp_pt(fun(B,A),fun(A,fun(B,bool)),F),Pivot),Xs)),filter(B,aa(A,fun(B,bool),aTP_Lamp_pu(fun(B,A),fun(A,fun(B,bool)),F),Pivot),Xs))) ) ).
% part_def
tff(fact_3774_merge__list__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ls: list(list(A)),As2: list(list(A))] :
( ! [L5: list(A)] :
( pp(member(list(A),L5,aa(list(list(A)),set(list(A)),set2(list(A)),Ls)))
=> ( distinct(A,L5)
& sorted_wrt(A,ord_less_eq(A),L5) ) )
=> ( ! [L5: list(A)] :
( pp(member(list(A),L5,aa(list(list(A)),set(list(A)),set2(list(A)),As2)))
=> ( distinct(A,L5)
& sorted_wrt(A,ord_less_eq(A),L5) ) )
=> ( distinct(A,merge_list(A,As2,Ls))
& sorted_wrt(A,ord_less_eq(A),merge_list(A,As2,Ls))
& ( aa(list(A),set(A),set2(A),merge_list(A,As2,Ls)) = aa(list(A),set(A),set2(A),concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As2),Ls))) ) ) ) ) ) ).
% merge_list_correct
tff(fact_3775_length__splice,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).
% length_splice
tff(fact_3776_splice__replicate,axiom,
! [A: $tType,M: nat,X2: A,N: nat] : splice(A,replicate(A,M,X2),replicate(A,N,X2)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),X2) ).
% splice_replicate
tff(fact_3777_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La: list(A),Acc22: list(list(A)),L: list(A)] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) ) ).
% merge_list.simps(4)
tff(fact_3778_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La: list(A),Acc22: list(list(A))] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) ) ).
% merge_list.simps(3)
tff(fact_3779_merge__list_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] : merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ).
% merge_list.simps(2)
tff(fact_3780_merge__list_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ).
% merge_list.simps(1)
tff(fact_3781_find__SomeD_I1_J,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X2: A] :
( ( find(A,P,Xs) = aa(A,option(A),some(A),X2) )
=> pp(aa(A,bool,P,X2)) ) ).
% find_SomeD(1)
tff(fact_3782_find_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X2: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X2))
=> ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(A,option(A),some(A),X2) ) )
& ( ~ pp(aa(A,bool,P,X2))
=> ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = find(A,P,Xs) ) ) ) ).
% find.simps(2)
tff(fact_3783_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: fun(A,bool)] : find(A,Uu,nil(A)) = none(A) ).
% find.simps(1)
tff(fact_3784_find__SomeD_I2_J,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X2: A] :
( ( find(A,P,Xs) = aa(A,option(A),some(A),X2) )
=> pp(member(A,X2,aa(list(A),set(A),set2(A),Xs))) ) ).
% find_SomeD(2)
tff(fact_3785_find__None__iff2,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( none(A) = find(A,P,Xs) )
<=> ~ ? [X4: A] :
( pp(member(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) ) ) ).
% find_None_iff2
tff(fact_3786_find__None__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( ( find(A,P,Xs) = none(A) )
<=> ~ ? [X4: A] :
( pp(member(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(aa(A,bool,P,X4)) ) ) ).
% find_None_iff
tff(fact_3787_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Pivot: A] : linorder_part(B,A,F,Pivot,nil(B)) = aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),nil(B)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),nil(B)),nil(B))) ) ).
% part_code(1)
tff(fact_3788_find__Some__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X2: A] :
( ( find(A,P,Xs) = aa(A,option(A),some(A),X2) )
<=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3)))
& ( X2 = aa(nat,A,nth(A,Xs),I3) )
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I3))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J2))) ) ) ) ).
% find_Some_iff
tff(fact_3789_find__Some__iff2,axiom,
! [A: $tType,X2: A,P: fun(A,bool),Xs: list(A)] :
( ( aa(A,option(A),some(A),X2) = find(A,P,Xs) )
<=> ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3)))
& ( X2 = aa(nat,A,nth(A,Xs),I3) )
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I3))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J2))) ) ) ) ).
% find_Some_iff2
tff(fact_3790_mergesort__remdups__def,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),mergesort_remdups(A),Xs) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_pv(A,list(A))),Xs)) ) ).
% mergesort_remdups_def
tff(fact_3791_splice_Opsimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list(A),Ys: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Ys)))
=> ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),splice(A,Ys,Xs)) ) ) ).
% splice.psimps(2)
tff(fact_3792_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys)))
=> ( splice(A,nil(A),Ys) = Ys ) ) ).
% splice.psimps(1)
tff(fact_3793_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Pivot: A,X2: B,Xs: list(B)] : linorder_part(B,A,F,Pivot,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs)) = aa(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B))),aa(fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),fun(product_prod(list(B),product_prod(list(B),list(B))),product_prod(list(B),product_prod(list(B),list(B)))),product_case_prod(list(B),product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_px(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),F),Pivot),X2)),linorder_part(B,A,F,Pivot,Xs)) ) ).
% part_code(2)
tff(fact_3794_extract__SomeE,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
& pp(aa(A,bool,P,Y))
& ~ ? [X: A] :
( pp(member(A,X,aa(list(A),set(A),set2(A),Ys)))
& pp(aa(A,bool,P,X)) ) ) ) ).
% extract_SomeE
tff(fact_3795_extract__Some__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
<=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
& pp(aa(A,bool,P,Y))
& ~ ? [X4: A] :
( pp(member(A,X4,aa(list(A),set(A),set2(A),Ys)))
& pp(aa(A,bool,P,X4)) ) ) ) ).
% extract_Some_iff
tff(fact_3796_extract__Cons__code,axiom,
! [A: $tType,P: fun(A,bool),X2: A,Xs: list(A)] :
( ( pp(aa(A,bool,P,X2))
=> ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),X2),Xs))) ) )
& ( ~ pp(aa(A,bool,P,X2))
=> ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A)))),aa(fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A))))),aa(option(product_prod(list(A),product_prod(A,list(A)))),fun(fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A)))))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A)))),none(product_prod(list(A),product_prod(A,list(A))))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_pz(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X2))),extract(A,P,Xs)) ) ) ) ).
% extract_Cons_code
tff(fact_3797_merge__list_Ocases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: product_prod(list(list(A)),list(list(A)))] :
( ( X2 != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
=> ( ! [L5: list(A)] : X2 != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))))
=> ( ! [La2: list(A),Acc23: list(list(A))] : X2 != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A)))
=> ( ! [La2: list(A),Acc23: list(list(A)),L5: list(A)] : X2 != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))))
=> ~ ! [Acc23: list(list(A)),L12: list(A),L22: list(A),Ls2: list(list(A))] : X2 != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2))) ) ) ) ) ) ).
% merge_list.cases
tff(fact_3798_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X2: list(A),Xa: list(A),Y: option(bool)] :
( ( subset_eq_mset_impl(A,X2,Xa) = Y )
=> ( ( ( X2 = nil(A) )
=> ( Y != aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa),nil(A)))) ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( Y != aa(option(product_prod(list(A),product_prod(A,list(A)))),option(bool),aa(fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(bool)),aa(option(bool),fun(fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(bool))),case_option(option(bool),product_prod(list(A),product_prod(A,list(A)))),none(bool)),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_qb(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2))),extract(A,aa(A,fun(A,bool),fequal(A),X3),Xa)) ) ) ) ) ).
% subset_eq_mset_impl.elims
tff(fact_3799_lexord__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lexord(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_qc(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lexord_def
tff(fact_3800_list__collect__set__alt,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),bool),aTP_Lamp_qd(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F),L))) ).
% list_collect_set_alt
tff(fact_3801_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X2: list(A),Xa: list(A),Y: option(bool)] :
( ( subset_eq_mset_impl(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Xa)))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = aa(bool,option(bool),some(bool),aa(bool,bool,fNot,aa(list(A),bool,aa(list(A),fun(list(A),bool),fequal(list(A)),Xa),nil(A)))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa))) ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( ( Y = aa(option(product_prod(list(A),product_prod(A,list(A)))),option(bool),aa(fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(bool)),aa(option(bool),fun(fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(bool))),case_option(option(bool),product_prod(list(A),product_prod(A,list(A)))),none(bool)),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_qb(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs2))),extract(A,aa(A,fun(A,bool),fequal(A),X3),Xa)) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xa))) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
tff(fact_3802_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B] : list_collect_set(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),nil(B))) = aa(B,set(A),F,A3) ).
% list_collect_set_simps(2)
tff(fact_3803_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,set(A))] : list_collect_set(B,A,F,nil(B)) = bot_bot(set(A)) ).
% list_collect_set_simps(1)
tff(fact_3804_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B,L: list(B)] : list_collect_set(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),L)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,A3)),list_collect_set(B,A,F,L)) ).
% list_collect_set_simps(3)
tff(fact_3805_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B),L4: list(B)] : list_collect_set(B,A,F,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L),L4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F,L)),list_collect_set(B,A,F,L4)) ).
% list_collect_set_simps(4)
tff(fact_3806_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X2: list(A),B2: A,Y: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)),lexord(A,R)))
<=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
| ( ( A3 = B2 )
& pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lexord(A,R))) ) ) ) ).
% lexord_cons_cons
tff(fact_3807_lexord__Nil__left,axiom,
! [A: $tType,Y: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Y),lexord(A,R)))
<=> ? [A7: A,X4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),X4) ) ).
% lexord_Nil_left
tff(fact_3808_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X2: fun(C,B),A3: C] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),nil(C)))) = aa(B,set(A),F,aa(C,B,X2,A3)) ).
% list_collect_set_map_simps(2)
tff(fact_3809_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),X2: fun(C,B)] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),nil(C))) = bot_bot(set(A)) ).
% list_collect_set_map_simps(1)
tff(fact_3810_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X2: fun(C,B),A3: C,L: list(C)] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),A3),L))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,aa(C,B,X2,A3))),list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),L))) ).
% list_collect_set_map_simps(3)
tff(fact_3811_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,set(A)),X2: fun(C,B),L: list(C),L4: list(C)] : list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),aa(list(C),list(C),aa(list(C),fun(list(C),list(C)),append(C),L),L4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),L))),list_collect_set(B,A,F,aa(list(C),list(B),map(C,B,X2),L4))) ).
% list_collect_set_map_simps(4)
tff(fact_3812_lexord__irreflexive,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
( ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),R))
=> ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lexord(A,R))) ) ).
% lexord_irreflexive
tff(fact_3813_lexord__linear,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: list(A),Y: list(A)] :
( ! [A5: A,B3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3),R))
| ( A5 = B3 )
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),R)) )
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lexord(A,R)))
| ( X2 = Y )
| pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),X2),lexord(A,R))) ) ) ).
% lexord_linear
tff(fact_3814_lexord__Nil__right,axiom,
! [A: $tType,X2: list(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),nil(A)),lexord(A,R))) ).
% lexord_Nil_right
tff(fact_3815_lexord__append__leftI,axiom,
! [A: $tType,U: list(A),V: list(A),R: set(product_prod(A,A)),X2: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X2),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X2),V)),lexord(A,R))) ) ).
% lexord_append_leftI
tff(fact_3816_lexord__partial__trans,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
( ! [X3: A,Y4: A,Z3: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3),R)) ) ) )
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,R)))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs),lexord(A,R))) ) ) ) ).
% lexord_partial_trans
tff(fact_3817_lexord__append__leftD,axiom,
! [A: $tType,X2: list(A),U: list(A),V: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X2),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X2),V)),lexord(A,R)))
=> ( ! [A5: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),R))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V),lexord(A,R))) ) ) ).
% lexord_append_leftD
tff(fact_3818_lexord__append__rightI,axiom,
! [A: $tType,Y: list(A),X2: list(A),R: set(product_prod(A,A))] :
( ? [B6: A,Z5: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B6),Z5)
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X2),Y)),lexord(A,R))) ) ).
% lexord_append_rightI
tff(fact_3819_lexord__sufE,axiom,
! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs)),lexord(A,R)))
=> ( ( Xs != Ys )
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
=> ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,R))) ) ) ) ) ).
% lexord_sufE
tff(fact_3820_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),aa(list(B),list(set(A)),map(B,set(A),F),L))) ).
% list_collect_set_as_map
tff(fact_3821_lexord__lex,axiom,
! [A: $tType,X2: list(A),Y: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lex(A,R)))
<=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lexord(A,R)))
& ( aa(list(A),nat,size_size(list(A)),X2) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).
% lexord_lex
tff(fact_3822_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),U: list(A),X2: list(A),Y: list(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R))) ) ).
% lexord_append_left_rightI
tff(fact_3823_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lexord(A,R)))
<=> ( ? [X4: A] :
( pp(member(A,X4,aa(list(A),set(A),set2(A),Xs)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R)) )
| pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R))) ) ) ).
% lexord_same_pref_iff
tff(fact_3824_lexord__sufI,axiom,
! [A: $tType,U: list(A),W: list(A),R: set(product_prod(A,A)),V: list(A),Z2: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W),lexord(A,R)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),aa(list(A),nat,size_size(list(A)),U)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),Z2)),lexord(A,R))) ) ) ).
% lexord_sufI
tff(fact_3825_list__collect__set__def,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),bool),aTP_Lamp_qe(fun(B,set(A)),fun(list(B),fun(set(A),bool)),F),L))) ).
% list_collect_set_def
tff(fact_3826_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list(A),Ys: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys) = aa(option(product_prod(list(A),product_prod(A,list(A)))),option(bool),aa(fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(bool)),aa(option(bool),fun(fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(bool))),case_option(option(bool),product_prod(list(A),product_prod(A,list(A)))),none(bool)),aa(fun(list(A),fun(product_prod(A,list(A)),option(bool))),fun(product_prod(list(A),product_prod(A,list(A))),option(bool)),product_case_prod(list(A),product_prod(A,list(A)),option(bool)),aTP_Lamp_qb(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Xs))),extract(A,aa(A,fun(A,bool),fequal(A),X2),Ys)) ).
% subset_eq_mset_impl.simps(2)
tff(fact_3827_List_Olexordp__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( lexordp(A,R,Xs,Ys)
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% List.lexordp_def
tff(fact_3828_upto_Opelims,axiom,
! [X2: int,Xa: int,Y: list(int)] :
( ( upto(X2,Xa) = Y )
=> ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X2),Xa)))
=> ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),Xa))
=> ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int)),Xa)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),Xa))
=> ( Y = nil(int) ) ) )
=> ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),X2),Xa))) ) ) ) ).
% upto.pelims
tff(fact_3829_upto_Opsimps,axiom,
! [I: int,J: int] :
( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J)))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = nil(int) ) ) ) ) ).
% upto.psimps
tff(fact_3830_listrel1__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : listrel1(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_qf(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% listrel1_def
tff(fact_3831_Cons__listrel1__Cons,axiom,
! [A: $tType,X2: A,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R)))
<=> ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
& ( Xs = Ys ) )
| ( ( X2 = Y )
& pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R))) ) ) ) ).
% Cons_listrel1_Cons
tff(fact_3832_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J))
=> ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).
% nth_upto
tff(fact_3833_length__upto,axiom,
! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),I)),one_one(int))) ).
% length_upto
tff(fact_3834_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).
% upto_rec_numeral(1)
tff(fact_3835_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).
% upto_rec_numeral(2)
tff(fact_3836_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).
% upto_rec_numeral(3)
tff(fact_3837_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
=> ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).
% upto_rec_numeral(4)
tff(fact_3838_listrel1I2,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),X2: A] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys)),listrel1(A,R))) ) ).
% listrel1I2
tff(fact_3839_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),X2: A] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R))))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys)),transitive_rtrancl(list(A),listrel1(A,R)))) ) ).
% rtrancl_listrel1_ConsI1
tff(fact_3840_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),nil(A)),listrel1(A,R))) ).
% not_listrel1_Nil
tff(fact_3841_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xs),listrel1(A,R))) ).
% not_Nil_listrel1
tff(fact_3842_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X2: list(A),Y: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),transitive_rtrancl(list(A),listrel1(A,R))))
=> ( aa(list(A),nat,size_size(list(A)),X2) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).
% rtrancl_listrel1_eq_len
tff(fact_3843_listrel1__eq__len,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).
% listrel1_eq_len
tff(fact_3844_append__listrel1I,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
( ( ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
& ( Us = Vs ) )
| ( ( Xs = Ys )
& pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Vs),listrel1(A,R))) ) )
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs)),listrel1(A,R))) ) ).
% append_listrel1I
tff(fact_3845_upto__split2,axiom,
! [I: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).
% upto_split2
tff(fact_3846_upto__rec1,axiom,
! [I: int,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).
% upto_rec1
tff(fact_3847_upto_Osimps,axiom,
! [I: int,J: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = nil(int) ) ) ) ).
% upto.simps
tff(fact_3848_upto_Oelims,axiom,
! [X2: int,Xa: int,Y: list(int)] :
( ( upto(X2,Xa) = Y )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),Xa))
=> ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int)),Xa)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),Xa))
=> ( Y = nil(int) ) ) ) ) ).
% upto.elims
tff(fact_3849_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R)))
=> ( ! [X3: A] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys) )
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y),R)) )
=> ~ ! [Zs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
=> ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys),listrel1(A,R))) ) ) ) ).
% Cons_listrel1E2
tff(fact_3850_Cons__listrel1E1,axiom,
! [A: $tType,X2: A,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Ys),listrel1(A,R)))
=> ( ! [Y4: A] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs) )
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),R)) )
=> ~ ! [Zs2: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2) )
=> ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2),listrel1(A,R))) ) ) ) ).
% Cons_listrel1E1
tff(fact_3851_listrel1I1,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),Xs: list(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)),listrel1(A,R))) ) ).
% listrel1I1
tff(fact_3852_greaterThanAtMost__upto,axiom,
! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).
% greaterThanAtMost_upto
tff(fact_3853_listrel1I,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Vs)) )
=> ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R))) ) ) ) ).
% listrel1I
tff(fact_3854_listrel1E,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
=> ~ ! [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R))
=> ! [Us2: list(A),Vs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs2)) )
=> ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Vs2)) ) ) ) ) ).
% listrel1E
tff(fact_3855_upto__split3,axiom,
! [I: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
=> ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).
% upto_split3
tff(fact_3856_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R))))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),transitive_rtrancl(list(A),listrel1(A,R)))) ) ) ).
% rtrancl_listrel1_ConsI2
tff(fact_3857_greaterThanLessThan__upto,axiom,
! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).
% greaterThanLessThan_upto
tff(fact_3858_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list(A),X2: A,Ys: list(A),Y: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))),listrel1(A,R)))
<=> ( ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
& ( X2 = Y ) )
| ( ( Xs = Ys )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R)) ) ) ) ).
% snoc_listrel1_snoc_iff
tff(fact_3859_listrel1p__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( listrel1p(A,R,Xs,Ys)
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% listrel1p_def
tff(fact_3860_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [L: list(product_prod(A,B))] :
( sorted_wrt(product_prod(A,B),aTP_Lamp_qg(product_prod(A,B),fun(product_prod(A,B),bool)),L)
<=> sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).
% sorted_wrt_map_rev_linord
tff(fact_3861_set__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_qh(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).
% set_zip
tff(fact_3862_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X2: fun(A,B),Xa: list(A),Y: A] :
( ( arg_min_list(A,B,X2,Xa) = Y )
=> ( pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X2),Xa)))
=> ( ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( ( Y = X3 )
=> ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) )
=> ( ! [X3: A,Y4: A,Zs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
=> ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X2,X3)),aa(A,B,X2,arg_min_list(A,B,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))),X3,arg_min_list(A,B,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))) )
=> ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))))) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ( Y = undefined(A) )
=> ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),X2),nil(A)))) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
tff(fact_3863_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),A6: list(A),B4: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B4) )
=> ( ( zip(A,B,A3,B2) = zip(A,B,A6,B4) )
<=> ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ) ) ).
% zip_eq_zip_same_len
tff(fact_3864_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,aTP_Lamp_qi(A,fun(A,bool)),L)
<=> sorted_wrt(A,ord_less_eq(A),rev(A,L)) ) ) ).
% sorted_wrt_rev_linord
tff(fact_3865_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),zip(A,B,Xs,Ys)) ).
% zip_Cons_Cons
tff(fact_3866_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),M))
=> ( map_of(A,B,rev(product_prod(A,B),M)) = map_of(A,B,M) ) ) ).
% map_of_rev_distinct
tff(fact_3867_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X2: A,J: nat,Y: B] : zip(A,B,replicate(A,I,X2),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,ord_min(nat,I),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)) ).
% zip_replicate
tff(fact_3868_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),X2: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X2) = none(B) )
<=> ~ pp(member(A,X2,aa(list(A),set(A),set2(A),Xs))) ) ) ).
% map_of_zip_is_None
tff(fact_3869_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).
% nth_zip
tff(fact_3870_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_qj(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).
% zip_assoc
tff(fact_3871_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
~ ! [L12: list(A),L22: list(B)] :
( ( L = zip(A,B,L12,L22) )
=> ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L22) )
=> ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L22) ) ) ) ).
% pair_list_split
tff(fact_3872_zip__inj,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),A6: list(A),B4: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ( aa(list(A),nat,size_size(list(A)),A6) = aa(list(B),nat,size_size(list(B)),B4) )
=> ( ( zip(A,B,A3,B2) = zip(A,B,A6,B4) )
=> ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ) ) ).
% zip_inj
tff(fact_3873_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_ql(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).
% zip_left_commute
tff(fact_3874_zip__same__conv__map,axiom,
! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_je(A,product_prod(A,A))),Xs) ).
% zip_same_conv_map
tff(fact_3875_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X2: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(member(B,Y,aa(list(B),set(B),set2(B),Ys))) ) ).
% set_zip_rightD
tff(fact_3876_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(member(A,X2,aa(list(A),set(A),set2(A),Xs))) ) ).
% set_zip_leftD
tff(fact_3877_in__set__zipE,axiom,
! [A: $tType,B: $tType,X2: A,Y: B,Xs: list(A),Ys: list(B)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> ~ ( pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ~ pp(member(B,Y,aa(list(B),set(B),set2(B),Ys))) ) ) ).
% in_set_zipE
tff(fact_3878_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs))))
<=> ( pp(member(A,A3,aa(list(A),set(A),set2(A),Xs)))
& ( A3 = B2 ) ) ) ).
% zip_same
tff(fact_3879_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
=> ~ ! [X3: A,Xs5: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5) )
=> ! [Y4: B,Ys3: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3) )
=> ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4) )
=> ( Xys != zip(A,B,Xs5,Ys3) ) ) ) ) ) ).
% zip_eq_ConsE
tff(fact_3880_set__zip__cart,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),L: list(A),L4: list(B)] :
( pp(member(product_prod(A,B),X2,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,L,L4))))
=> pp(member(product_prod(A,B),X2,product_Sigma(A,B,aa(list(A),set(A),set2(A),L),aTP_Lamp_qm(list(B),fun(A,set(B)),L4)))) ) ).
% set_zip_cart
tff(fact_3881_zip__commute,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).
% zip_commute
tff(fact_3882_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).
% last_zip
tff(fact_3883_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X2: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ~ ! [Y4: B] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y4),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).
% in_set_impl_in_set_zip1
tff(fact_3884_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(member(B,Y,aa(list(B),set(B),set2(B),Ys)))
=> ~ ! [X3: A] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).
% in_set_impl_in_set_zip2
tff(fact_3885_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X2: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
<=> ? [Y3: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X2) = aa(B,option(B),some(B),Y3) ) ) ).
% map_of_zip_is_Some
tff(fact_3886_map__upds__fold__map__upd,axiom,
! [A: $tType,B: $tType,M: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M,Ks,Vs) = aa(list(product_prod(A,B)),fun(A,option(B)),aa(fun(A,option(B)),fun(list(product_prod(A,B)),fun(A,option(B))),foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_qo(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))))),M),zip(A,B,Ks,Vs)) ).
% map_upds_fold_map_upd
tff(fact_3887_enumerate__eq__zip,axiom,
! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).
% enumerate_eq_zip
tff(fact_3888_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_qp(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F),G))),zip(D,C,Xs,Ys)) ).
% map_zip_map
tff(fact_3889_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_qq(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F),G))),zip(B,D,Xs,Ys)) ).
% map_zip_map2
tff(fact_3890_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F: fun(C,fun(A,C)),B2: C] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs)))
=> ( aa(list(product_prod(B,A)),C,aa(C,fun(list(product_prod(B,A)),C),foldl(C,product_prod(B,A),aTP_Lamp_qs(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F)),B2),zip(B,A,Xs,Ys)) = aa(list(A),C,aa(C,fun(list(A),C),foldl(C,A,F),B2),Ys) ) ) ).
% foldl_snd_zip
tff(fact_3891_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).
% map_zip2
tff(fact_3892_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))),K)),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).
% map_zip1
tff(fact_3893_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),Xs: list(C),Ys: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_kc(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),zip(C,D,Xs,Ys)) = zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),aa(list(D),list(B),map(D,B,G),Ys)) ).
% map_prod_fun_zip
tff(fact_3894_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_od(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),zip(A,C,Xs,Ys)) ).
% zip_map2
tff(fact_3895_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_qt(fun(C,A),fun(C,fun(B,product_prod(A,B))),F))),zip(C,B,Xs,Ys)) ).
% zip_map1
tff(fact_3896_map__of__zip__map,axiom,
! [A: $tType,B: $tType,Xs: list(A),F: fun(A,B),X: A] :
( ( pp(member(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F),Xs))),X) = aa(B,option(B),some(B),aa(A,B,F,X)) ) )
& ( ~ pp(member(A,X,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F),Xs))),X) = none(B) ) ) ) ).
% map_of_zip_map
tff(fact_3897_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list(B),Xs: list(A),Zs: list(B),X2: A,Y: B,Z2: B] :
( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(A),nat,size_size(list(A)),Xs) )
=> ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
=> ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X2,aa(B,option(B),some(B),Y)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X2,aa(B,option(B),some(B),Z2)) )
=> ( map_of(A,B,zip(A,B,Xs,Ys)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).
% map_of_zip_upd
tff(fact_3898_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( distinct(A,Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I)) ) ) ) ) ).
% map_of_zip_nth
tff(fact_3899_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F: fun(A,fun(C,C)),B2: C] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs)))
=> ( aa(C,C,foldr(product_prod(B,A),C,aa(fun(B,fun(A,fun(C,C))),fun(product_prod(B,A),fun(C,C)),product_case_prod(B,A,fun(C,C)),aTP_Lamp_qu(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F)),zip(B,A,Xs,Ys)),B2) = aa(C,C,foldr(A,C,F,Ys),B2) ) ) ).
% foldr_snd_zip
tff(fact_3900_nths__shift__lemma,axiom,
! [A: $tType,A4: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter(product_prod(A,nat),aTP_Lamp_qv(set(nat),fun(product_prod(A,nat),bool),A4),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_qw(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A4),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).
% nths_shift_lemma
tff(fact_3901_remove__rev__alt__def,axiom,
! [A: $tType,X2: A,Xs: list(A)] : aa(list(A),list(A),remove_rev(A,X2),Xs) = filter(A,aTP_Lamp_qx(A,fun(A,bool),X2),rev(A,Xs)) ).
% remove_rev_alt_def
tff(fact_3902_zip__Cons1,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys) = aa(list(B),list(product_prod(A,B)),case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_qy(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X2),Xs)),Ys) ).
% zip_Cons1
tff(fact_3903_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_qz(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).
% zip_Cons
tff(fact_3904_filter__rev__alt,axiom,
! [A: $tType,P: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),P),L) = filter(A,P,rev(A,L)) ).
% filter_rev_alt
tff(fact_3905_remove__rev__def,axiom,
! [A: $tType,X2: A] : remove_rev(A,X2) = aa(fun(A,bool),fun(list(A),list(A)),filter_rev(A),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),aa(A,fun(A,bool),fequal(A),X2))) ).
% remove_rev_def
tff(fact_3906_min__list_Opelims,axiom,
! [A: $tType] :
( ord(A)
=> ! [X2: list(A),Y: A] :
( ( min_list(A,X2) = Y )
=> ( pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),X2))
=> ( ! [X3: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( ( Y = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ra(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) )
=> ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) )
=> ~ ( ( X2 = nil(A) )
=> ( ( Y = undefined(A) )
=> ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),nil(A))) ) ) ) ) ) ) ).
% min_list.pelims
tff(fact_3907_merge__list_Opsimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A))))))
=> ( merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ) ) ).
% merge_list.psimps(2)
tff(fact_3908_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La: list(A),Acc22: list(list(A))] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))))
=> ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)) ) ) ) ).
% merge_list.psimps(3)
tff(fact_3909_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [La: list(A),Acc22: list(list(A)),L: list(A)] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A))))))
=> ( merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22))) ) ) ) ).
% merge_list.psimps(4)
tff(fact_3910_merge__list_Opsimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))))
=> ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ) ).
% merge_list.psimps(1)
tff(fact_3911_merge__list_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: list(list(A)),Xa: list(list(A)),Y: list(A)] :
( ( merge_list(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X2),Xa)))
=> ( ( ( X2 = nil(list(A)) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = nil(A) )
=> ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))))) ) ) )
=> ( ( ( X2 = nil(list(A)) )
=> ! [L5: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))) )
=> ( ( Y = L5 )
=> ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A)))))) ) ) )
=> ( ! [La2: list(A),Acc23: list(list(A))] :
( ( X2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23) )
=> ( ( Xa = nil(list(A)) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) )
=> ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A))))) ) ) )
=> ( ! [La2: list(A),Acc23: list(list(A))] :
( ( X2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23) )
=> ! [L5: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))) )
=> ( ( Y = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) )
=> ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A)))))) ) ) )
=> ~ ! [L12: list(A),L22: list(A),Ls2: list(list(A))] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2)) )
=> ( ( Y = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L22)),X2),Ls2) )
=> ~ pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),X2),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2))))) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
tff(fact_3912_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc22: list(list(A)),L1: list(A),L2: list(A),Ls: list(list(A))] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),Ls)))))
=> ( merge_list(A,Acc22,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),Ls))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L2)),Acc22),Ls) ) ) ) ).
% merge_list.psimps(5)
tff(fact_3913_partition__filter__conv,axiom,
! [A: $tType,F: fun(A,bool),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter(A,F,Xs)),filter(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),F),Xs)) ).
% partition_filter_conv
tff(fact_3914_partition__rev__filter__conv,axiom,
! [A: $tType,P: fun(A,bool),Yes2: list(A),No2: list(A),Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter(A,P,Xs))),Yes2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,filter(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P),Xs))),No2)) ).
% partition_rev_filter_conv
tff(fact_3915_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L2: list(A)] : merge(A,nil(A),L2) = L2 ) ).
% merge.simps(1)
tff(fact_3916_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,X22: A,L1: list(A),L2: list(A)] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X22))
=> ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L2))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),X22))
=> ( ( ( X1 = X22 )
=> ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),merge(A,L1,L2)) ) )
& ( ( X1 != X22 )
=> ( merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),L2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X22),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),L1),L2)) ) ) ) ) ) ) ).
% merge.simps(3)
tff(fact_3917_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [V: A,Va2: list(A)] : merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2) ) ).
% merge.simps(2)
tff(fact_3918_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),Yes2: list(A),No2: list(A),X2: A,Xs: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = partition_rev(A,P,if(product_prod(list(A),list(A)),aa(A,bool,P,X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Yes2)),No2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),No2))),Xs) ).
% partition_rev.simps(2)
tff(fact_3919_partition__rev_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool),Yes2: list(A),No2: list(A)] : partition_rev(A,P,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) ).
% partition_rev.simps(1)
tff(fact_3920_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Acc22: list(list(A)),L1: list(A),L2: list(A),Ls: list(list(A))] : merge_list(A,Acc22,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),Ls))) = merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L1,L2)),Acc22),Ls) ) ).
% merge_list.simps(5)
tff(fact_3921_partition_Osimps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)) ).
% partition.simps(1)
tff(fact_3922_partition__P,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
=> ( ! [X: A] :
( pp(member(A,X,aa(list(A),set(A),set2(A),Yes2)))
=> pp(aa(A,bool,P,X)) )
& ! [X: A] :
( pp(member(A,X,aa(list(A),set(A),set2(A),No2)))
=> ~ pp(aa(A,bool,P,X)) ) ) ) ).
% partition_P
tff(fact_3923_merge_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: list(A),Xa: list(A),Y: list(A)] :
( ( merge(A,X2,Xa) = Y )
=> ( ( ( X2 = nil(A) )
=> ( Y != Xa ) )
=> ( ! [V4: A,Va: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( ( Xa = nil(A) )
=> ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) ) ) )
=> ~ ! [X12: A,L12: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12) )
=> ! [X23: A,L22: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
=> ~ ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X23))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X23))
=> ( ( ( X12 = X23 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,L22)) ) )
& ( ( X12 != X23 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12),L22)) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
tff(fact_3924_partition_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X2: A,Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_rb(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),X2)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) ).
% partition.simps(2)
tff(fact_3925_partition__rev_Oelims,axiom,
! [A: $tType,X2: fun(A,bool),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X2,Xa,Xb) = Y )
=> ( ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ( ( Xb = nil(A) )
=> ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ! [X3: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( Y != partition_rev(A,X2,if(product_prod(list(A),list(A)),aa(A,bool,X2,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),No))),Xs2) ) ) ) ) ) ).
% partition_rev.elims
tff(fact_3926_partition__set,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),Yes2: list(A),No2: list(A)] :
( ( aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes2),No2) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes2)),aa(list(A),set(A),set2(A),No2)) = aa(list(A),set(A),set2(A),Xs) ) ) ).
% partition_set
tff(fact_3927_inv__image__partition,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
( ! [X3: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,P,X3)) )
=> ( ! [Y4: A] :
( pp(member(A,Y4,aa(list(A),set(A),set2(A),Ys)))
=> ~ pp(aa(A,bool,P,Y4)) )
=> ( aa(set(product_prod(list(A),list(A))),set(list(A)),vimage(list(A),product_prod(list(A),list(A)),partition(A,P)),aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),insert2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).
% inv_image_partition
tff(fact_3928_merge__list_Oelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: list(list(A)),Xa: list(list(A)),Y: list(A)] :
( ( merge_list(A,X2,Xa) = Y )
=> ( ( ( X2 = nil(list(A)) )
=> ( ( Xa = nil(list(A)) )
=> ( Y != nil(A) ) ) )
=> ( ( ( X2 = nil(list(A)) )
=> ! [L5: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))) )
=> ( Y != L5 ) ) )
=> ( ! [La2: list(A),Acc23: list(list(A))] :
( ( X2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23) )
=> ( ( Xa = nil(list(A)) )
=> ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ) )
=> ( ! [La2: list(A),Acc23: list(list(A))] :
( ( X2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23) )
=> ! [L5: list(A)] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))) )
=> ( Y != merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ) )
=> ~ ! [L12: list(A),L22: list(A),Ls2: list(list(A))] :
( ( Xa = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2)) )
=> ( Y != merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L22)),X2),Ls2) ) ) ) ) ) ) ) ) ).
% merge_list.elims
tff(fact_3929_merge__correct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L1: list(A),L2: list(A)] :
( ( distinct(A,L1)
& sorted_wrt(A,ord_less_eq(A),L1) )
=> ( ( distinct(A,L2)
& sorted_wrt(A,ord_less_eq(A),L2) )
=> ( distinct(A,merge(A,L1,L2))
& sorted_wrt(A,ord_less_eq(A),merge(A,L1,L2))
& ( aa(list(A),set(A),set2(A),merge(A,L1,L2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),L1)),aa(list(A),set(A),set2(A),L2)) ) ) ) ) ) ).
% merge_correct
tff(fact_3930_merge__list_Opinduct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A0: list(list(A)),A1: list(list(A)),P: fun(list(list(A)),fun(list(list(A)),bool))] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),A0),A1)))
=> ( ( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A)))))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),nil(list(A)))) )
=> ( ! [L5: list(A)] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))))))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))))) )
=> ( ! [La2: list(A),Acc23: list(list(A))] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A)))))
=> ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),nil(list(A)))) ) )
=> ( ! [La2: list(A),Acc23: list(list(A)),L5: list(A)] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))))))
=> ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L5),nil(list(A))))) ) )
=> ( ! [Acc23: list(list(A)),L12: list(A),L22: list(A),Ls2: list(list(A))] :
( pp(aa(product_prod(list(list(A)),list(list(A))),bool,accp(product_prod(list(list(A)),list(list(A))),merge_list_rel(A)),aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2)))))
=> ( pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),merge(A,L12,L22)),Acc23)),Ls2))
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2)))) ) )
=> pp(aa(list(list(A)),bool,aa(list(list(A)),fun(list(list(A)),bool),P,A0),A1)) ) ) ) ) ) ) ) ).
% merge_list.pinduct
tff(fact_3931_partition__rev_Opelims,axiom,
! [A: $tType,X2: fun(A,bool),Xa: product_prod(list(A),list(A)),Xb: list(A),Y: product_prod(list(A),list(A))] :
( ( partition_rev(A,X2,Xa,Xb) = Y )
=> ( pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa),Xb))))
=> ( ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ( ( Xb = nil(A) )
=> ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ~ pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A))))) ) ) )
=> ~ ! [Yes: list(A),No: list(A)] :
( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No) )
=> ! [X3: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( ( Y = partition_rev(A,X2,if(product_prod(list(A),list(A)),aa(A,bool,X2,X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Yes)),No),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),No))),Xs2) )
=> ~ pp(aa(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),bool,accp(product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),partition_rev_rel(A)),aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,bool),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,bool),product_prod(product_prod(list(A),list(A)),list(A))),X2),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))))) ) ) ) ) ) ) ).
% partition_rev.pelims
tff(fact_3932_merge_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: list(A),Xa: list(A),Y: list(A)] :
( ( merge(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Xa)))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = Xa )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa))) ) )
=> ( ! [V4: A,Va: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( ( Xa = nil(A) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(A)))) ) ) )
=> ~ ! [X12: A,L12: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12) )
=> ! [X23: A,L22: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22) )
=> ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X23))
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X12),X23))
=> ( ( ( X12 = X23 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),merge(A,L12,L22)) ) )
& ( ( X12 != X23 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),merge(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12),L22)) ) ) ) ) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),merge_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X12),L12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22)))) ) ) ) ) ) ) ) ) ).
% merge.pelims
tff(fact_3933_transpose_Opinduct,axiom,
! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),bool)] :
( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),A0))
=> ( ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A))))
=> pp(aa(list(list(A)),bool,P,nil(list(A)))) )
=> ( ! [Xss2: list(list(A))] :
( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)))
=> ( pp(aa(list(list(A)),bool,P,Xss2))
=> pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
=> ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2)))
=> ( pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_rc(A,fun(list(A),list(list(A)))))),Xss2)))))
=> pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) )
=> pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).
% transpose.pinduct
tff(fact_3934_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( aa(list(A),list(A),quicksort_by_rel(A,X2,Xa),Xb) = Y )
=> ( ( ( Xb = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( Y != aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_rd(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X2),Xa),X3)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X2),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)) ) ) ) ) ).
% quicksort_by_rel.elims
tff(fact_3935_set__quicksort__by__rel,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).
% set_quicksort_by_rel
tff(fact_3936_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),nil(A)) = Sl2 ).
% quicksort_by_rel.simps(1)
tff(fact_3937_quicksort__by__rel__remove__acc,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R4,nil(A)),Xs)),Sl2) ).
% quicksort_by_rel_remove_acc
tff(fact_3938_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list(A),R4: fun(A,fun(A,bool)),Xs: list(A)] :
( ( Sl2 != nil(A) )
=> ( aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),quicksort_by_rel(A,R4,nil(A)),Xs)),Sl2) ) ) ).
% quicksort_by_rel_remove_acc_guared
tff(fact_3939_sorted__wrt__quicksort__by__rel,axiom,
! [X9: $tType,R4: fun(X9,fun(X9,bool)),Xs: list(X9)] :
( ! [X3: X9,Y4: X9] :
( pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,X3),Y4))
| pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,Y4),X3)) )
=> ( ! [X3: X9,Y4: X9,Z3: X9] :
( pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,X3),Y4))
=> ( pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,Y4),Z3))
=> pp(aa(X9,bool,aa(X9,fun(X9,bool),R4,X3),Z3)) ) )
=> sorted_wrt(X9,R4,aa(list(X9),list(X9),quicksort_by_rel(X9,R4,nil(X9)),Xs)) ) ) ).
% sorted_wrt_quicksort_by_rel
tff(fact_3940_sorted__quicksort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),quicksort_by_rel(A,ord_less_eq(A),nil(A)),Xs)) ) ).
% sorted_quicksort_by_rel
tff(fact_3941_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A),X2: A,Xs: list(A)] : aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_rd(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R4),Sl2),X2)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R4),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ).
% quicksort_by_rel.simps(2)
tff(fact_3942_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list(A),Xss: list(list(A))] :
( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Xss)))
=> ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_rf(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_rc(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).
% transpose.psimps(3)
tff(fact_3943_transpose_Opelims,axiom,
! [A: $tType,X2: list(list(A)),Y: list(list(A))] :
( ( transpose(A,X2) = Y )
=> ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),X2))
=> ( ( ( X2 = nil(list(A)) )
=> ( ( Y = nil(list(A)) )
=> ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A)))) ) )
=> ( ! [Xss2: list(list(A))] :
( ( X2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
=> ( ( Y = transpose(A,Xss2) )
=> ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
=> ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
( ( X2 = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
=> ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_rf(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_rc(A,fun(list(A),list(list(A)))))),Xss2))))) )
=> ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) ) ) ) ) ) ).
% transpose.pelims
tff(fact_3944_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A),X2: A,Xs: list(A)] :
( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)))))
=> ( aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_rd(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),R4),Sl2),X2)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R4),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs)) ) ) ).
% quicksort_by_rel.psimps(2)
tff(fact_3945_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Xb: list(A),Y: list(A)] :
( ( aa(list(A),list(A),quicksort_by_rel(A,X2,Xa),Xb) = Y )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Xb))))
=> ( ( ( Xb = nil(A) )
=> ( ( Y = Xa )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),nil(A))))) ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( ( Y = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_rd(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),X2),Xa),X3)),partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),X2),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2)) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),X2),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))))) ) ) ) ) ) ).
% quicksort_by_rel.pelims
tff(fact_3946_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A))))
=> ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).
% transpose.psimps(1)
tff(fact_3947_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list(list(A))] :
( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)))
=> ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) = transpose(A,Xss) ) ) ).
% transpose.psimps(2)
tff(fact_3948_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A)] :
( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R4),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl2),nil(A)))))
=> ( aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),nil(A)) = Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
tff(fact_3949_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: fun(A,fun(A,bool)),A1: list(A),A22: list(A),P: fun(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)))] :
( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),A0),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A1),A22))))
=> ( ! [R7: fun(A,fun(A,bool)),Sl: list(A)] :
( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R7),Sl),nil(A))) )
=> ( ! [R7: fun(A,fun(A,bool)),Sl: list(A),X3: A,Xs2: list(A)] :
( pp(aa(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),bool,accp(product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),quicksort_by_rel_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A))),aa(fun(A,fun(A,bool)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,bool)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,bool)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)))))
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y5: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R7),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y5) = Xa3 )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R7),Sl),Y5)) ) )
=> ( ! [Xa3: product_prod(list(A),list(A)),Xb2: list(A),Y5: list(A)] :
( ( Xa3 = partition_rev(A,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R7),X3),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2) )
=> ( ( aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb2),Y5) = Xa3 )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R7),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),quicksort_by_rel(A,R7,Sl),Y5))),Xb2)) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,R7),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),bool)),P,A0),A1),A22)) ) ) ) ).
% quicksort_by_rel.pinduct
tff(fact_3950_filter__rev__aux__alt,axiom,
! [A: $tType,A3: list(A),P: fun(A,bool),L: list(A)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),filter(A,P,rev(A,L))),A3) ).
% filter_rev_aux_alt
tff(fact_3951_lexord__take__index__conv,axiom,
! [A: $tType,X2: list(A),Y: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lexord(A,R)))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X2)),aa(list(A),nat,size_size(list(A)),Y)))
& ( take(A,aa(list(A),nat,size_size(list(A)),X2),Y) = X2 ) )
| ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,ord_min(nat,aa(list(A),nat,size_size(list(A)),X2)),aa(list(A),nat,size_size(list(A)),Y))))
& ( take(A,I3,X2) = take(A,I3,Y) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,X2),I3)),aa(nat,A,nth(A,Y),I3)),R)) ) ) ) ).
% lexord_take_index_conv
tff(fact_3952_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L)))
=> ( ( N != zero_zero(nat) )
=> ( last(A,take(A,N,L)) = aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).
% last_take_nth_conv
tff(fact_3953_revg__fun,axiom,
! [A: $tType,A3: list(A),B2: list(A)] : revg(A,A3,B2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,A3)),B2) ).
% revg_fun
tff(fact_3954_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)),N))
=> ( take(nat,M,upt(I,N)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).
% take_upt
tff(fact_3955_take__Cons__numeral,axiom,
! [A: $tType,V: num,X2: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).
% take_Cons_numeral
tff(fact_3956_take__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list(A)] : take(A,N,drop(A,M,Xs)) = drop(A,M,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs)) ).
% take_drop
tff(fact_3957_revg_Osimps_I2_J,axiom,
! [A: $tType,A3: A,As2: list(A),B2: list(A)] : revg(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),B2) = revg(A,As2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),B2)) ).
% revg.simps(2)
tff(fact_3958_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list(A)] : revg(A,nil(A),B2) = B2 ).
% revg.simps(1)
tff(fact_3959_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X2: A,A3: list(A),Xs: list(A)] :
( ( pp(aa(A,bool,P,X2))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),A3)),P),Xs) ) )
& ( ~ pp(aa(A,bool,P,X2))
=> ( aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),Xs) ) ) ) ).
% filter_rev_aux.simps(2)
tff(fact_3960_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A3: list(A),P: fun(A,bool)] : aa(list(A),list(A),aa(fun(A,bool),fun(list(A),list(A)),filter_rev_aux(A,A3),P),nil(A)) = A3 ).
% filter_rev_aux.simps(1)
tff(fact_3961_drop__take__drop__unsplit,axiom,
! [A: $tType,I: nat,J: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,I,take(A,J,L))),drop(A,J,L)) = drop(A,I,L) ) ) ).
% drop_take_drop_unsplit
tff(fact_3962_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),take(A,J,drop(A,I,Xs))) ).
% take_add
tff(fact_3963_slice__def,axiom,
! [A: $tType,From: nat,To: nat,List: list(A)] : slice(A,From,To,List) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From),drop(A,From,List)) ).
% slice_def
tff(fact_3964_filter__upt__take__conv,axiom,
! [A: $tType,P: fun(A,bool),M: nat,L: list(A),N: nat] : filter(nat,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_rg(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),P),M),L),upt(N,M)) = filter(nat,aa(list(A),fun(nat,bool),aTP_Lamp_pg(fun(A,bool),fun(list(A),fun(nat,bool)),P),L),upt(N,M)) ).
% filter_upt_take_conv
tff(fact_3965_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X2: A,Ys: list(B)] : zip(A,B,replicate(A,N,X2),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2)),take(B,N,Ys)) ).
% zip_replicate1
tff(fact_3966_revg_Oelims,axiom,
! [A: $tType,X2: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X2,Xa) = Y )
=> ( ( ( X2 = nil(A) )
=> ( Y != Xa ) )
=> ~ ! [A5: A,As: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( Y != revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa)) ) ) ) ) ).
% revg.elims
tff(fact_3967_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,replicate(B,N,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))),Y)),take(A,N,Xs)) ).
% zip_replicate2
tff(fact_3968_take__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = nil(A) ) )
& ( ( N != zero_zero(nat) )
=> ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs)) ) ) ) ).
% take_Cons'
tff(fact_3969_filter__nth__ex__nth,axiom,
! [A: $tType,N: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),filter(A,P,Xs))))
=> ? [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M4))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,filter(A,P,Xs)),N) = aa(nat,A,nth(A,Xs),M4) )
& ( filter(A,P,take(A,M4,Xs)) = take(A,N,filter(A,P,Xs)) ) ) ) ).
% filter_nth_ex_nth
tff(fact_3970_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X2: A,Ys: list(B),Xs: list(A),F: fun(A,option(B)),Y: B] :
( ( pp(member(A,X2,aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))))
=> ( map_upds(A,B,fun_upd(A,option(B),F,X2,aa(B,option(B),some(B),Y)),Xs,Ys) = map_upds(A,B,F,Xs,Ys) ) )
& ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))))
=> ( map_upds(A,B,fun_upd(A,option(B),F,X2,aa(B,option(B),some(B),Y)),Xs,Ys) = fun_upd(A,option(B),map_upds(A,B,F,Xs,Ys),X2,aa(B,option(B),some(B),Y)) ) ) ) ).
% map_upd_upds_conv_if
tff(fact_3971_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),N,L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),take(set(A),N,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),L)) ).
% Union_take_drop_id
tff(fact_3972_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R)))
=> ~ ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys)))
=> ( ( take(A,I2,Xs) = take(A,I2,Ys) )
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys),I2)),R)) ) ) ) ) ).
% lex_take_index
tff(fact_3973_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N5: nat,L: list(A),M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N5),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(list(nat),list(A),map(nat,A,nth(A,L)),upt(M2,N5)) = drop(A,M2,take(A,N5,L)) ) ) ).
% map_nth_upt_drop_take_conv
tff(fact_3974_filter__rev__def,axiom,
! [A: $tType] : filter_rev(A) = filter_rev_aux(A,nil(A)) ).
% filter_rev_def
tff(fact_3975_revg_Opelims,axiom,
! [A: $tType,X2: list(A),Xa: list(A),Y: list(A)] :
( ( revg(A,X2,Xa) = Y )
=> ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Xa)))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = Xa )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa))) ) )
=> ~ ! [A5: A,As: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( ( Y = revg(A,As,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),Xa)) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),revg_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),Xa))) ) ) ) ) ) ).
% revg.pelims
tff(fact_3976_take__butlast__conv,axiom,
! [A: $tType,L: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,L) ).
% take_butlast_conv
tff(fact_3977_take__minus__one__conv__butlast,axiom,
! [A: $tType,N: nat,L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),L)))
=> ( take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,take(A,N,L)) ) ) ).
% take_minus_one_conv_butlast
tff(fact_3978_take__update__last,axiom,
! [A: $tType,N: nat,List: list(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),List)))
=> ( list_update(A,take(A,aa(nat,nat,suc,N),List),N,X2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) ) ) ).
% take_update_last
tff(fact_3979_take__update,axiom,
! [A: $tType,N: nat,L: list(A),I: nat,X2: A] : take(A,N,list_update(A,L,I,X2)) = list_update(A,take(A,N,L),I,X2) ).
% take_update
tff(fact_3980_drop__upd__irrelevant,axiom,
! [A: $tType,M: nat,N: nat,L: list(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( drop(A,N,list_update(A,L,M,X2)) = drop(A,N,L) ) ) ).
% drop_upd_irrelevant
tff(fact_3981_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list(A),J: nat,X2: A] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(nat,A,nth(A,list_update(A,L,J,X2)),I) = aa(nat,A,nth(A,L),I) ) ) ).
% nth_update_invalid
tff(fact_3982_length__butlast,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),butlast(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).
% length_butlast
tff(fact_3983_butlast__upd__last__eq,axiom,
! [A: $tType,L: list(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(list(A),nat,size_size(list(A)),L)))
=> ( list_update(A,butlast(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),X2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),L)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) ) ) ).
% butlast_upd_last_eq
tff(fact_3984_butlast__update_H,axiom,
! [A: $tType,L: list(A),I: nat,X2: A] : list_update(A,butlast(A,L),I,X2) = butlast(A,list_update(A,L,I,X2)) ).
% butlast_update'
tff(fact_3985_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list(A),I: nat] :
( distinct(A,Pq)
=> distinct(A,butlast(A,list_update(A,Pq,I,last(A,Pq)))) ) ).
% distinct_butlast_swap
tff(fact_3986_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list(A),I: nat,X2: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I,X2),list_update(B,Ys,I,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)) ).
% zip_update
tff(fact_3987_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list(A),X2: A] :
( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
=> ( butlast(A,list_update(A,Xs,K,X2)) = butlast(A,Xs) ) )
& ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
=> ( butlast(A,list_update(A,Xs,K,X2)) = list_update(A,butlast(A,Xs),K,X2) ) ) ) ).
% butlast_list_update
tff(fact_3988_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list(A),X2: A,Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),list_update(A,L,I,Y))))
<=> ( ( X2 = Y )
| ! [Y3: A] : pp(member(A,X2,aa(list(A),set(A),set2(A),list_update(A,L,I,Y3)))) ) ) ) ).
% in_set_upd_eq_aux
tff(fact_3989_in__set__upd__cases,axiom,
! [A: $tType,X2: A,L: list(A),I: nat,Y: A] :
( pp(member(A,X2,aa(list(A),set(A),set2(A),list_update(A,L,I,Y))))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( X2 != Y ) )
=> pp(member(A,X2,aa(list(A),set(A),set2(A),L))) ) ) ).
% in_set_upd_cases
tff(fact_3990_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list(A),X2: A,Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),list_update(A,L,I,Y))))
<=> ( ( X2 = Y )
| ( pp(member(A,X2,aa(list(A),set(A),set2(A),L)))
& ! [Y3: A] : pp(member(A,X2,aa(list(A),set(A),set2(A),list_update(A,L,I,Y3)))) ) ) ) ) ).
% in_set_upd_eq
tff(fact_3991_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list(A),X2: A] :
( ( ( ( I = J )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))) )
=> ( aa(nat,A,nth(A,list_update(A,L,I,X2)),J) = X2 ) )
& ( ~ ( ( I = J )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))) )
=> ( aa(nat,A,nth(A,list_update(A,L,I,X2)),J) = aa(nat,A,nth(A,L),J) ) ) ) ).
% nth_list_update'
tff(fact_3992_butlast__subset,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( ( Xs != nil(A) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),butlast(A,Xs))),A4)) ) ) ).
% butlast_subset
tff(fact_3993_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list(A),X2: A,Xs: list(A)] :
( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) )
<=> ? [Xl: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl),nil(A)))) ) ).
% butlast_eq_cons_conv
tff(fact_3994_butlast__eq__consE,axiom,
! [A: $tType,L: list(A),X2: A,Xs: list(A)] :
( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) )
=> ~ ! [Xl2: A] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl2),nil(A)))) ) ).
% butlast_eq_consE
tff(fact_3995_butlast__upt,axiom,
! [M: nat,N: nat] : butlast(nat,upt(M,N)) = upt(M,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).
% butlast_upt
tff(fact_3996_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),X2: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(A,B,F,aa(nat,A,nth(A,L),I)) = aa(A,B,F,X2) ) )
=> ( aa(list(A),list(B),map(A,B,F),list_update(A,L,I,X2)) = aa(list(A),list(B),map(A,B,F),L) ) ) ).
% map_upd_eq
tff(fact_3997_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list(A),Xs: list(A),X2: A] :
( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))) )
<=> ( ( Ys != nil(A) )
& ( butlast(A,Ys) = Xs )
& ( last(A,Ys) = X2 ) ) ) ).
% snoc_eq_iff_butlast'
tff(fact_3998_butlast__conv__take,axiom,
! [A: $tType,Xs: list(A)] : butlast(A,Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).
% butlast_conv_take
tff(fact_3999_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( aa(set(A),set(A),insert2(A,aa(nat,A,nth(A,L),I)),aa(list(A),set(A),set2(A),list_update(A,L,I,X2))) = aa(set(A),set(A),insert2(A,X2),aa(list(A),set(A),set2(A),L)) ) ) ).
% insert_swap_set_eq
tff(fact_4000_rev__update,axiom,
! [A: $tType,K: nat,Xs: list(A),Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( rev(A,list_update(A,Xs,K,Y)) = list_update(A,rev(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).
% rev_update
tff(fact_4001_last__list__update,axiom,
! [A: $tType,Xs: list(A),K: nat,X2: A] :
( ( Xs != nil(A) )
=> ( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
=> ( last(A,list_update(A,Xs,K,X2)) = X2 ) )
& ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
=> ( last(A,list_update(A,Xs,K,X2)) = last(A,Xs) ) ) ) ) ).
% last_list_update
tff(fact_4002_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( butlast(A,take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ).
% butlast_take
tff(fact_4003_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
<=> ? [Y3: A,N7: nat] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N7)),Y3),R))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N7),aa(list(A),nat,size_size(list(A)),Xs)))
& ( Ys = list_update(A,Xs,N7,Y3) ) ) ) ).
% listrel1_iff_update
tff(fact_4004_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),X2: B] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ( distinct(A,L)
=> ( list_update(B,aa(list(A),list(B),map(A,B,F),L),I,X2) = aa(list(A),list(B),map(A,B,fun_upd(A,B,F,aa(nat,A,nth(A,L),I),X2)),L) ) ) ) ).
% map_distinct_upd_conv
tff(fact_4005_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list(A),F: fun(B,fun(A,B)),A3: B,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F),A3),list_update(A,Xs,N,X2)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F),aa(A,B,aa(B,fun(A,B),F,aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F),A3),take(A,N,Xs))),X2)),drop(A,aa(nat,nat,suc,N),Xs)) ) ) ).
% foldl_list_update
tff(fact_4006_case__natural__def,axiom,
! [T: $tType,X: T,Xa3: fun(code_natural,T)] : code_case_natural(T,X,Xa3) = code_rec_natural(T,X,aTP_Lamp_rh(fun(code_natural,T),fun(code_natural,fun(T,T)),Xa3)) ).
% case_natural_def
tff(fact_4007_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list(A),M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% nth_rotate
tff(fact_4008_list_Osize__gen_I2_J,axiom,
! [A: $tType,X2: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X2,X21)),size_list(A,X2,X222))),aa(nat,nat,suc,zero_zero(nat))) ).
% list.size_gen(2)
tff(fact_4009_rotate1__length01,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> ( rotate1(A,Xs) = Xs ) ) ).
% rotate1_length01
tff(fact_4010_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: fun(code_natural,fun(T,T))] : aa(code_natural,T,code_rec_natural(T,F1,F22),zero_zero(code_natural)) = F1 ).
% natural.simps(6)
tff(fact_4011_size__list__append,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A),Ys: list(A)] : size_list(A,F,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(A,F,Xs)),size_list(A,F,Ys)) ).
% size_list_append
tff(fact_4012_rotate__length01,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
=> ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).
% rotate_length01
tff(fact_4013_rotate__rotate,axiom,
! [A: $tType,M: nat,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,M),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),Xs) ).
% rotate_rotate
tff(fact_4014_rotate__add,axiom,
! [A: $tType,M: nat,N: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(fun(list(A),list(A)),fun(list(A),list(A))),comp(list(A),list(A),list(A)),rotate(A,M)),rotate(A,N)) ).
% rotate_add
tff(fact_4015_rec__natural__def,axiom,
! [T: $tType,X: T,Xa3: fun(code_natural,fun(T,T)),Xb2: code_natural] : aa(code_natural,T,code_rec_natural(T,X,Xa3),Xb2) = the(T,code_rec_set_natural(T,X,Xa3,Xb2)) ).
% rec_natural_def
tff(fact_4016_lenlex__append2,axiom,
! [A: $tType,R4: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
( irrefl(A,R4)
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys)),lenlex(A,R4)))
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lenlex(A,R4))) ) ) ).
% lenlex_append2
tff(fact_4017_zipf__zip,axiom,
! [A: $tType,B: $tType,L1: list(A),L2: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L2) )
=> ( zipf(A,B,product_prod(A,B),product_Pair(A,B),L1,L2) = zip(A,B,L1,L2) ) ) ).
% zipf_zip
tff(fact_4018_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X2: fun(A,B),Xa: list(A),Xb: list(B),Y: list(B)] :
( ( map_tailrec_rev(A,B,X2,Xa,Xb) = Y )
=> ( pp(aa(product_prod(fun(A,B),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb))))
=> ( ( ( Xa = nil(A) )
=> ( ( Y = Xb )
=> ~ pp(aa(product_prod(fun(A,B),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xb)))) ) )
=> ~ ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ( ( Y = map_tailrec_rev(A,B,X2,As,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,X2,A5)),Xb)) )
=> ~ pp(aa(product_prod(fun(A,B),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,B),product_prod(list(A),list(B))),map_tailrec_rev_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),Xb)))) ) ) ) ) ) ).
% map_tailrec_rev.pelims
tff(fact_4019_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( irrefl(A,R)
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lexord(A,R)))
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R))) ) ) ).
% lexord_same_pref_if_irrefl
tff(fact_4020_irreflI,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( ! [A5: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5),R4))
=> irrefl(A,R4) ) ).
% irreflI
tff(fact_4021_irrefl__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irrefl(A,R)
<=> ! [A7: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),A7),R)) ) ).
% irrefl_def
tff(fact_4022_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(B,C)),A3: A,As2: list(A),B2: B,Bs2: list(B)] : zipf(A,B,C,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2)) = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),F,A3),B2)),zipf(A,B,C,F,As2,Bs2)) ).
% zipf.simps(2)
tff(fact_4023_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(B,C))] : zipf(A,B,C,F,nil(A),nil(B)) = nil(C) ).
% zipf.simps(1)
tff(fact_4024_lexl__not__refl,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: list(A)] :
( irrefl(A,R)
=> ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),X2),lex(A,R))) ) ).
% lexl_not_refl
tff(fact_4025_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X2: fun(A,fun(B,C)),Xa: list(A),Xb: list(B),Y: list(C)] :
( ( zipf(A,B,C,X2,Xa,Xb) = Y )
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( Y != nil(C) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( Y != aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X2,A5),B3)),zipf(A,B,C,X2,As,Bs)) ) ) )
=> ( ( ? [V4: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)
=> ( ( Xb = nil(B) )
=> ( Y != undefined(list(C)) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V4: B,Va: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va)
=> ( Y != undefined(list(C)) ) ) ) ) ) ) ) ).
% zipf.elims
tff(fact_4026_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(A,fun(B,C)),Xa: list(A),Xb: list(B),Y: list(C)] :
( ( zipf(A,B,C,X2,Xa,Xb) = Y )
=> ( pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb))))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( ( Y = nil(C) )
=> ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))))) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( ( Y = aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),aa(B,C,aa(A,fun(B,C),X2,A5),B3)),zipf(A,B,C,X2,As,Bs)) )
=> ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs))))) ) ) )
=> ( ! [V4: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( ( Xb = nil(B) )
=> ( ( Y = undefined(list(C)) )
=> ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(B))))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V4: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
=> ( ( Y = undefined(list(C)) )
=> ~ pp(aa(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),zipf_rel(A,B,C)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va))))) ) ) ) ) ) ) ) ) ).
% zipf.pelims
tff(fact_4027_count__list_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Y: A,Xs: list(A)] :
( ( ( X2 = Y )
=> ( count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),count_list(A,Xs,Y)),one_one(nat)) ) )
& ( ( X2 != Y )
=> ( count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Y) = count_list(A,Xs,Y) ) ) ) ).
% count_list.simps(2)
tff(fact_4028_effect__makeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: array(A),N2: nat] :
( heap_Time_effect(array(A),array_make(A,N,F),H,H3,R,N2)
=> ~ ( ( R = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N)),H)) )
=> ( ( H3 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N)),H)) )
=> ( ( array_get(A,H3,R) = aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N)) )
=> ( pp(aa(array(A),bool,array_present(A,H3),R))
=> ( ~ pp(aa(array(A),bool,array_present(A,H),R))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N2 ) ) ) ) ) ) ) ) ).
% effect_makeE
tff(fact_4029_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [N7: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N7),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),N7)),aa(nat,B,nth(B,Ys),N7)),R)) ) ) ) ).
% listrel_iff_nth
tff(fact_4030_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list(A),R: set(product_prod(A,A))] : pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),listrel(A,A,transitive_rtrancl(A,R)))) ).
% listrel_rtrancl_refl
tff(fact_4031_Array__Time_Oget__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit),Ys: list(A)] : array_get(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Ys,H))) = Xs ) ).
% Array_Time.get_alloc
tff(fact_4032_Array__Time_Opresent__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : pp(aa(array(A),bool,array_present(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)))) ) ).
% Array_Time.present_alloc
tff(fact_4033_Array__Time_Onot__present__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),Xs: list(A)] : ~ pp(aa(array(A),bool,array_present(A,H),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)))) ) ).
% Array_Time.not_present_alloc
tff(fact_4034_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] : pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)),listrel(A,B,R))) ).
% listrel.Nil
tff(fact_4035_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Xs),listrel(A,B,R)))
=> ( Xs = nil(B) ) ) ).
% listrel_Nil1
tff(fact_4036_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list(A),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),nil(B)),listrel(A,B,R)))
=> ( Xs = nil(A) ) ) ).
% listrel_Nil2
tff(fact_4037_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R)))
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).
% listrel_eq_len
tff(fact_4038_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Zs: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,transitive_rtrancl(A,R))))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),listrel(A,A,transitive_rtrancl(A,R))))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs),listrel(A,A,transitive_rtrancl(A,R)))) ) ) ).
% listrel_rtrancl_trans
tff(fact_4039_listrel_OCons,axiom,
! [B: $tType,A: $tType,X2: A,Y: B,R: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),R))
=> ( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R)))
=> pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R))) ) ) ).
% listrel.Cons
tff(fact_4040_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs),listrel(A,B,R)))
=> ~ ! [Y4: B,Ys2: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys2) )
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Y4),R))
=> ~ pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys2),listrel(A,B,R))) ) ) ) ).
% listrel_Cons1
tff(fact_4041_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)),listrel(A,B,R)))
=> ~ ! [X3: A,Xs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y),R))
=> ~ pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys),listrel(A,B,R))) ) ) ) ).
% listrel_Cons2
tff(fact_4042_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,transitive_rtrancl(A,R)))) ) ).
% listrel_reflcl_if_listrel1
tff(fact_4043_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel(A,A,R)))
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),transitive_rtrancl(list(A),listrel1(A,R)))) ) ).
% rtrancl_listrel1_if_listrel
tff(fact_4044_effect__of__listE,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: array(A),N2: nat] :
( heap_Time_effect(array(A),array_of_list(A,Xs),H,H3,R,N2)
=> ~ ( ( R = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)) )
=> ( ( H3 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H)) )
=> ( ( array_get(A,H3,R) = Xs )
=> ( pp(aa(array(A),bool,array_present(A,H3),R))
=> ( ~ pp(aa(array(A),bool,array_present(A,H),R))
=> ( N2 != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ) ) ) ) ) ) ).
% effect_of_listE
tff(fact_4045_effect__newE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: array(A),N2: nat] :
( heap_Time_effect(array(A),aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2),H,H3,R,N2)
=> ~ ( ( R = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,replicate(A,N,X2),H)) )
=> ( ( H3 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,replicate(A,N,X2),H)) )
=> ( ( array_get(A,H3,R) = replicate(A,N,X2) )
=> ( pp(aa(array(A),bool,array_present(A,H3),R))
=> ( ~ pp(aa(array(A),bool,array_present(A,H),R))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N2 ) ) ) ) ) ) ) ) ).
% effect_newE
tff(fact_4046_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R)))
=> ( ( ( A1 = nil(A) )
=> ( A22 != nil(B) ) )
=> ~ ! [X3: A,Y4: B,Xs2: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ! [Ys2: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys2) )
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4),R))
=> ~ pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs2),Ys2),listrel(A,B,R))) ) ) ) ) ) ).
% listrel.cases
tff(fact_4047_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22),listrel(A,B,R)))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X4: A,Y3: B,Xs4: list(A),Ys4: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs4) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys4) )
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R))
& pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs4),Ys4),listrel(A,B,R))) ) ) ) ).
% listrel.simps
tff(fact_4048_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
( pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys),listrel(A,B,R)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X4: product_prod(A,B)] :
( pp(member(product_prod(A,B),X4,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X4)) ) ) ) ).
% listrel_iff_zip
tff(fact_4049_present__alloc__noteq,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [H: heap_ext(product_unit),A3: array(A),Xs: list(B)] :
( pp(aa(array(A),bool,array_present(A,H),A3))
=> array_noteq(A,B,A3,aa(product_prod(array(B),heap_ext(product_unit)),array(B),product_fst(array(B),heap_ext(product_unit)),array_alloc(B,Xs,H))) ) ) ).
% present_alloc_noteq
tff(fact_4050_listrel__def,axiom,
! [B: $tType,A: $tType,X: set(product_prod(A,B))] : listrel(A,B,X) = aa(fun(product_prod(list(A),list(B)),bool),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),bool)),fun(product_prod(list(A),list(B)),bool),product_case_prod(list(A),list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),X)))) ).
% listrel_def
tff(fact_4051_set,axiom,
! [A: $tType] :
( heap(A)
=> ! [Ls: list(A),H: heap_ext(product_unit),New_ls: list(A)] : aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Ls,H)),New_ls),aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Ls,H))) = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,New_ls,H)) ) ).
% set
tff(fact_4052_Array__Time_Oset__same,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X2: list(A),Y: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,X2),aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,Y),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,X2),H) ) ).
% Array_Time.set_same
tff(fact_4053_Array__Time_Ounequal,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),A6: array(A)] :
( ( A3 != A6 )
<=> array_noteq(A,A,A3,A6) ) ) ).
% Array_Time.unequal
tff(fact_4054_Array__Time_Oget__set__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X2: list(A),H: heap_ext(product_unit)] : array_get(A,aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,X2),H),R) = X2 ) ).
% Array_Time.get_set_eq
tff(fact_4055_Array__Time_Oget__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: array(A),S: array(B),X2: list(B),H: heap_ext(product_unit)] :
( array_noteq(A,B,R,S)
=> ( array_get(A,aa(heap_ext(product_unit),heap_ext(product_unit),array_set(B,S,X2),H),R) = array_get(A,H,R) ) ) ) ).
% Array_Time.get_set_neq
tff(fact_4056_Array__Time_Oset__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: array(A),R3: array(B),X2: list(A),X6: list(B),H: heap_ext(product_unit)] :
( array_noteq(A,B,R,R3)
=> ( aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,X2),aa(heap_ext(product_unit),heap_ext(product_unit),array_set(B,R3,X6),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),array_set(B,R3,X6),aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,X2),H)) ) ) ) ).
% Array_Time.set_set_swap
tff(fact_4057_Array__Time_Onoteq__irrefl,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A)] : ~ array_noteq(A,A,R,R) ) ).
% Array_Time.noteq_irrefl
tff(fact_4058_Array__Time_Onoteq__sym,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [A3: array(A),B2: array(B)] :
( array_noteq(A,B,A3,B2)
=> array_noteq(B,A,B2,A3) ) ) ).
% Array_Time.noteq_sym
tff(fact_4059_Field__not__elem,axiom,
! [A: $tType,V: A,R4: set(product_prod(A,A))] :
( ~ pp(member(A,V,field2(A,R4)))
=> ! [X: product_prod(A,A)] :
( pp(member(product_prod(A,A),X,R4))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ri(A,fun(A,fun(A,bool)),V)),X)) ) ) ).
% Field_not_elem
tff(fact_4060_refl__on__def_H,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( refl_on(A,A4,R)
<=> ( ! [X4: product_prod(A,A)] :
( pp(member(product_prod(A,A),X4,R))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_rj(set(A),fun(A,fun(A,bool)),A4)),X4)) )
& ! [X4: A] :
( pp(member(A,X4,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),R)) ) ) ) ).
% refl_on_def'
tff(fact_4061_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X: list(A),Xa3: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X),Xa3))
<=> pp(member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X),Xa3),listrel(A,B,R))) ) ).
% listrelp_listrel_eq
tff(fact_4062_Under__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_Under(A,R,A4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_rk(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A4)) ).
% Under_def
tff(fact_4063_UnderS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_UnderS(A,R,A4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_rl(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A4)) ).
% UnderS_def
tff(fact_4064_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A)] :
( sorted_wrt(A,X2,Xa)
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ( ( ( Xa = nil(A) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),nil(A)))) )
=> ~ ! [X3: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2) )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2))))
=> ~ ( ! [Xa3: A] :
( pp(member(A,Xa3,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Xa3)) )
& sorted_wrt(A,X2,Ys2) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
tff(fact_4065_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
( ( sorted_wrt(A,X2,Xa)
<=> pp(Y) )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ( ( ( Xa = nil(A) )
=> ( pp(Y)
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),nil(A)))) ) )
=> ~ ! [X3: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2) )
=> ( ( pp(Y)
<=> ( ! [Xa2: A] :
( pp(member(A,Xa2,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Xa2)) )
& sorted_wrt(A,X2,Ys2) ) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2)))) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
tff(fact_4066_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ sorted_wrt(A,X2,Xa)
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ~ ! [X3: A,Ys2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2) )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys2))))
=> ( ! [Xa4: A] :
( pp(member(A,Xa4,aa(list(A),set(A),set2(A),Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Xa4)) )
& sorted_wrt(A,X2,Ys2) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
tff(fact_4067_Above__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_Above(A,R,A4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_rm(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A4)) ).
% Above_def
tff(fact_4068_execute__nth_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),aa(nat,heap_Time_Heap(A),array_nth(A,A3),I)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H,A3)),I)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ) ) ).
% execute_nth(1)
tff(fact_4069_Array__Time_Olength__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit),Ys: list(A)] : aa(array(A),nat,array_length(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Ys,H))) = aa(list(A),nat,size_size(list(A)),Xs) ) ).
% Array_Time.length_alloc
tff(fact_4070_Array__Time_Onth__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat] : aa(nat,heap_Time_Heap(A),array_nth(A,A3),I) = heap_Time_guard(A,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_rn(array(A),fun(nat,fun(heap_ext(product_unit),bool)),A3),I),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ro(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),A3),I)) ) ).
% Array_Time.nth_def
tff(fact_4071_success__nthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> heap_Time_success(A,aa(nat,heap_Time_Heap(A),array_nth(A,A3),I),H) ) ) ).
% success_nthI
tff(fact_4072_fails__array__nth,axiom,
! [A: $tType] :
( heap(A)
=> ! [P3: array(A),I: nat,H: heap_ext(product_unit)] :
( time_fails(A,aa(nat,heap_Time_Heap(A),array_nth(A,P3),I),H)
<=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),P3))) ) ) ).
% fails_array_nth
tff(fact_4073_execute__nth_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,array_length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),aa(nat,heap_Time_Heap(A),array_nth(A,A3),I)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ).
% execute_nth(2)
tff(fact_4074_length__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A)] : aa(array(A),nat,array_length(A,H),A3) = aa(list(A),nat,size_size(list(A)),array_get(A,H,A3)) ) ).
% length_def
tff(fact_4075_effect__nthE,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,aa(nat,heap_Time_Heap(A),array_nth(A,A3),I),H,H3,R,N)
=> ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( ( R = aa(nat,A,nth(A,array_get(A,H,A3)),I) )
=> ( ( H3 = H )
=> ( N != one_one(nat) ) ) ) ) ) ) ).
% effect_nthE
tff(fact_4076_effect__nthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),H3: heap_ext(product_unit),R: A,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( ( H3 = H )
=> ( ( R = aa(nat,A,nth(A,array_get(A,H,A3)),I) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,aa(nat,heap_Time_Heap(A),array_nth(A,A3),I),H,H3,R,N) ) ) ) ) ) ).
% effect_nthI
tff(fact_4077_nth_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_nth2(A,A3) = aa(fun(code_integer,nat),fun(code_integer,heap_Time_Heap(A)),aa(fun(nat,heap_Time_Heap(A)),fun(fun(code_integer,nat),fun(code_integer,heap_Time_Heap(A))),comp(nat,heap_Time_Heap(A),code_integer),array_nth(A,A3)),code_nat_of_integer) ) ).
% nth'_def
tff(fact_4078_freeze__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_freeze(A,A3) = heap_Time_heap(list(A),aTP_Lamp_rp(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),A3)) ) ).
% freeze_def
tff(fact_4079_execute__map__entry_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat,F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,array_length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_map_entry(A,I,F,A3)),H) = none(product_prod(array(A),product_prod(heap_ext(product_unit),nat))) ) ) ) ).
% execute_map_entry(2)
tff(fact_4080_execute__freeze,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(list(A)),fun(heap_ext(product_unit),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(list(A)),array_freeze(A,A3)),H) = aa(product_prod(list(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(list(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,H,A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(array(A),nat,array_length(A,H),A3))))) ) ).
% execute_freeze
tff(fact_4081_success__freezeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit)] : heap_Time_success(list(A),array_freeze(A,A3),H) ) ).
% success_freezeI
tff(fact_4082_success__map__entryI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> heap_Time_success(array(A),array_map_entry(A,I,F,A3),H) ) ) ).
% success_map_entryI
tff(fact_4083_effect__freezeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: list(A),N: nat] :
( heap_Time_effect(list(A),array_freeze(A,A3),H,H3,R,N)
=> ~ ( ( H3 = H )
=> ( ( R = array_get(A,H,A3) )
=> ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(array(A),nat,array_length(A,H),A3)),one_one(nat)) ) ) ) ) ) ).
% effect_freezeE
tff(fact_4084_effect__freezeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H3: heap_ext(product_unit),H: heap_ext(product_unit),R: list(A),A3: array(A),N: nat] :
( ( H3 = H )
=> ( ( R = array_get(A,H,A3) )
=> ( ( N = aa(array(A),nat,array_length(A,H),A3) )
=> heap_Time_effect(list(A),array_freeze(A,A3),H,H3,R,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ) ) ).
% effect_freezeI
tff(fact_4085_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_map_entry(A,I,F,A3)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I,aa(A,A,F,aa(nat,A,nth(A,array_get(A,H,A3)),I)),H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).
% execute_map_entry(1)
tff(fact_4086_map__entry__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,F: fun(A,A),A3: array(A)] : array_map_entry(A,I,F,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_rq(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rr(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I),F),A3)) ) ).
% map_entry_def
tff(fact_4087_effect__map__entryI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),H3: heap_ext(product_unit),F: fun(A,A),R: array(A),N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( ( H3 = array_update(A,A3,I,aa(A,A,F,aa(nat,A,nth(A,array_get(A,H,A3)),I)),H) )
=> ( ( R = A3 )
=> ( ( N = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
=> heap_Time_effect(array(A),array_map_entry(A,I,F,A3),H,H3,R,N) ) ) ) ) ) ).
% effect_map_entryI
tff(fact_4088_effect__map__entryE,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,F: fun(A,A),A3: array(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: array(A),N: nat] :
( heap_Time_effect(array(A),array_map_entry(A,I,F,A3),H,H3,R,N)
=> ~ ( ( R = A3 )
=> ( ( H3 = array_update(A,A3,I,aa(A,A,F,aa(nat,A,nth(A,array_get(A,H,A3)),I)),H) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( N != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) ) ) ) ) ) ).
% effect_map_entryE
tff(fact_4089_length__update,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [B2: array(B),I: nat,V: B,H: heap_ext(product_unit)] : array_length(A,array_update(B,B2,I,V,H)) = array_length(A,H) ) ).
% length_update
tff(fact_4090_Array__Time_Opresent__update,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [B2: array(B),I: nat,V: B,H: heap_ext(product_unit)] : array_present(A,array_update(B,B2,I,V,H)) = array_present(A,H) ) ).
% Array_Time.present_update
tff(fact_4091_get__update__elem__neqIndex,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,J: nat,A3: array(A),V: A,H: heap_ext(product_unit)] :
( ( I != J )
=> ( aa(nat,A,nth(A,array_get(A,array_update(A,A3,J,V,H),A3)),I) = aa(nat,A,nth(A,array_get(A,H,A3)),I) ) ) ) ).
% get_update_elem_neqIndex
tff(fact_4092_get__update__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,V: A,H: heap_ext(product_unit)] : array_get(A,array_update(A,A3,I,V,H),A3) = list_update(A,array_get(A,H,A3),I,V) ) ).
% get_update_eq
tff(fact_4093_nth__update__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [A3: array(A),B2: array(B),J: nat,V: B,H: heap_ext(product_unit),I: nat] :
( array_noteq(A,B,A3,B2)
=> ( aa(nat,A,nth(A,array_get(A,array_update(B,B2,J,V,H),A3)),I) = aa(nat,A,nth(A,array_get(A,H,A3)),I) ) ) ) ).
% nth_update_neq
tff(fact_4094_update__swap__neqIndex,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,I6: nat,A3: array(A),V: A,V6: A,H: heap_ext(product_unit)] :
( ( I != I6 )
=> ( array_update(A,A3,I,V,array_update(A,A3,I6,V6,H)) = array_update(A,A3,I6,V6,array_update(A,A3,I,V,H)) ) ) ) ).
% update_swap_neqIndex
tff(fact_4095_update__swap__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [A3: array(A),A6: array(B),I: nat,V: A,I6: nat,V6: B,H: heap_ext(product_unit)] :
( array_noteq(A,B,A3,A6)
=> ( array_update(A,A3,I,V,array_update(B,A6,I6,V6,H)) = array_update(B,A6,I6,V6,array_update(A,A3,I,V,H)) ) ) ) ).
% update_swap_neq
tff(fact_4096_Array__Time_Oupdate__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,X2: A,H: heap_ext(product_unit)] : array_update(A,A3,I,X2,H) = aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,A3,list_update(A,array_get(A,H,A3),I,X2)),H) ) ).
% Array_Time.update_def
tff(fact_4097_execute__swap_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),array_swap(A,I,X2,A3)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,H,A3)),I)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I,X2,H)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).
% execute_swap(1)
tff(fact_4098_swap__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,A3: array(A)] : array_swap(A,I,X2,A3) = heap_Time_guard(A,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_rq(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rs(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),I),X2),A3)) ) ).
% swap_def
tff(fact_4099_effect__swapI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),H3: heap_ext(product_unit),X2: A,R: A,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( ( H3 = array_update(A,A3,I,X2,H) )
=> ( ( R = aa(nat,A,nth(A,array_get(A,H,A3)),I) )
=> ( ( N = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
=> heap_Time_effect(A,array_swap(A,I,X2,A3),H,H3,R,N) ) ) ) ) ) ).
% effect_swapI
tff(fact_4100_effect__swapE,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,A3: array(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,array_swap(A,I,X2,A3),H,H3,R,N)
=> ~ ( ( R = aa(nat,A,nth(A,array_get(A,H,A3)),I) )
=> ( ( H3 = array_update(A,A3,I,X2,H) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( N != aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ) ) ) ) ) ) ).
% effect_swapE
tff(fact_4101_success__swapI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> heap_Time_success(A,array_swap(A,I,X2,A3),H) ) ) ).
% success_swapI
tff(fact_4102_execute__swap_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,array_length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),array_swap(A,I,X2,A3)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ).
% execute_swap(2)
tff(fact_4103_execute__upd_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_upd(A,I,X2,A3)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),A3),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,A3,I,X2,H)),one_one(nat)))) ) ) ) ).
% execute_upd(1)
tff(fact_4104_upd__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,A3: array(A)] : array_upd(A,I,X2,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_rq(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rt(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I),X2),A3)) ) ).
% upd_def
tff(fact_4105_execute__upd_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,array_length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_upd(A,I,X2,A3)),H) = none(product_prod(array(A),product_prod(heap_ext(product_unit),nat))) ) ) ) ).
% execute_upd(2)
tff(fact_4106_effect__updI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),H3: heap_ext(product_unit),V: A,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( ( H3 = array_update(A,A3,I,V,H) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(array(A),array_upd(A,I,V,A3),H,H3,A3,N) ) ) ) ) ).
% effect_updI
tff(fact_4107_upd__ureturn,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,A3: array(A)] : heap_Time_bind(array(A),array(A),array_upd(A,I,X2,A3),aTP_Lamp_ru(array(A),fun(array(A),heap_Time_Heap(array(A))),A3)) = array_upd(A,I,X2,A3) ) ).
% upd_ureturn
tff(fact_4108_success__updI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> heap_Time_success(array(A),array_upd(A,I,X2,A3),H) ) ) ).
% success_updI
tff(fact_4109_fails__array__upd,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,P3: array(A),H: heap_ext(product_unit)] :
( time_fails(array(A),array_upd(A,I,X2,P3),H)
<=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),P3))) ) ) ).
% fails_array_upd
tff(fact_4110_effect__updE,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,V: A,A3: array(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: array(A),N: nat] :
( heap_Time_effect(array(A),array_upd(A,I,V,A3),H,H3,R,N)
=> ~ ( ( R = A3 )
=> ( ( H3 = array_update(A,A3,I,V,H) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,array_length(A,H),A3)))
=> ( N != one_one(nat) ) ) ) ) ) ) ).
% effect_updE
tff(fact_4111_upd_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: code_integer,X2: A] : array_upd2(A,A3,I,X2) = heap_Time_bind(array(A),product_unit,array_upd(A,aa(code_integer,nat,code_nat_of_integer,I),X2,A3),aTP_Lamp_rv(array(A),heap_Time_Heap(product_unit))) ) ).
% upd'_def
tff(fact_4112_execute__len,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(nat),fun(heap_ext(product_unit),option(product_prod(nat,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(nat),array_len(A,A3)),H) = aa(product_prod(nat,product_prod(heap_ext(product_unit),nat)),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),some(product_prod(nat,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(product_prod(heap_ext(product_unit),nat),product_prod(nat,product_prod(heap_ext(product_unit),nat))),product_Pair(nat,product_prod(heap_ext(product_unit),nat)),aa(array(A),nat,array_length(A,H),A3)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ).
% execute_len
tff(fact_4113_in__measures_I2_J,axiom,
! [A: $tType,X2: A,Y: A,F: fun(A,nat),Fs: list(fun(A,nat))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y)))
| ( ( aa(A,nat,F,X2) = aa(A,nat,F,Y) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measures(A,Fs))) ) ) ) ).
% in_measures(2)
tff(fact_4114_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list(A),Xs22: list(A),Xs: list(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs)))),mset(A,aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Xs1))),mset(A,Xs22)) ).
% mset_mergesort_by_rel_split
tff(fact_4115_union__eq__empty,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5) = zero_zero(multiset(A)) )
<=> ( ( M2 = zero_zero(multiset(A)) )
& ( N5 = zero_zero(multiset(A)) ) ) ) ).
% union_eq_empty
tff(fact_4116_empty__eq__union,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A)] :
( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5) )
<=> ( ( M2 = zero_zero(multiset(A)) )
& ( N5 = zero_zero(multiset(A)) ) ) ) ).
% empty_eq_union
tff(fact_4117_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A)] :
( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Y) )
<=> ( ( X2 = zero_zero(multiset(A)) )
& ( Y = zero_zero(multiset(A)) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_4118_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Y) = zero_zero(multiset(A)) )
<=> ( ( X2 = zero_zero(multiset(A)) )
& ( Y = zero_zero(multiset(A)) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_4119_diff__diff__add__mset,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A),P: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),N5)),P) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),P)) ).
% diff_diff_add_mset
tff(fact_4120_mergesort__by__rel__permutes,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A)] : mset(A,aa(list(A),list(A),mergesort_by_rel(A,R4),Xs)) = mset(A,Xs) ).
% mergesort_by_rel_permutes
tff(fact_4121_in__measures_I1_J,axiom,
! [A: $tType,X2: A,Y: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measures(A,nil(fun(A,nat))))) ).
% in_measures(1)
tff(fact_4122_mset__append,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] : mset(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys)) ).
% mset_append
tff(fact_4123_quicksort__by__rel__permutes,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Sl2: list(A),Xs: list(A)] : mset(A,aa(list(A),list(A),quicksort_by_rel(A,R4,Sl2),Xs)) = mset(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Sl2)) ).
% quicksort_by_rel_permutes
tff(fact_4124_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R4: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] : mset(A,merges9089515139780605204_merge(A,R4,Xs,Ys)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys)) ).
% mset_mergesort_by_rel_merge
tff(fact_4125_success__lenI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit)] : heap_Time_success(nat,array_len(A,A3),H) ) ).
% success_lenI
tff(fact_4126_fails__array__len,axiom,
! [A: $tType] :
( heap(A)
=> ! [P3: array(A),H: heap_ext(product_unit)] : ~ time_fails(nat,array_len(A,P3),H) ) ).
% fails_array_len
tff(fact_4127_mset__distrib,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),M2: multiset(A),N5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5) )
=> ~ ! [Am: multiset(A),An: multiset(A)] :
( ( A4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
=> ! [Bm: multiset(A),Bn: multiset(A)] :
( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
=> ( ( M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
=> ( N5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).
% mset_distrib
tff(fact_4128_mset__shuffles,axiom,
! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
( pp(member(list(A),Zs,shuffles(A,Xs,Ys)))
=> ( mset(A,Zs) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset(A,Xs)),mset(A,Ys)) ) ) ).
% mset_shuffles
tff(fact_4129_union__less__mono,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A4: multiset(A),C5: multiset(A),B5: multiset(A),D4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),A4),C5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B5),D4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D4))) ) ) ) ).
% union_less_mono
tff(fact_4130_union__le__mono2,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B5: multiset(A),D4: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B5),D4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D4))) ) ) ).
% union_le_mono2
tff(fact_4131_union__le__mono1,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B5: multiset(A),D4: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),B5),D4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D4),C5))) ) ) ).
% union_le_mono1
tff(fact_4132_multi__union__self__other__eq,axiom,
! [A: $tType,A4: multiset(A),X7: multiset(A),Y7: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),X7) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),Y7) )
=> ( X7 = Y7 ) ) ).
% multi_union_self_other_eq
tff(fact_4133_union__right__cancel,axiom,
! [A: $tType,M2: multiset(A),K2: multiset(A),N5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),K2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),K2) )
<=> ( M2 = N5 ) ) ).
% union_right_cancel
tff(fact_4134_union__left__cancel,axiom,
! [A: $tType,K2: multiset(A),M2: multiset(A),N5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K2),M2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K2),N5) )
<=> ( M2 = N5 ) ) ).
% union_left_cancel
tff(fact_4135_union__commute,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),M2) ).
% union_commute
tff(fact_4136_union__lcomm,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A),K2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),K2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),K2)) ).
% union_lcomm
tff(fact_4137_union__assoc,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A),K2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)),K2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),K2)) ).
% union_assoc
tff(fact_4138_empty__neutral_I1_J,axiom,
! [A: $tType,X2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),X2) = X2 ).
% empty_neutral(1)
tff(fact_4139_empty__neutral_I2_J,axiom,
! [A: $tType,X2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),zero_zero(multiset(A))) = X2 ).
% empty_neutral(2)
tff(fact_4140_diff__union__cancelR,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)),N5) = M2 ).
% diff_union_cancelR
tff(fact_4141_diff__union__cancelL,axiom,
! [A: $tType,N5: multiset(A),M2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),M2)),N5) = M2 ).
% diff_union_cancelL
tff(fact_4142_Multiset_Odiff__add,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N5),Q)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),N5)),Q) ).
% Multiset.diff_add
tff(fact_4143_union__diff__assoc,axiom,
! [A: $tType,C5: multiset(A),B5: multiset(A),A4: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C5),B5) = zero_zero(multiset(A)) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C5)) ) ) ).
% union_diff_assoc
tff(fact_4144_comp__fun__commute__plus__mset,axiom,
! [A: $tType] : finite6289374366891150609ommute(multiset(A),multiset(A),plus_plus(multiset(A))) ).
% comp_fun_commute_plus_mset
tff(fact_4145_len__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_len(A,A3) = heap_Time_tap(nat,aTP_Lamp_rw(array(A),fun(heap_ext(product_unit),nat),A3)) ) ).
% len_def
tff(fact_4146_measures__less,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs)))) ) ).
% measures_less
tff(fact_4147_measures__lesseq,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measures(A,Fs)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs)))) ) ) ).
% measures_lesseq
tff(fact_4148_effect__lengthE,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R: nat,N: nat] :
( heap_Time_effect(nat,array_len(A,A3),H,H3,R,N)
=> ~ ( ( R = aa(array(A),nat,array_length(A,H3),A3) )
=> ( ( H3 = H )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_lengthE
tff(fact_4149_effect__lengthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H3: heap_ext(product_unit),H: heap_ext(product_unit),R: nat,A3: array(A),N: nat] :
( ( H3 = H )
=> ( ( R = aa(array(A),nat,array_length(A,H),A3) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(nat,array_len(A,A3),H,H3,R,N) ) ) ) ) ).
% effect_lengthI
tff(fact_4150_len_H__def,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1(A)
& heap(B) )
=> ! [A3: array(B)] : array_len2(B,A,A3) = heap_Time_bind(nat,A,array_len(B,A3),aTP_Lamp_rx(nat,heap_Time_Heap(A))) ) ).
% len'_def
tff(fact_4151_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),J: nat,X2: A,Y: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( mset(product_prod(A,B),zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,J,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),drop(A,J,Xs))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),take(B,J,Ys)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),drop(B,J,Ys))))) = aa(multiset(product_prod(A,B)),multiset(product_prod(A,B)),aa(product_prod(A,B),fun(multiset(product_prod(A,B)),multiset(product_prod(A,B))),add_mset(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),mset(product_prod(A,B),zip(A,B,Xs,Ys))) ) ) ) ).
% mset_zip_take_Cons_drop_twice
tff(fact_4152_image__mset__map__of,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( aa(multiset(A),multiset(B),image_mset(A,B,aTP_Lamp_ry(list(product_prod(A,B)),fun(A,B),Xs)),mset(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))) = mset(B,aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs)) ) ) ).
% image_mset_map_of
tff(fact_4153_mset__set__Union,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( finite_finite2(A,A4)
=> ( finite_finite2(A,B5)
=> ( ( aa(set(A),set(A),inf_inf(set(A),A4),B5) = bot_bot(set(A)) )
=> ( mset_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),mset_set(A,A4)),mset_set(A,B5)) ) ) ) ) ).
% mset_set_Union
tff(fact_4154_union__mset__add__mset__right,axiom,
! [A: $tType,A4: multiset(A),A3: A,B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),B5)) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)) ).
% union_mset_add_mset_right
tff(fact_4155_union__mset__add__mset__left,axiom,
! [A: $tType,A3: A,A4: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A4)),B5) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)) ).
% union_mset_add_mset_left
tff(fact_4156_image__mset__union,axiom,
! [A: $tType,B: $tType,F: fun(B,A),M2: multiset(B),N5: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,F),aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),M2),N5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(B),multiset(A),image_mset(B,A,F),M2)),aa(multiset(B),multiset(A),image_mset(B,A,F),N5)) ).
% image_mset_union
tff(fact_4157_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F: fun(B,A),P: multiset(B),M13: multiset(A),M24: multiset(A)] :
( ( aa(multiset(B),multiset(A),image_mset(B,A,F),P) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M13),M24) )
=> ~ ! [P1: multiset(B),P22: multiset(B)] :
( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P1),P22) )
=> ( ( aa(multiset(B),multiset(A),image_mset(B,A,F),P1) = M13 )
=> ( aa(multiset(B),multiset(A),image_mset(B,A,F),P22) != M24 ) ) ) ) ).
% mset_map_split_orig
tff(fact_4158_add__mset__add__single,axiom,
! [A: $tType,A3: A,A4: multiset(A)] : aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))) ).
% add_mset_add_single
tff(fact_4159_union__is__single,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A),A3: A] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) )
<=> ( ( ( M2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) )
& ( N5 = zero_zero(multiset(A)) ) )
| ( ( M2 = zero_zero(multiset(A)) )
& ( N5 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ) ) ).
% union_is_single
tff(fact_4160_single__is__union,axiom,
! [A: $tType,A3: A,M2: multiset(A),N5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5) )
<=> ( ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = M2 )
& ( N5 = zero_zero(multiset(A)) ) )
| ( ( M2 = zero_zero(multiset(A)) )
& ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = N5 ) ) ) ) ).
% single_is_union
tff(fact_4161_add__mset__replicate__mset__safe,axiom,
! [A: $tType,B: $tType,M2: multiset(B),A3: B] :
( nO_MATCH(multiset(A),multiset(B),zero_zero(multiset(A)),M2)
=> ( aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),M2) = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),A3),zero_zero(multiset(B)))),M2) ) ) ).
% add_mset_replicate_mset_safe
tff(fact_4162_mset__single__cases,axiom,
! [A: $tType,S: A,C2: multiset(A),R3: A,C6: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),C6) )
=> ( ( ( S = R3 )
=> ( C2 != C6 ) )
=> ~ ( ( C6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A))))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))) ) ) ) ) ) ).
% mset_single_cases
tff(fact_4163_mset__single__cases_H,axiom,
! [A: $tType,S: A,C2: multiset(A),R3: A,C6: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),C6) )
=> ( ( ( S = R3 )
=> ( C2 != C6 ) )
=> ~ ! [Cc: multiset(A)] :
( ( C6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),Cc) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))),Cc) )
=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))) = Cc )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).
% mset_single_cases'
tff(fact_4164_mset__single__cases2,axiom,
! [A: $tType,S: A,C2: multiset(A),R3: A,C6: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),C6) )
=> ( ( ( S = R3 )
=> ( C2 != C6 ) )
=> ~ ( ( C6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))) ) ) ) ) ) ).
% mset_single_cases2
tff(fact_4165_mset__single__cases2_H,axiom,
! [A: $tType,S: A,C2: multiset(A),R3: A,C6: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),C2) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),C6) )
=> ( ( ( S = R3 )
=> ( C2 != C6 ) )
=> ~ ! [Cc: multiset(A)] :
( ( C6 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))) )
=> ( ( C2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))) )
=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C6),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))) = Cc )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),R3),zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).
% mset_single_cases2'
tff(fact_4166_mset__unplusm__dist__cases,axiom,
! [A: $tType,S: A,A4: multiset(A),B5: multiset(A),C5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5) )
=> ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))),C5) ) )
=> ~ ( ( C5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) ) ) ) ) ).
% mset_unplusm_dist_cases
tff(fact_4167_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B5: multiset(A),C5: multiset(A),S: A,A4: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),A4) )
=> ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))),C5) ) )
=> ~ ( ( C5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) )
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),S),zero_zero(multiset(A))))) ) ) ) ) ).
% mset_unplusm_dist_cases2
tff(fact_4168_image__mset_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),Multiset: multiset(A)] : aa(multiset(B),multiset(C),image_mset(B,C,F),aa(multiset(A),multiset(B),image_mset(A,B,G),Multiset)) = aa(multiset(A),multiset(C),image_mset(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),Multiset) ).
% image_mset.compositionality
tff(fact_4169_multiset_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: multiset(A)] : aa(multiset(B),multiset(C),image_mset(B,C,G),aa(multiset(A),multiset(B),image_mset(A,B,F),V)) = aa(multiset(A),multiset(C),image_mset(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% multiset.map_comp
tff(fact_4170_image__mset_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(multiset(A),multiset(B)),fun(multiset(A),multiset(C)),aa(fun(multiset(B),multiset(C)),fun(fun(multiset(A),multiset(B)),fun(multiset(A),multiset(C))),comp(multiset(B),multiset(C),multiset(A)),image_mset(B,C,F)),image_mset(A,B,G)) = image_mset(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).
% image_mset.comp
tff(fact_4171_mset__map__id,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B),X7: multiset(A)] :
( ! [X3: A] : aa(B,A,F,aa(A,B,G,X3)) = X3
=> ( aa(multiset(B),multiset(A),image_mset(B,A,F),aa(multiset(A),multiset(B),image_mset(A,B,G),X7)) = X7 ) ) ).
% mset_map_id
tff(fact_4172_mset__union__2__elem,axiom,
! [A: $tType,A3: A,B2: A,C2: A,M2: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),M2) )
=> ( ( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) = M2 )
& ( B2 = C2 ) )
| ( ( A3 = C2 )
& ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))) = M2 ) ) ) ) ).
% mset_union_2_elem
tff(fact_4173_wcount__add__mset,axiom,
! [A: $tType,F: fun(A,nat),X2: A,M2: multiset(A),A3: A] : wcount(A,F,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),M2),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),if(nat,aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),A3),aa(nat,nat,suc,aa(A,nat,F,A3)),zero_zero(nat))),wcount(A,F,M2,A3)) ).
% wcount_add_mset
tff(fact_4174_subset__mset_Osum__list__update,axiom,
! [A: $tType,K: nat,Xs: list(multiset(A)),X2: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Xs)))
=> ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),list_update(multiset(A),Xs,K,X2)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs)),X2)),aa(nat,multiset(A),nth(multiset(A),Xs),K)) ) ) ).
% subset_mset.sum_list_update
tff(fact_4175_size__multiset__add__mset,axiom,
! [A: $tType,F: fun(A,nat),A3: A,M2: multiset(A)] : size_multiset(A,F,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,aa(A,nat,F,A3))),size_multiset(A,F,M2)) ).
% size_multiset_add_mset
tff(fact_4176_subset__mset_Osum__eq__0__iff,axiom,
! [B: $tType,A: $tType,F5: set(B),F: fun(B,multiset(A))] :
( finite_finite2(B,F5)
=> ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,F5) = zero_zero(multiset(A)) )
<=> ! [X4: B] :
( pp(member(B,X4,F5))
=> ( aa(B,multiset(A),F,X4) = zero_zero(multiset(A)) ) ) ) ) ).
% subset_mset.sum_eq_0_iff
tff(fact_4177_size__multiset__union,axiom,
! [A: $tType,F: fun(A,nat),M2: multiset(A),N5: multiset(A)] : size_multiset(A,F,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_multiset(A,F,M2)),size_multiset(A,F,N5)) ).
% size_multiset_union
tff(fact_4178_subset__mset_Osum__list__eq__0__iff,axiom,
! [A: $tType,Ns: list(multiset(A))] :
( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns) = zero_zero(multiset(A)) )
<=> ! [X4: multiset(A)] :
( pp(member(multiset(A),X4,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Ns)))
=> ( X4 = zero_zero(multiset(A)) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
tff(fact_4179_wcount__union,axiom,
! [A: $tType,F: fun(A,nat),M2: multiset(A),N5: multiset(A),A3: A] : wcount(A,F,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),wcount(A,F,M2,A3)),wcount(A,F,N5,A3)) ).
% wcount_union
tff(fact_4180_subset__mset_Oelem__le__sum__list,axiom,
! [A: $tType,K: nat,Ns: list(multiset(A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Ns)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(nat,multiset(A),nth(multiset(A),Ns),K)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns))) ) ).
% subset_mset.elem_le_sum_list
tff(fact_4181_subset__mset_Osum__mono2,axiom,
! [A: $tType,B: $tType,B5: set(B),A4: set(B),F: fun(B,multiset(A))] :
( finite_finite2(B,B5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A4),B5))
=> ( ! [B3: B] :
( pp(member(B,B3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A4)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F,B3))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,A4)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,B5))) ) ) ) ).
% subset_mset.sum_mono2
tff(fact_4182_next_Osimps,axiom,
! [V: code_natural,W: code_natural] : aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),V),W)) = aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),V),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2)))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),W),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))),one_one(code_natural)))),one_one(code_natural))),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),V),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),V),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,one2))))))))))))))))),minus_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),W),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))))))))),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),W),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,one2)))))))))))))))))),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))) ).
% next.simps
tff(fact_4183_find__dropWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_rz(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P),Xs)) ).
% find_dropWhile
tff(fact_4184_mset__subset__eq__mono__add__right__cancel,axiom,
! [A: $tType,A4: multiset(A),C5: multiset(A),B5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),B5)) ) ).
% mset_subset_eq_mono_add_right_cancel
tff(fact_4185_mset__subset__eq__mono__add__left__cancel,axiom,
! [A: $tType,C5: multiset(A),A4: multiset(A),B5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),A4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B5)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),B5)) ) ).
% mset_subset_eq_mono_add_left_cancel
tff(fact_4186_subset__mset_Oadd__le__cancel__right,axiom,
! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% subset_mset.add_le_cancel_right
tff(fact_4187_subset__mset_Oadd__le__cancel__left,axiom,
! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% subset_mset.add_le_cancel_left
tff(fact_4188_subset__mset_Ole__add__same__cancel2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2)) ) ).
% subset_mset.le_add_same_cancel2
tff(fact_4189_subset__mset_Ole__add__same__cancel1,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2)) ) ).
% subset_mset.le_add_same_cancel1
tff(fact_4190_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),B2))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A)))) ) ).
% subset_mset.add_le_same_cancel2
tff(fact_4191_subset__mset_Oadd__le__same__cancel1,axiom,
! [A: $tType,B2: multiset(A),A3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3)),B2))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A)))) ) ).
% subset_mset.add_le_same_cancel1
tff(fact_4192_mset__subset__eq__multiset__union__diff__commute,axiom,
! [A: $tType,B5: multiset(A),A4: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),A4))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),B5)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C5)),B5) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
tff(fact_4193_subset__mset_Oadd__diff__assoc2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)),C2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)),A3) ) ) ).
% subset_mset.add_diff_assoc2
tff(fact_4194_subset__mset_Oadd__diff__assoc,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2)),A3) ) ) ).
% subset_mset.add_diff_assoc
tff(fact_4195_subset__mset_Odiff__add,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)),A3) = B2 ) ) ).
% subset_mset.diff_add
tff(fact_4196_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),A3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% mset_le_add_mset_decr_left1
tff(fact_4197_mset__le__incr__right2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))) ) ).
% mset_le_incr_right2
tff(fact_4198_mset__le__incr__right1,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))) ) ).
% mset_le_incr_right1
tff(fact_4199_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% mset_le_decr_left2
tff(fact_4200_mset__le__decr__left1,axiom,
! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% mset_le_decr_left1
tff(fact_4201_mset__union__subset,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),C5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),C5))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),C5)) ) ) ).
% mset_union_subset
tff(fact_4202_mset__le__distrib,axiom,
! [A: $tType,X7: multiset(A),A4: multiset(A),B5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)))
=> ~ ! [Xa5: multiset(A),Xb5: multiset(A)] :
( ( X7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa5),Xb5) )
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xa5),A4))
=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xb5),B5)) ) ) ) ).
% mset_le_distrib
tff(fact_4203_mset__le__addE,axiom,
! [A: $tType,Xs: multiset(A),Ys: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Xs),Ys))
=> ~ ! [Zs2: multiset(A)] : Ys != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xs),Zs2) ) ).
% mset_le_addE
tff(fact_4204_mset__subset__eq__exists__conv,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),B5))
<=> ? [C7: multiset(A)] : B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C7) ) ).
% mset_subset_eq_exists_conv
tff(fact_4205_mset__subset__eq__add__right,axiom,
! [A: $tType,B5: multiset(A),A4: multiset(A)] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))) ).
% mset_subset_eq_add_right
tff(fact_4206_mset__subset__eq__mono__add,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A),D4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),B5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C5),D4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),D4))) ) ) ).
% mset_subset_eq_mono_add
tff(fact_4207_mset__subset__eq__add__left,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))) ).
% mset_subset_eq_add_left
tff(fact_4208_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% subset_mset.add_le_imp_le_right
tff(fact_4209_subset__mset_Oadd__le__imp__le__left,axiom,
! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2)) ) ).
% subset_mset.add_le_imp_le_left
tff(fact_4210_subset__mset_Oadd__right__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))) ) ).
% subset_mset.add_right_mono
tff(fact_4211_subset__mset_Oadd__left__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))) ) ).
% subset_mset.add_left_mono
tff(fact_4212_subset__mset_Ole__iff__add,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
<=> ? [C4: multiset(A)] : B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C4) ) ).
% subset_mset.le_iff_add
tff(fact_4213_subset__mset_Oless__eqE,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ~ ! [C3: multiset(A)] : B2 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3) ) ).
% subset_mset.less_eqE
tff(fact_4214_subset__mset_Oadd__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),D3))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),D3))) ) ) ).
% subset_mset.add_mono
tff(fact_4215_full__exhaustive__natural_H_Ocases,axiom,
! [X2: product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: code_natural,I2: code_natural] : X2 != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),D2),I2)) ).
% full_exhaustive_natural'.cases
tff(fact_4216_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( quickc3360725361186068524ustive(B)
& cl_HOL_Oequal(A)
& quickc3360725361186068524ustive(A) )
=> ! [X2: product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),I2: code_natural,D2: code_natural] : X2 != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I2),D2)) ) ).
% full_exhaustive_fun'.cases
tff(fact_4217_exhaustive__natural_H_Ocases,axiom,
! [X2: product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(code_natural,option(product_prod(bool,list(code_term)))),D2: code_natural,I2: code_natural] : X2 != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(code_natural,option(product_prod(bool,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),D2),I2)) ).
% exhaustive_natural'.cases
tff(fact_4218_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( quickc658316121487927005ustive(B)
& cl_HOL_Oequal(A)
& quickc658316121487927005ustive(A) )
=> ! [X2: product_prod(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(fun(A,B),option(product_prod(bool,list(code_term)))),I2: code_natural,D2: code_natural] : X2 != aa(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),aa(fun(fun(A,B),option(product_prod(bool,list(code_term)))),fun(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))),product_Pair(fun(fun(A,B),option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I2),D2)) ) ).
% exhaustive_fun'.cases
tff(fact_4219_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))] :
~ ! [F3: fun(code_natural,A),N3: code_natural,M4: code_natural] : X2 != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N3),M4)) ).
% Lazy_Sequence.iterate_upto.cases
tff(fact_4220_iter_Ocases,axiom,
! [A: $tType,X2: product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))] :
~ ! [Random: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),Nrandom: code_natural,Seed: product_prod(code_natural,code_natural)] : X2 != aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),aa(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))),product_prod(code_natural,product_prod(code_natural,code_natural))),Random),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),Nrandom),Seed)) ).
% iter.cases
tff(fact_4221_log_Ocases,axiom,
! [X2: product_prod(code_natural,code_natural)] :
~ ! [B3: code_natural,I2: code_natural] : X2 != aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),B3),I2) ).
% log.cases
tff(fact_4222_mset__le__subtract,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),B5))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C5))) ) ).
% mset_le_subtract
tff(fact_4223_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),A3)),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),C2),zero_zero(multiset(A)))),B2)) ) ).
% mset_le_add_mset_decr_left2
tff(fact_4224_mset__le__single__cases,axiom,
! [A: $tType,M2: multiset(A),A3: A] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))))
=> ( ( M2 != zero_zero(multiset(A)) )
=> ( M2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ) ).
% mset_le_single_cases
tff(fact_4225_mset__le__add__mset,axiom,
! [A: $tType,X2: A,B5: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),B5)),C5))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A)))),C5))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),C5)) ) ) ).
% mset_le_add_mset
tff(fact_4226_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X2),zero_zero(multiset(A))))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),Y),zero_zero(multiset(A))))
=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Y) = zero_zero(multiset(A)) )
<=> ( ( X2 = zero_zero(multiset(A)) )
& ( Y = zero_zero(multiset(A)) ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
tff(fact_4227_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),X2))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),Y))
=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Y) = zero_zero(multiset(A)) )
<=> ( ( X2 = zero_zero(multiset(A)) )
& ( Y = zero_zero(multiset(A)) ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
tff(fact_4228_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A))))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),zero_zero(multiset(A))))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)),zero_zero(multiset(A)))) ) ) ).
% subset_mset.add_nonpos_nonpos
tff(fact_4229_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2))) ) ) ).
% subset_mset.add_nonneg_nonneg
tff(fact_4230_subset__mset_Oadd__increasing2,axiom,
! [A: $tType,C2: multiset(A),B2: multiset(A),A3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),C2))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),A3))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2))) ) ) ).
% subset_mset.add_increasing2
tff(fact_4231_subset__mset_Oadd__decreasing2,axiom,
! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),zero_zero(multiset(A))))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),B2)) ) ) ).
% subset_mset.add_decreasing2
tff(fact_4232_subset__mset_Oadd__increasing,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),C2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2))) ) ) ).
% subset_mset.add_increasing
tff(fact_4233_subset__mset_Oadd__decreasing,axiom,
! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A))))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)),B2)) ) ) ).
% subset_mset.add_decreasing
tff(fact_4234_mset__le__subtract__left,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),X7: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),X7))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X7),A4)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),X7)) ) ) ).
% mset_le_subtract_left
tff(fact_4235_mset__le__subtract__right,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),X7: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),X7))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X7),B5)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),X7)) ) ) ).
% mset_le_subtract_right
tff(fact_4236_multiset__diff__union__assoc,axiom,
! [A: $tType,C5: multiset(A),B5: multiset(A),A4: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C5),B5))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C5)) ) ) ).
% multiset_diff_union_assoc
tff(fact_4237_subset__eq__diff__conv,axiom,
! [A: $tType,A4: multiset(A),C5: multiset(A),B5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),C5)),B5))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5))) ) ).
% subset_eq_diff_conv
tff(fact_4238_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3) = C2 )
<=> ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
tff(fact_4239_subset__mset_Oadd__diff__inverse,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) = B2 ) ) ).
% subset_mset.add_diff_inverse
tff(fact_4240_subset__mset_Odiff__diff__right,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),B2) ) ) ).
% subset_mset.diff_diff_right
tff(fact_4241_subset__mset_Odiff__add__assoc2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)),A3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)),C2) ) ) ).
% subset_mset.diff_add_assoc2
tff(fact_4242_subset__mset_Odiff__add__assoc,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2)),A3) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)) ) ) ).
% subset_mset.diff_add_assoc
tff(fact_4243_subset__mset_Ole__diff__conv2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B2),A3)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3)),B2)) ) ) ).
% subset_mset.le_diff_conv2
tff(fact_4244_subset__mset_Ole__add__diff,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)),A3))) ) ).
% subset_mset.le_add_diff
tff(fact_4245_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F: fun(B,A),P: multiset(B),M13: multiset(A),M24: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F),P)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M13),M24)))
=> ~ ! [P1: multiset(B),P22: multiset(B)] :
( ( P = aa(multiset(B),multiset(B),aa(multiset(B),fun(multiset(B),multiset(B)),plus_plus(multiset(B)),P1),P22) )
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F),P1)),M13))
=> ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,F),P22)),M24)) ) ) ) ).
% mset_map_split_orig_le
tff(fact_4246_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X2: A,B5: multiset(A),X7: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),B5)),X7))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X7),B5)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),X7)) ) ) ).
% mset_le_subtract_add_mset_right
tff(fact_4247_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X2: A,B5: multiset(A),X7: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),B5)),X7))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X7),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A))))))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A)))),X7)) ) ) ).
% mset_le_subtract_add_mset_left
tff(fact_4248_subset__mset_Osum__nonneg,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A))] :
( ! [X3: B] :
( pp(member(B,X3,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F,X3))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,A4))) ) ).
% subset_mset.sum_nonneg
tff(fact_4249_subset__mset_Osum__nonpos,axiom,
! [B: $tType,A: $tType,A4: set(B),F: fun(B,multiset(A))] :
( ! [X3: B] :
( pp(member(B,X3,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,X3)),zero_zero(multiset(A)))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,A4)),zero_zero(multiset(A)))) ) ).
% subset_mset.sum_nonpos
tff(fact_4250_subset__mset_Osum__list__nonneg,axiom,
! [A: $tType,Xs: list(multiset(A))] :
( ! [X3: multiset(A)] :
( pp(member(multiset(A),X3,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),X3)) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs))) ) ).
% subset_mset.sum_list_nonneg
tff(fact_4251_subset__mset_Osum__list__nonpos,axiom,
! [A: $tType,Xs: list(multiset(A))] :
( ! [X3: multiset(A)] :
( pp(member(multiset(A),X3,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X3),zero_zero(multiset(A)))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs)),zero_zero(multiset(A)))) ) ).
% subset_mset.sum_list_nonpos
tff(fact_4252_subset__mset_Omember__le__sum__list,axiom,
! [A: $tType,X2: multiset(A),Xs: list(multiset(A))] :
( pp(member(multiset(A),X2,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),X2),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs))) ) ).
% subset_mset.member_le_sum_list
tff(fact_4253_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [A: $tType,Xs: list(multiset(A))] :
( ! [X3: multiset(A)] :
( pp(member(multiset(A),X3,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),X3)) )
=> ( ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs) = zero_zero(multiset(A)) )
<=> ! [X4: multiset(A)] :
( pp(member(multiset(A),X4,aa(list(multiset(A)),set(multiset(A)),set2(multiset(A)),Xs)))
=> ( X4 = zero_zero(multiset(A)) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
tff(fact_4254_subset__mset_Osum__mono,axiom,
! [A: $tType,B: $tType,K2: set(B),F: fun(B,multiset(A)),G: fun(B,multiset(A))] :
( ! [I2: B] :
( pp(member(B,I2,K2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,I2)),aa(B,multiset(A),G,I2))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,K2)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),G,K2))) ) ).
% subset_mset.sum_mono
tff(fact_4255_subset__mset_Osum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType,S: set(B),T2: set(C),G: fun(C,multiset(A)),I: fun(C,B),F: fun(B,multiset(A))] :
( finite_finite2(B,S)
=> ( finite_finite2(C,T2)
=> ( ! [X3: C] :
( pp(member(C,X3,T2))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(C,multiset(A),G,X3))) )
=> ( ! [X3: B] :
( pp(member(B,X3,S))
=> ? [Xa3: C] :
( pp(member(C,Xa3,T2))
& ( aa(C,B,I,Xa3) = X3 )
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,X3)),aa(C,multiset(A),G,Xa3))) ) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,S)),groups3894954378712506084id_sum(multiset(A),C,plus_plus(multiset(A)),zero_zero(multiset(A)),G,T2))) ) ) ) ) ).
% subset_mset.sum_le_included
tff(fact_4256_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [B: $tType,A: $tType,A4: set(B),F: fun(B,multiset(A))] :
( finite_finite2(B,A4)
=> ( ! [X3: B] :
( pp(member(B,X3,A4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F,X3))) )
=> ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,A4) = zero_zero(multiset(A)) )
<=> ! [X4: B] :
( pp(member(B,X4,A4))
=> ( aa(B,multiset(A),F,X4) = zero_zero(multiset(A)) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
tff(fact_4257_subset__mset_Osum__nonneg__0,axiom,
! [B: $tType,A: $tType,S: set(B),F: fun(B,multiset(A)),I: B] :
( finite_finite2(B,S)
=> ( ! [I2: B] :
( pp(member(B,I2,S))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F,I2))) )
=> ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,S) = zero_zero(multiset(A)) )
=> ( pp(member(B,I,S))
=> ( aa(B,multiset(A),F,I) = zero_zero(multiset(A)) ) ) ) ) ) ).
% subset_mset.sum_nonneg_0
tff(fact_4258_subset__mset_Osum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType,S: set(B),F: fun(B,multiset(A)),B5: multiset(A),I: B] :
( finite_finite2(B,S)
=> ( ! [I2: B] :
( pp(member(B,I2,S))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F,I2))) )
=> ( ( groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,S) = B5 )
=> ( pp(member(B,I,S))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,I)),B5)) ) ) ) ) ).
% subset_mset.sum_nonneg_leq_bound
tff(fact_4259_minus__shift__def,axiom,
! [K: code_natural,L: code_natural,R: code_natural] :
( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K),L))
=> ( minus_shift(R,K,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),R),K)),L) ) )
& ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),K),L))
=> ( minus_shift(R,K,L) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),L) ) ) ) ).
% minus_shift_def
tff(fact_4260_split__seed__def,axiom,
! [S: product_prod(code_natural,code_natural)] : split_seed(S) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_sb(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),S)),S) ).
% split_seed_def
tff(fact_4261_log_Osimps,axiom,
! [B2: code_natural,I: code_natural] :
( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),B2)) )
=> ( log(B2,I) = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),B2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),B2)) )
=> ( log(B2,I) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(B2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),I),B2))) ) ) ) ).
% log.simps
tff(fact_4262_log_Oelims,axiom,
! [X2: code_natural,Xa: code_natural,Y: code_natural] :
( ( log(X2,Xa) = Y )
=> ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X2)) )
=> ( Y = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X2)) )
=> ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X2))) ) ) ) ) ).
% log.elims
tff(fact_4263_extract__def,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_sc(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P),Xs)) ).
% extract_def
tff(fact_4264_drop__takeWhile,axiom,
! [A: $tType,I: nat,P: fun(A,bool),L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L))))
=> ( drop(A,I,takeWhile(A,P,L)) = takeWhile(A,P,drop(A,I,L)) ) ) ).
% drop_takeWhile
tff(fact_4265_eq__len__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: fun(A,bool),L: list(A)] :
( ( I = aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L)) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),L)))
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),I))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,L),J2))) )
& ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,L),I))) ) ) ) ).
% eq_len_takeWhile_conv
tff(fact_4266_less__length__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: fun(A,bool),L: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,L))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)))
& ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J2),I))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,L),J2))) ) ) ) ).
% less_length_takeWhile_conv
tff(fact_4267_length__dropWhile__takeWhile,axiom,
! [A: $tType,X2: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% length_dropWhile_takeWhile
tff(fact_4268_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))))
=> ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).
% dropWhile_nth
tff(fact_4269_inc__shift__def,axiom,
! [V: code_natural,K: code_natural] :
( ( ( V = K )
=> ( inc_shift(V,K) = one_one(code_natural) ) )
& ( ( V != K )
=> ( inc_shift(V,K) = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),K),one_one(code_natural)) ) ) ) ).
% inc_shift_def
tff(fact_4270_log_Opelims,axiom,
! [X2: code_natural,Xa: code_natural,Y: code_natural] :
( ( log(X2,Xa) = Y )
=> ( pp(aa(product_prod(code_natural,code_natural),bool,accp(product_prod(code_natural,code_natural),log_rel),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X2),Xa)))
=> ~ ( ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X2)) )
=> ( Y = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X2),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa),X2)) )
=> ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X2,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X2))) ) ) )
=> ~ pp(aa(product_prod(code_natural,code_natural),bool,accp(product_prod(code_natural,code_natural),log_rel),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),X2),Xa))) ) ) ) ).
% log.pelims
tff(fact_4271_Random_Orange__def,axiom,
! [K: code_natural] : range(K) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),iterate(code_natural,product_prod(code_natural,code_natural),log(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),K),aTP_Lamp_se(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))))),one_one(code_natural)),aTP_Lamp_sf(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),K)) ).
% Random.range_def
tff(fact_4272_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: fun(code_natural,A),A1: code_natural,A22: code_natural,P: fun(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)))] :
( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),A0),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),A1),A22))))
=> ( ! [F3: fun(code_natural,A),N3: code_natural,M4: code_natural] :
( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N3),M4))))
=> ( ! [X: product_unit] :
( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),M4),N3))
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,F3),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N3),one_one(code_natural))),M4)) )
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,F3),N3),M4)) ) )
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),aa(fun(code_natural,A),fun(code_natural,fun(code_natural,bool)),P,A0),A1),A22)) ) ) ).
% Predicate.iterate_upto.pinduct
tff(fact_4273_mset__size2elem,axiom,
! [A: $tType,P: multiset(A),Q3: A,Q4: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q4),zero_zero(multiset(A))))),P))
=> ( P = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q4),zero_zero(multiset(A)))) ) ) ) ).
% mset_size2elem
tff(fact_4274_scomp__apply,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(B,product_prod(C,D)),G: fun(C,fun(D,A)),X2: B] : aa(B,A,product_scomp(B,C,D,A,F,G),X2) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),G),aa(B,product_prod(C,D),F,X2)) ).
% scomp_apply
tff(fact_4275_size__union,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M2)),aa(multiset(A),nat,size_size(multiset(A)),N5)) ).
% size_union
tff(fact_4276_scomp__scomp,axiom,
! [A: $tType,E: $tType,F4: $tType,B: $tType,D: $tType,C: $tType,F: fun(A,product_prod(E,F4)),G: fun(E,fun(F4,product_prod(C,D))),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F4,product_prod(C,D),F,G),H) = product_scomp(A,E,F4,B,F,aa(fun(C,fun(D,B)),fun(E,fun(F4,B)),aTP_Lamp_sg(fun(E,fun(F4,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F4,B))),G),H)) ).
% scomp_scomp
tff(fact_4277_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X2: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X2,product_Pair(B,C)) = X2 ).
% scomp_Pair
tff(fact_4278_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X2: C,F: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X2),F) = aa(C,fun(A,B),F,X2) ).
% Pair_scomp
tff(fact_4279_scomp__def,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(B,fun(C,D)),X: A] : aa(A,D,product_scomp(A,B,C,D,F,G),X) = aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G),aa(A,product_prod(B,C),F,X)) ).
% scomp_def
tff(fact_4280_size__1__singleton__mset,axiom,
! [A: $tType,M2: multiset(A)] :
( ( aa(multiset(A),nat,size_size(multiset(A)),M2) = one_one(nat) )
=> ? [A5: A] : M2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),zero_zero(multiset(A))) ) ).
% size_1_singleton_mset
tff(fact_4281_size__single,axiom,
! [A: $tType,B2: A] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))) = one_one(nat) ).
% size_single
tff(fact_4282_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X2,Xa),Xb) = Y )
=> ( ( ( X2 = zero_zero(code_natural) )
=> ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
& ( ( X2 != zero_zero(code_natural) )
=> ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X2),one_one(code_natural)),Xa)) ) ) ) ) ).
% iterate.elims
tff(fact_4283_iterate_Osimps,axiom,
! [A: $tType,B: $tType,K: code_natural,F: fun(B,fun(A,product_prod(B,A))),X2: B] :
( ( ( K = zero_zero(code_natural) )
=> ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F),X2) = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2) ) )
& ( ( K != zero_zero(code_natural) )
=> ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F),X2) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F,X2),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F)) ) ) ) ).
% iterate.simps
tff(fact_4284_size__mset__SucE,axiom,
! [A: $tType,A4: multiset(A),N: nat] :
( ( aa(multiset(A),nat,size_size(multiset(A)),A4) = aa(nat,nat,suc,N) )
=> ~ ! [A5: A,B8: multiset(A)] :
( ( A4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),zero_zero(multiset(A)))),B8) )
=> ( aa(multiset(A),nat,size_size(multiset(A)),B8) != N ) ) ) ).
% size_mset_SucE
tff(fact_4285_scomp__unfold,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,X: fun(A,product_prod(B,C)),Xa3: fun(B,fun(C,D)),Xb2: A] : aa(A,D,product_scomp(A,B,C,D,X,Xa3),Xb2) = aa(C,D,aa(B,fun(C,D),Xa3,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X,Xb2))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X,Xb2))) ).
% scomp_unfold
tff(fact_4286_mset__size__le1__cases,axiom,
! [A: $tType,M2: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M2)),aa(nat,nat,suc,zero_zero(nat))))
=> ( ( M2 != zero_zero(multiset(A)) )
=> ~ ! [M4: A] : M2 != aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M4),zero_zero(multiset(A))) ) ) ).
% mset_size_le1_cases
tff(fact_4287_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb: B,Y: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X2,Xa),Xb) = Y )
=> ( pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X2),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb))))
=> ~ ( ( ( ( X2 = zero_zero(code_natural) )
=> ( Y = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb) ) )
& ( ( X2 != zero_zero(code_natural) )
=> ( Y = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X2),one_one(code_natural)),Xa)) ) ) )
=> ~ pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X2),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa),Xb)))) ) ) ) ).
% iterate.pelims
tff(fact_4288_size__Diff__singleton__if,axiom,
! [A: $tType,X2: A,A4: multiset(A)] :
( ( pp(member(A,X2,set_mset(A,A4)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(A),nat,size_size(multiset(A)),A4)),one_one(nat)) ) )
& ( ~ pp(member(A,X2,set_mset(A,A4)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A))))) = aa(multiset(A),nat,size_size(multiset(A)),A4) ) ) ) ).
% size_Diff_singleton_if
tff(fact_4289_size__Diff__singleton,axiom,
! [A: $tType,X2: A,M2: multiset(A)] :
( pp(member(A,X2,set_mset(A,M2)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(A),nat,size_size(multiset(A)),M2)),one_one(nat)) ) ) ).
% size_Diff_singleton
tff(fact_4290_size__diff__se,axiom,
! [A: $tType,T2: A,S2: multiset(A)] :
( pp(member(A,T2,set_mset(A,S2)))
=> ( aa(multiset(A),nat,size_size(multiset(A)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T2),zero_zero(multiset(A)))))),one_one(nat)) ) ) ).
% size_diff_se
tff(fact_4291_set__mset__union,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A)] : set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_mset(A,M2)),set_mset(A,N5)) ).
% set_mset_union
tff(fact_4292_mset__diff__cancel1elem,axiom,
! [A: $tType,A3: A,B5: multiset(A)] :
( ~ pp(member(A,A3,set_mset(A,B5)))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),B5) = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))) ) ) ).
% mset_diff_cancel1elem
tff(fact_4293_ex__Melem__conv,axiom,
! [A: $tType,A4: multiset(A)] :
( ? [X4: A] : pp(member(A,X4,set_mset(A,A4)))
<=> ( A4 != zero_zero(multiset(A)) ) ) ).
% ex_Melem_conv
tff(fact_4294_union__iff,axiom,
! [A: $tType,A3: A,A4: multiset(A),B5: multiset(A)] :
( pp(member(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))))
<=> ( pp(member(A,A3,set_mset(A,A4)))
| pp(member(A,A3,set_mset(A,B5))) ) ) ).
% union_iff
tff(fact_4295_mset__un__cases,axiom,
! [A: $tType,A3: A,A4: multiset(A),B5: multiset(A)] :
( pp(member(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))))
=> ( ~ pp(member(A,A3,set_mset(A,A4)))
=> pp(member(A,A3,set_mset(A,B5))) ) ) ).
% mset_un_cases
tff(fact_4296_mset__left__cancel__union,axiom,
! [A: $tType,A3: A,A4: multiset(A),B5: multiset(A)] :
( pp(member(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))))
=> ( ~ pp(member(A,A3,set_mset(A,A4)))
=> pp(member(A,A3,set_mset(A,B5))) ) ) ).
% mset_left_cancel_union
tff(fact_4297_mset__right__cancel__union,axiom,
! [A: $tType,A3: A,A4: multiset(A),B5: multiset(A)] :
( pp(member(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))))
=> ( ~ pp(member(A,A3,set_mset(A,B5)))
=> pp(member(A,A3,set_mset(A,A4))) ) ) ).
% mset_right_cancel_union
tff(fact_4298_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))] :
~ ! [K3: code_natural,F3: fun(B,fun(A,product_prod(B,A))),X3: B] : X2 != aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),K3),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),F3),X3)) ).
% iterate.cases
tff(fact_4299_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M2: multiset(product_prod(A,B)),F: fun(A,fun(B,C)),G: fun(A,fun(B,C))] :
( ! [X3: A,Y4: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4),set_mset(product_prod(A,B),M2)))
=> ( aa(B,C,aa(A,fun(B,C),F,X3),Y4) = aa(B,C,aa(A,fun(B,C),G,X3),Y4) ) )
=> ( aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),M2) = aa(multiset(product_prod(A,B)),multiset(C),image_mset(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),M2) ) ) ).
% image_mset_cong_pair
tff(fact_4300_multi__member__skip,axiom,
! [A: $tType,X2: A,XS: multiset(A),Y: A] :
( pp(member(A,X2,set_mset(A,XS)))
=> pp(member(A,X2,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Y),zero_zero(multiset(A)))),XS)))) ) ).
% multi_member_skip
tff(fact_4301_multi__member__this,axiom,
! [A: $tType,X2: A,XS: multiset(A)] : pp(member(A,X2,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A)))),XS)))) ).
% multi_member_this
tff(fact_4302_mset__right__cancel__elem,axiom,
! [A: $tType,A3: A,A4: multiset(A),B2: A] :
( pp(member(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))))))
=> ( ( A3 != B2 )
=> pp(member(A,A3,set_mset(A,A4))) ) ) ).
% mset_right_cancel_elem
tff(fact_4303_mset__left__cancel__elem,axiom,
! [A: $tType,A3: A,B2: A,A4: multiset(A)] :
( pp(member(A,A3,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))),A4))))
=> ( ( A3 != B2 )
=> pp(member(A,A3,set_mset(A,A4))) ) ) ).
% mset_left_cancel_elem
tff(fact_4304_mset__le__mono__add__single,axiom,
! [A: $tType,A3: A,Ys: multiset(A),B2: A,Ws: multiset(A)] :
( pp(member(A,A3,set_mset(A,Ys)))
=> ( pp(member(A,B2,set_mset(A,Ws)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Ys),Ws))) ) ) ).
% mset_le_mono_add_single
tff(fact_4305_mset__union__subset__s,axiom,
! [A: $tType,A3: A,B5: multiset(A),C5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),B5)),C5))
=> ( pp(member(A,A3,set_mset(A,C5)))
& pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B5),C5)) ) ) ).
% mset_union_subset_s
tff(fact_4306_mset__2dist2__cases,axiom,
! [A: $tType,A3: A,B2: A,A4: multiset(A),B5: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)))
=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),A4))
=> ( ~ pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A))))),B5))
=> ( ( pp(member(A,A3,set_mset(A,A4)))
=> ~ pp(member(A,B2,set_mset(A,B5))) )
=> ~ ( pp(member(A,A3,set_mset(A,B5)))
=> ~ pp(member(A,B2,set_mset(A,A4))) ) ) ) ) ) ).
% mset_2dist2_cases
tff(fact_4307_insert__DiffM2,axiom,
! [A: $tType,X2: A,M2: multiset(A)] :
( pp(member(A,X2,set_mset(A,M2)))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A))))),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A)))) = M2 ) ) ).
% insert_DiffM2
tff(fact_4308_diff__union__single__conv,axiom,
! [A: $tType,A3: A,J4: multiset(A),I4: multiset(A)] :
( pp(member(A,A3,set_mset(A,J4)))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),J4)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),J4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) ) ) ).
% diff_union_single_conv
tff(fact_4309_mset__un__single__un__cases,axiom,
! [A: $tType,A3: A,A4: multiset(A),B5: multiset(A),C5: multiset(A)] :
( ( aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5) )
=> ( ( pp(member(A,A3,set_mset(A,B5)))
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))),C5) ) )
=> ~ ( pp(member(A,A3,set_mset(A,C5)))
=> ( A4 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))) ) ) ) ) ).
% mset_un_single_un_cases
tff(fact_4310_diff__union__single__conv2,axiom,
! [A: $tType,A3: A,J4: multiset(A),I4: multiset(A)] :
( pp(member(A,A3,set_mset(A,J4)))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),J4),I4)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A)))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),J4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),zero_zero(multiset(A))))),I4) ) ) ).
% diff_union_single_conv2
tff(fact_4311_mset__union__diff__comm,axiom,
! [A: $tType,T2: A,S2: multiset(A),T5: multiset(A)] :
( pp(member(A,T2,set_mset(A,S2)))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T2),zero_zero(multiset(A))))) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),T5),S2)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T2),zero_zero(multiset(A)))) ) ) ).
% mset_union_diff_comm
tff(fact_4312_mset__contains__eq,axiom,
! [A: $tType,M: A,M2: multiset(A)] :
( pp(member(A,M,set_mset(A,M2)))
<=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M),zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),M),zero_zero(multiset(A))))) = M2 ) ) ).
% mset_contains_eq
tff(fact_4313_mset__size1elem,axiom,
! [A: $tType,P: multiset(A),Q3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P)),one_one(nat)))
=> ( pp(member(A,Q3,set_mset(A,P)))
=> ( P = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Q3),zero_zero(multiset(A))) ) ) ) ).
% mset_size1elem
tff(fact_4314_subset__mset_Osum__mset__mono,axiom,
! [A: $tType,B: $tType,K2: multiset(B),F: fun(B,multiset(A)),G: fun(B,multiset(A))] :
( ! [I2: B] :
( pp(member(B,I2,set_mset(B,K2)))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(B,multiset(A),F,I2)),aa(B,multiset(A),G,I2))) )
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),comm_monoid_sum_mset(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),aa(multiset(B),multiset(multiset(A)),image_mset(B,multiset(A),F),K2))),comm_monoid_sum_mset(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),aa(multiset(B),multiset(multiset(A)),image_mset(B,multiset(A),G),K2)))) ) ).
% subset_mset.sum_mset_mono
tff(fact_4315_subset__mset_Osum__mset__0__iff,axiom,
! [A: $tType,M2: multiset(multiset(A))] :
( ( comm_monoid_sum_mset(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),M2) = zero_zero(multiset(A)) )
<=> ! [X4: multiset(A)] :
( pp(member(multiset(A),X4,set_mset(multiset(A),M2)))
=> ( X4 = zero_zero(multiset(A)) ) ) ) ).
% subset_mset.sum_mset_0_iff
tff(fact_4316_wmsI,axiom,
! [A4: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat)),Z7: multiset(product_prod(nat,nat))] :
( ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),set_mset(product_prod(nat,nat),A4)),set_mset(product_prod(nat,nat),B5)),fun_max_strict))
| ( ( A4 = zero_zero(multiset(product_prod(nat,nat))) )
& ( B5 = zero_zero(multiset(product_prod(nat,nat))) ) ) )
=> pp(member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),B5)),ms_weak)) ) ).
% wmsI
tff(fact_4317_smsI,axiom,
! [A4: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat)),Z7: multiset(product_prod(nat,nat))] :
( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),set_mset(product_prod(nat,nat),A4)),set_mset(product_prod(nat,nat),B5)),fun_max_strict))
=> pp(member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),B5)),ms_strict)) ) ).
% smsI
tff(fact_4318_ms__reduction__pair,axiom,
fun_reduction_pair(multiset(product_prod(nat,nat)),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_prod(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),aa(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),fun(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_prod(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))))),product_Pair(set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),set(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))))),ms_strict),ms_weak)) ).
% ms_reduction_pair
tff(fact_4319_ms__strictI,axiom,
! [Z7: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat)),A4: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat))] :
( pw_leq(Z7,Z8)
=> ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),set_mset(product_prod(nat,nat),A4)),set_mset(product_prod(nat,nat),B5)),fun_max_strict))
=> pp(member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z8),B5)),ms_strict)) ) ) ).
% ms_strictI
tff(fact_4320_ms__weakI1,axiom,
! [Z7: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat)),A4: multiset(product_prod(nat,nat)),B5: multiset(product_prod(nat,nat))] :
( pw_leq(Z7,Z8)
=> ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),set_mset(product_prod(nat,nat),A4)),set_mset(product_prod(nat,nat),B5)),fun_max_strict))
=> pp(member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),A4)),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z8),B5)),ms_weak)) ) ) ).
% ms_weakI1
tff(fact_4321_pw__leq__split,axiom,
! [X7: multiset(product_prod(nat,nat)),Y7: multiset(product_prod(nat,nat))] :
( pw_leq(X7,Y7)
=> ? [A10: multiset(product_prod(nat,nat)),B8: multiset(product_prod(nat,nat)),Z9: multiset(product_prod(nat,nat))] :
( ( X7 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),A10),Z9) )
& ( Y7 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),B8),Z9) )
& ( pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),set_mset(product_prod(nat,nat),A10)),set_mset(product_prod(nat,nat),B8)),fun_max_strict))
| ( ( B8 = zero_zero(multiset(product_prod(nat,nat))) )
& ( A10 = zero_zero(multiset(product_prod(nat,nat))) ) ) ) ) ) ).
% pw_leq_split
tff(fact_4322_ms__weakI2,axiom,
! [Z7: multiset(product_prod(nat,nat)),Z8: multiset(product_prod(nat,nat))] :
( pw_leq(Z7,Z8)
=> pp(member(product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),product_prod(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)))),product_Pair(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z7),zero_zero(multiset(product_prod(nat,nat))))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),Z8),zero_zero(multiset(product_prod(nat,nat))))),ms_weak)) ) ).
% ms_weakI2
tff(fact_4323_pw__leq__lstep,axiom,
! [X2: product_prod(nat,nat),Y: product_prod(nat,nat)] :
( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y),fun_pair_leq))
=> pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X2),zero_zero(multiset(product_prod(nat,nat)))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))) ) ).
% pw_leq_lstep
tff(fact_4324_pw__leq__step,axiom,
! [X2: product_prod(nat,nat),Y: product_prod(nat,nat),X7: multiset(product_prod(nat,nat)),Y7: multiset(product_prod(nat,nat))] :
( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y),fun_pair_leq))
=> ( pw_leq(X7,Y7)
=> pw_leq(aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X2),zero_zero(multiset(product_prod(nat,nat))))),X7),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y),zero_zero(multiset(product_prod(nat,nat))))),Y7)) ) ) ).
% pw_leq_step
tff(fact_4325_pw__leq_Osimps,axiom,
! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
( pw_leq(A1,A22)
<=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
& ( A22 = zero_zero(multiset(product_prod(nat,nat))) ) )
| ? [X4: product_prod(nat,nat),Y3: product_prod(nat,nat),X10: multiset(product_prod(nat,nat)),Y8: multiset(product_prod(nat,nat))] :
( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X4),zero_zero(multiset(product_prod(nat,nat))))),X10) )
& ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y3),zero_zero(multiset(product_prod(nat,nat))))),Y8) )
& pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X4),Y3),fun_pair_leq))
& pw_leq(X10,Y8) ) ) ) ).
% pw_leq.simps
tff(fact_4326_pw__leq_Ocases,axiom,
! [A1: multiset(product_prod(nat,nat)),A22: multiset(product_prod(nat,nat))] :
( pw_leq(A1,A22)
=> ( ( ( A1 = zero_zero(multiset(product_prod(nat,nat))) )
=> ( A22 != zero_zero(multiset(product_prod(nat,nat))) ) )
=> ~ ! [X3: product_prod(nat,nat),Y4: product_prod(nat,nat),X11: multiset(product_prod(nat,nat))] :
( ( A1 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),X3),zero_zero(multiset(product_prod(nat,nat))))),X11) )
=> ! [Y9: multiset(product_prod(nat,nat))] :
( ( A22 = aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(multiset(product_prod(nat,nat)),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),plus_plus(multiset(product_prod(nat,nat))),aa(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(multiset(product_prod(nat,nat)),multiset(product_prod(nat,nat))),add_mset(product_prod(nat,nat)),Y4),zero_zero(multiset(product_prod(nat,nat))))),Y9) )
=> ( pp(member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X3),Y4),fun_pair_leq))
=> ~ pw_leq(X11,Y9) ) ) ) ) ) ).
% pw_leq.cases
tff(fact_4327_mult1E,axiom,
! [A: $tType,N5: multiset(A),M2: multiset(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N5),M2),mult1(A,R)))
=> ~ ! [A5: A,M0: multiset(A)] :
( ( M2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),M0) )
=> ! [K9: multiset(A)] :
( ( N5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K9) )
=> ~ ! [B6: A] :
( pp(member(A,B6,set_mset(A,K9)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),A5),R)) ) ) ) ) ).
% mult1E
tff(fact_4328_mult1I,axiom,
! [A: $tType,M2: multiset(A),A3: A,M02: multiset(A),N5: multiset(A),K2: multiset(A),R: set(product_prod(A,A))] :
( ( M2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M02) )
=> ( ( N5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K2) )
=> ( ! [B3: A] :
( pp(member(A,B3,set_mset(A,K2)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A3),R)) )
=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N5),M2),mult1(A,R))) ) ) ) ).
% mult1I
tff(fact_4329_less__add,axiom,
! [A: $tType,N5: multiset(A),A3: A,M02: multiset(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N5),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M02)),mult1(A,R)))
=> ( ? [M8: multiset(A)] :
( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M8),M02),mult1(A,R)))
& ( N5 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),M8) ) )
| ? [K9: multiset(A)] :
( ! [B6: A] :
( pp(member(A,B6,set_mset(A,K9)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),A3),R)) )
& ( N5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M02),K9) ) ) ) ) ).
% less_add
tff(fact_4330_one__step__implies__mult,axiom,
! [A: $tType,J4: multiset(A),K2: multiset(A),R: set(product_prod(A,A)),I4: multiset(A)] :
( ( J4 != zero_zero(multiset(A)) )
=> ( ! [X3: A] :
( pp(member(A,X3,set_mset(A,K2)))
=> ? [Xa3: A] :
( pp(member(A,Xa3,set_mset(A,J4)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R)) ) )
=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),K2)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),J4)),mult(A,R))) ) ) ).
% one_step_implies_mult
tff(fact_4331_not__less__empty,axiom,
! [A: $tType,M2: multiset(A),R: set(product_prod(A,A))] : ~ pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M2),zero_zero(multiset(A))),mult1(A,R))) ).
% not_less_empty
tff(fact_4332_mult1__union,axiom,
! [A: $tType,B5: multiset(A),D4: multiset(A),R: set(product_prod(A,A)),C5: multiset(A)] :
( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),B5),D4),mult1(A,R)))
=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D4)),mult1(A,R))) ) ).
% mult1_union
tff(fact_4333_mult1__lessE,axiom,
! [A: $tType,N5: multiset(A),M2: multiset(A),R: fun(A,fun(A,bool))] :
( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N5),M2),mult1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R)))))
=> ( asymp(A,R)
=> ~ ! [A5: A,M0: multiset(A)] :
( ( M2 = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A5),M0) )
=> ! [K9: multiset(A)] :
( ( N5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M0),K9) )
=> ( ~ pp(member(A,A5,set_mset(A,K9)))
=> ~ ! [B6: A] :
( pp(member(A,B6,set_mset(A,K9)))
=> pp(aa(A,bool,aa(A,fun(A,bool),R,B6),A5)) ) ) ) ) ) ) ).
% mult1_lessE
tff(fact_4334_in__mset__fold__plus__iff,axiom,
! [A: $tType,X2: A,M2: multiset(A),NN: multiset(multiset(A))] :
( pp(member(A,X2,set_mset(A,fold_mset(multiset(A),multiset(A),plus_plus(multiset(A)),M2,NN))))
<=> ( pp(member(A,X2,set_mset(A,M2)))
| ? [N8: multiset(A)] :
( pp(member(multiset(A),N8,set_mset(multiset(A),NN)))
& pp(member(A,X2,set_mset(A,N8))) ) ) ) ).
% in_mset_fold_plus_iff
tff(fact_4335_mult__implies__one__step,axiom,
! [A: $tType,R: set(product_prod(A,A)),M2: multiset(A),N5: multiset(A)] :
( trans(A,R)
=> ( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M2),N5),mult(A,R)))
=> ? [I7: multiset(A),J5: multiset(A)] :
( ( N5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),J5) )
& ? [K9: multiset(A)] :
( ( M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),K9) )
& ( J5 != zero_zero(multiset(A)) )
& ! [X: A] :
( pp(member(A,X,set_mset(A,K9)))
=> ? [Xa4: A] :
( pp(member(A,Xa4,set_mset(A,J5)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4),R)) ) ) ) ) ) ) ).
% mult_implies_one_step
tff(fact_4336_mult1__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : mult1(A,R) = aa(fun(product_prod(multiset(A),multiset(A)),bool),set(product_prod(multiset(A),multiset(A))),collect(product_prod(multiset(A),multiset(A))),aa(fun(multiset(A),fun(multiset(A),bool)),fun(product_prod(multiset(A),multiset(A)),bool),product_case_prod(multiset(A),multiset(A),bool),aTP_Lamp_sh(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),bool)),R))) ).
% mult1_def
tff(fact_4337_trans__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( trans(A,R)
<=> ! [X4: A,Y3: A,Z4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z4),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Z4),R)) ) ) ) ).
% trans_def
tff(fact_4338_transI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X3: A,Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Z3),R)) ) )
=> trans(A,R) ) ).
% transI
tff(fact_4339_transE,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y: A,Z2: A] :
( trans(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),R)) ) ) ) ).
% transE
tff(fact_4340_transD,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y: A,Z2: A] :
( trans(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),R)) ) ) ) ).
% transD
tff(fact_4341_lexord__trans,axiom,
! [A: $tType,X2: list(A),Y: list(A),R: set(product_prod(A,A)),Z2: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lexord(A,R)))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2),lexord(A,R)))
=> ( trans(A,R)
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Z2),lexord(A,R))) ) ) ) ).
% lexord_trans
tff(fact_4342_lenlex__trans,axiom,
! [A: $tType,X2: list(A),Y: list(A),R: set(product_prod(A,A)),Z2: list(A)] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),lenlex(A,R)))
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Z2),lenlex(A,R)))
=> ( trans(A,R)
=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Z2),lenlex(A,R))) ) ) ) ).
% lenlex_trans
tff(fact_4343_union__fold__mset__add__mset,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5) = fold_mset(A,multiset(A),add_mset(A),A4,B5) ).
% union_fold_mset_add_mset
tff(fact_4344_trans__singleton,axiom,
! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),bot_bot(set(product_prod(A,A))))) ).
% trans_singleton
tff(fact_4345_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A4: set(product_prod(A,A))] :
( trans(A,A4)
=> ( transitive_rtrancl(A,A4) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A4),id2(A)) ) ) ).
% trans_rtrancl_eq_reflcl
tff(fact_4346_comp__fun__commute_Ofold__mset__union,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),S: B,M2: multiset(A),N5: multiset(A)] :
( finite6289374366891150609ommute(A,B,F)
=> ( fold_mset(A,B,F,S,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = fold_mset(A,B,F,fold_mset(A,B,F,S,M2),N5) ) ) ).
% comp_fun_commute.fold_mset_union
tff(fact_4347_trans__join,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( trans(A,R)
<=> ! [X4: product_prod(A,A)] :
( pp(member(product_prod(A,A),X4,R))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_sj(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X4)) ) ) ).
% trans_join
tff(fact_4348_Image__absorb__rtrancl,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B5: set(A),C5: set(A)] :
( trans(A,A4)
=> ( refl_on(A,B5,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),B5))
=> ( image(A,A,transitive_rtrancl(A,A4),C5) = image(A,A,A4,C5) ) ) ) ) ).
% Image_absorb_rtrancl
tff(fact_4349_wf__finite__segments,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irrefl(A,R)
=> ( trans(A,R)
=> ( ! [X3: A] : finite_finite2(A,aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_sk(set(product_prod(A,A)),fun(A,fun(A,bool)),R),X3)))
=> wf(A,R) ) ) ) ).
% wf_finite_segments
tff(fact_4350_mult__cancel__add__mset,axiom,
! [A: $tType,S: set(product_prod(A,A)),Uu: A,X7: multiset(A),Y7: multiset(A)] :
( trans(A,S)
=> ( irrefl(A,S)
=> ( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),X7)),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),Uu),Y7)),mult(A,S)))
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X7),Y7),mult(A,S))) ) ) ) ).
% mult_cancel_add_mset
tff(fact_4351_mult__cancel,axiom,
! [A: $tType,S: set(product_prod(A,A)),X7: multiset(A),Z7: multiset(A),Y7: multiset(A)] :
( trans(A,S)
=> ( irrefl(A,S)
=> ( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X7),Z7)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y7),Z7)),mult(A,S)))
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X7),Y7),mult(A,S))) ) ) ) ).
% mult_cancel
tff(fact_4352_mult__cancel__max,axiom,
! [A: $tType,S: set(product_prod(A,A)),X7: multiset(A),Y7: multiset(A)] :
( trans(A,S)
=> ( irrefl(A,S)
=> ( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X7),Y7),mult(A,S)))
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X7),Y7)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y7),X7)),mult(A,S))) ) ) ) ).
% mult_cancel_max
tff(fact_4353_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R4: set(product_prod(A,A)),P: fun(A,fun(A,bool)),N5: multiset(A),M2: multiset(A)] :
( irrefl(A,R4)
=> ( trans(A,R4)
=> ( ! [X3: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y4))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R4)) )
=> ( multeqp_code(A,P,N5,M2)
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N5),M2),aa(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A))),aa(set(product_prod(multiset(A),multiset(A))),fun(set(product_prod(multiset(A),multiset(A))),set(product_prod(multiset(A),multiset(A)))),sup_sup(set(product_prod(multiset(A),multiset(A)))),mult(A,R4)),id2(multiset(A))))) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
tff(fact_4354_multp__code__iff__mult,axiom,
! [A: $tType,R4: set(product_prod(A,A)),P: fun(A,fun(A,bool)),N5: multiset(A),M2: multiset(A)] :
( irrefl(A,R4)
=> ( trans(A,R4)
=> ( ! [X3: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y4))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R4)) )
=> ( multp_code(A,P,N5,M2)
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),N5),M2),mult(A,R4))) ) ) ) ) ).
% multp_code_iff_mult
tff(fact_4355_max__ext__eq,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : max_ext(A,R4) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_sl(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R4))) ).
% max_ext_eq
tff(fact_4356_init__seg__of__def,axiom,
! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool),aTP_Lamp_sm(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).
% init_seg_of_def
tff(fact_4357_refl__on__init__seg__of,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R),init_seg_of(A))) ).
% refl_on_init_seg_of
tff(fact_4358_antisym__init__seg__of,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),S),init_seg_of(A)))
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S),R),init_seg_of(A)))
=> ( R = S ) ) ) ).
% antisym_init_seg_of
tff(fact_4359_trans__init__seg__of,axiom,
! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),T2: set(product_prod(A,A))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),S),init_seg_of(A)))
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),S),T2),init_seg_of(A)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),T2),init_seg_of(A))) ) ) ).
% trans_init_seg_of
tff(fact_4360_Image__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] : image(A,B,R,S) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_sn(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R),S)) ).
% Image_def
tff(fact_4361_initial__segment__of__Diff,axiom,
! [A: $tType,P3: set(product_prod(A,A)),Q3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P3),Q3),init_seg_of(A)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),P3),S)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Q3),S)),init_seg_of(A))) ) ).
% initial_segment_of_Diff
tff(fact_4362_min__ext__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : min_ext(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_so(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R)) ).
% min_ext_def
tff(fact_4363_map__project__def,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),A4: set(A)] : map_project(A,B,F,A4) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_sp(fun(A,option(B)),fun(set(A),fun(B,bool)),F),A4)) ).
% map_project_def
tff(fact_4364_Chains__init__seg__of__Union,axiom,
! [A: $tType,R4: set(set(product_prod(A,A))),R: set(product_prod(A,A))] :
( pp(member(set(set(product_prod(A,A))),R4,chains(set(product_prod(A,A)),init_seg_of(A))))
=> ( pp(member(set(product_prod(A,A)),R,R4))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R4)),init_seg_of(A))) ) ) ).
% Chains_init_seg_of_Union
tff(fact_4365_mult__cancel__max0,axiom,
! [A: $tType,S: set(product_prod(A,A)),X7: multiset(A),Y7: multiset(A)] :
( trans(A,S)
=> ( irrefl(A,S)
=> ( pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),X7),Y7),mult(A,S)))
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X7),inter_mset(A,X7,Y7))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),Y7),inter_mset(A,X7,Y7))),mult(A,S))) ) ) ) ).
% mult_cancel_max0
tff(fact_4366_sum__mset_Oremove,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [X2: A,A4: multiset(A)] :
( pp(member(A,X2,set_mset(A,A4)))
=> ( aa(multiset(A),A,comm_m7189776963980413722m_mset(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),zero_zero(multiset(A)))))) ) ) ) ).
% sum_mset.remove
tff(fact_4367_sum__mset_Oadd__mset,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [X2: A,N5: multiset(A)] : aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),X2),N5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),N5)) ) ).
% sum_mset.add_mset
tff(fact_4368_sum__mset_Ounion,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M2: multiset(A),N5: multiset(A)] : aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),M2)),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),N5)) ) ).
% sum_mset.union
tff(fact_4369_sum__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),X2: B,A4: multiset(B)] : aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),aa(multiset(B),multiset(B),aa(B,fun(multiset(B),multiset(B)),add_mset(B),X2),A4))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X2)),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A4))) ) ).
% sum_mset.insert
tff(fact_4370_inter__union__distrib__right,axiom,
! [A: $tType,C5: multiset(A),A4: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),inter_mset(A,A4,B5)) = inter_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B5)) ).
% inter_union_distrib_right
tff(fact_4371_inter__union__distrib__left,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),inter_mset(A,A4,B5)),C5) = inter_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5)) ).
% inter_union_distrib_left
tff(fact_4372_inter__subset__eq__union,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),inter_mset(A,A4,B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5))) ).
% inter_subset_eq_union
tff(fact_4373_inter__mset__empty__distrib__left,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A)] :
( ( inter_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5),C5) = zero_zero(multiset(A)) )
<=> ( ( inter_mset(A,A4,C5) = zero_zero(multiset(A)) )
& ( inter_mset(A,B5,C5) = zero_zero(multiset(A)) ) ) ) ).
% inter_mset_empty_distrib_left
tff(fact_4374_inter__mset__empty__distrib__right,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A)] :
( ( inter_mset(A,A4,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5)) = zero_zero(multiset(A)) )
<=> ( ( inter_mset(A,A4,B5) = zero_zero(multiset(A)) )
& ( inter_mset(A,A4,C5) = zero_zero(multiset(A)) ) ) ) ).
% inter_mset_empty_distrib_right
tff(fact_4375_sum__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),H: fun(B,A),A4: multiset(B)] : aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bz(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A4))),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,H),A4))) ) ).
% sum_mset.distrib
tff(fact_4376_size__eq__sum__mset,axiom,
! [A: $tType,M2: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),M2) = aa(multiset(nat),nat,comm_m7189776963980413722m_mset(nat),aa(multiset(A),multiset(nat),image_mset(A,nat,aTP_Lamp_ia(A,nat)),M2)) ).
% size_eq_sum_mset
tff(fact_4377_multiset__union__diff__commute,axiom,
! [A: $tType,B5: multiset(A),C5: multiset(A),A4: multiset(A)] :
( ( inter_mset(A,B5,C5) = zero_zero(multiset(A)) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A4),C5)),B5) ) ) ).
% multiset_union_diff_commute
tff(fact_4378_sum__mset_Oeq__fold,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M2: multiset(A)] : aa(multiset(A),A,comm_m7189776963980413722m_mset(A),M2) = fold_mset(A,A,plus_plus(A),zero_zero(A),M2) ) ).
% sum_mset.eq_fold
tff(fact_4379_Chains__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_sq(set(product_prod(A,A)),fun(set(A),bool),R)) ).
% Chains_def
tff(fact_4380_Chains__subset_H,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R)
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R)))),chains(A,R))) ) ).
% Chains_subset'
tff(fact_4381_Chains__subset,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R))))) ).
% Chains_subset
tff(fact_4382_Chains__alt__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R)
=> ( chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R))) ) ) ).
% Chains_alt_def
tff(fact_4383_iter_H_Ocases,axiom,
! [A: $tType] :
( quickcheck_random(A)
=> ! [X2: product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))] :
~ ! [T3: itself(A),Nrandom: code_natural,Sz: code_natural,Seed: product_prod(code_natural,code_natural)] : X2 != aa(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),aa(itself(A),fun(product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),product_prod(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))))),product_Pair(itself(A),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),T3),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),aa(code_natural,fun(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural)))),product_Pair(code_natural,product_prod(code_natural,product_prod(code_natural,code_natural))),Nrandom),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),Sz),Seed))) ) ).
% iter'.cases
tff(fact_4384_multp__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),M2: multiset(A),N5: multiset(A)] :
( multp(A,R,M2,N5)
<=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),M2),N5),mult(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% multp_def
tff(fact_4385_AboveS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] : order_AboveS(A,R,A4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_sr(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A4)) ).
% AboveS_def
tff(fact_4386_lexordp__conv__lexord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( ord_lexordp(A,Xs,Ys)
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))))) ) ) ).
% lexordp_conv_lexord
tff(fact_4387_select__weight__cons__zero,axiom,
! [A: $tType,X2: A,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),aa(product_prod(code_natural,A),fun(list(product_prod(code_natural,A)),list(product_prod(code_natural,A))),cons(product_prod(code_natural,A)),aa(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),zero_zero(code_natural)),X2)),Xs)) = select_weight(A,Xs) ).
% select_weight_cons_zero
tff(fact_4388_one__step__implies__multp,axiom,
! [A: $tType,J4: multiset(A),K2: multiset(A),R: fun(A,fun(A,bool)),I4: multiset(A)] :
( ( J4 != zero_zero(multiset(A)) )
=> ( ! [X3: A] :
( pp(member(A,X3,set_mset(A,K2)))
=> ? [Xa3: A] :
( pp(member(A,Xa3,set_mset(A,J4)))
& pp(aa(A,bool,aa(A,fun(A,bool),R,X3),Xa3)) ) )
=> multp(A,R,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),K2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I4),J4)) ) ) ).
% one_step_implies_multp
tff(fact_4389_select__weight__select,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( select_weight(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural))),Xs)) = select(A,Xs) ) ) ).
% select_weight_select
tff(fact_4390_wo__rel_Osuc__greater,axiom,
! [A: $tType,R: set(product_prod(A,A)),B5: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R)))
=> ( ( order_AboveS(A,R,B5) != bot_bot(set(A)) )
=> ( pp(member(A,B2,B5))
=> ( ( bNF_Wellorder_wo_suc(A,R,B5) != B2 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,B5)),R)) ) ) ) ) ) ).
% wo_rel.suc_greater
tff(fact_4391_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R: set(product_prod(A,A)),B5: set(A),A3: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),field2(A,R)))
=> ( pp(member(A,A3,order_AboveS(A,R,B5)))
=> ( ! [A11: A] :
( pp(member(A,A11,order_AboveS(A,R,B5)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A11),R)) )
=> ( A3 = bNF_Wellorder_wo_suc(A,R,B5) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
tff(fact_4392_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B5: set(A)] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(member(A,A3,order_AboveS(A,R,B5)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R,B5)),A3),R)) ) ) ).
% wo_rel.suc_least_AboveS
tff(fact_4393_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( order_ofilter(A,R,A4)
=> ( ( order_AboveS(A,R,A4) != bot_bot(set(A)) )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,A4)),R))
=> ( ( B2 != bNF_Wellorder_wo_suc(A,R,A4) )
=> pp(member(A,B2,A4)) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
tff(fact_4394_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),From),To))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( hd(A,slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).
% slice_head
tff(fact_4395_gen__length__def,axiom,
! [A: $tType,N: nat,Xs: list(A)] : gen_length(A,N,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) ).
% gen_length_def
tff(fact_4396_subset__mset_Osum__pos2,axiom,
! [A: $tType,B: $tType,I4: set(B),I: B,F: fun(B,multiset(A))] :
( finite_finite2(B,I4)
=> ( pp(member(B,I,I4))
=> ( subset_mset(A,zero_zero(multiset(A)),aa(B,multiset(A),F,I))
=> ( ! [I2: B] :
( pp(member(B,I2,I4))
=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),aa(B,multiset(A),F,I2))) )
=> subset_mset(A,zero_zero(multiset(A)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,I4)) ) ) ) ) ).
% subset_mset.sum_pos2
tff(fact_4397_subset__mset_Oadd__less__cancel__right,axiom,
! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))
<=> subset_mset(A,A3,B2) ) ).
% subset_mset.add_less_cancel_right
tff(fact_4398_subset__mset_Oadd__less__cancel__left,axiom,
! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))
<=> subset_mset(A,A3,B2) ) ).
% subset_mset.add_less_cancel_left
tff(fact_4399_subset__mset_Oadd__less__same__cancel1,axiom,
! [A: $tType,B2: multiset(A),A3: multiset(A)] :
( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3),B2)
<=> subset_mset(A,A3,zero_zero(multiset(A))) ) ).
% subset_mset.add_less_same_cancel1
tff(fact_4400_subset__mset_Oadd__less__same__cancel2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2),B2)
<=> subset_mset(A,A3,zero_zero(multiset(A))) ) ).
% subset_mset.add_less_same_cancel2
tff(fact_4401_subset__mset_Oless__add__same__cancel1,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2))
<=> subset_mset(A,zero_zero(multiset(A)),B2) ) ).
% subset_mset.less_add_same_cancel1
tff(fact_4402_subset__mset_Oless__add__same__cancel2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,A3,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),A3))
<=> subset_mset(A,zero_zero(multiset(A)),B2) ) ).
% subset_mset.less_add_same_cancel2
tff(fact_4403_subset__mset_Oadd__less__imp__less__right,axiom,
! [A: $tType,A3: multiset(A),C2: multiset(A),B2: multiset(A)] :
( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2))
=> subset_mset(A,A3,B2) ) ).
% subset_mset.add_less_imp_less_right
tff(fact_4404_subset__mset_Oadd__less__imp__less__left,axiom,
! [A: $tType,C2: multiset(A),A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2))
=> subset_mset(A,A3,B2) ) ).
% subset_mset.add_less_imp_less_left
tff(fact_4405_subset__mset_Oadd__strict__right__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( subset_mset(A,A3,B2)
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C2)) ) ).
% subset_mset.add_strict_right_mono
tff(fact_4406_subset__mset_Oadd__strict__left__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( subset_mset(A,A3,B2)
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),A3),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C2),B2)) ) ).
% subset_mset.add_strict_left_mono
tff(fact_4407_subset__mset_Oadd__strict__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
( subset_mset(A,A3,B2)
=> ( subset_mset(A,C2,D3)
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),D3)) ) ) ).
% subset_mset.add_strict_mono
tff(fact_4408_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( hd(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),hd(A,Xs)),hd(B,Ys)) ) ) ) ).
% hd_zip
tff(fact_4409_subset__mset_Oadd__le__less__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),B2))
=> ( subset_mset(A,C2,D3)
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),D3)) ) ) ).
% subset_mset.add_le_less_mono
tff(fact_4410_subset__mset_Oadd__less__le__mono,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A),D3: multiset(A)] :
( subset_mset(A,A3,B2)
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),C2),D3))
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),D3)) ) ) ).
% subset_mset.add_less_le_mono
tff(fact_4411_subset__mset_OlessE,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,A3,B2)
=> ~ ! [C3: multiset(A)] :
( ( B2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C3) )
=> ( C3 = zero_zero(multiset(A)) ) ) ) ).
% subset_mset.lessE
tff(fact_4412_subset__mset_Oadd__neg__neg,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,A3,zero_zero(multiset(A)))
=> ( subset_mset(A,B2,zero_zero(multiset(A)))
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2),zero_zero(multiset(A))) ) ) ).
% subset_mset.add_neg_neg
tff(fact_4413_subset__mset_Oadd__pos__pos,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,zero_zero(multiset(A)),A3)
=> ( subset_mset(A,zero_zero(multiset(A)),B2)
=> subset_mset(A,zero_zero(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)) ) ) ).
% subset_mset.add_pos_pos
tff(fact_4414_subset__mset_Opos__add__strict,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( subset_mset(A,zero_zero(multiset(A)),A3)
=> ( subset_mset(A,B2,C2)
=> subset_mset(A,B2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)) ) ) ).
% subset_mset.pos_add_strict
tff(fact_4415_subset__implies__mult,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),R: set(product_prod(A,A))] :
( subset_mset(A,A4,B5)
=> pp(member(product_prod(multiset(A),multiset(A)),aa(multiset(A),product_prod(multiset(A),multiset(A)),aa(multiset(A),fun(multiset(A),product_prod(multiset(A),multiset(A))),product_Pair(multiset(A),multiset(A)),A4),B5),mult(A,R))) ) ).
% subset_implies_mult
tff(fact_4416_subset__mset_Oadd__strict__increasing2,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
=> ( subset_mset(A,B2,C2)
=> subset_mset(A,B2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)) ) ) ).
% subset_mset.add_strict_increasing2
tff(fact_4417_subset__mset_Oadd__strict__increasing,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A),C2: multiset(A)] :
( subset_mset(A,zero_zero(multiset(A)),A3)
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),C2))
=> subset_mset(A,B2,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),C2)) ) ) ).
% subset_mset.add_strict_increasing
tff(fact_4418_subset__mset_Oadd__pos__nonneg,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,zero_zero(multiset(A)),A3)
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),B2))
=> subset_mset(A,zero_zero(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)) ) ) ).
% subset_mset.add_pos_nonneg
tff(fact_4419_subset__mset_Oadd__nonpos__neg,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),A3),zero_zero(multiset(A))))
=> ( subset_mset(A,B2,zero_zero(multiset(A)))
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2),zero_zero(multiset(A))) ) ) ).
% subset_mset.add_nonpos_neg
tff(fact_4420_subset__mset_Oadd__nonneg__pos,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),zero_zero(multiset(A))),A3))
=> ( subset_mset(A,zero_zero(multiset(A)),B2)
=> subset_mset(A,zero_zero(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2)) ) ) ).
% subset_mset.add_nonneg_pos
tff(fact_4421_subset__mset_Oadd__neg__nonpos,axiom,
! [A: $tType,A3: multiset(A),B2: multiset(A)] :
( subset_mset(A,A3,zero_zero(multiset(A)))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),B2),zero_zero(multiset(A))))
=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B2),zero_zero(multiset(A))) ) ) ).
% subset_mset.add_neg_nonpos
tff(fact_4422_sorted__hd__min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( ( Xs != nil(A) )
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> ! [X: A] :
( pp(member(A,X,aa(list(A),set(A),set2(A),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),hd(A,Xs)),X)) ) ) ) ) ).
% sorted_hd_min
tff(fact_4423_sorted__hd__last,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,ord_less_eq(A),L)
=> ( ( L != nil(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),hd(A,L)),last(A,L))) ) ) ) ).
% sorted_hd_last
tff(fact_4424_hd__last__singletonI,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ( hd(A,Xs) = last(A,Xs) )
=> ( distinct(A,Xs)
=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,Xs)),nil(A)) ) ) ) ) ).
% hd_last_singletonI
tff(fact_4425_hd__butlast,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( hd(A,butlast(A,Xs)) = hd(A,Xs) ) ) ).
% hd_butlast
tff(fact_4426_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I4: set(B),F: fun(B,multiset(A))] :
( finite_finite2(B,I4)
=> ( ( I4 != bot_bot(set(B)) )
=> ( ! [I2: B] :
( pp(member(B,I2,I4))
=> subset_mset(A,zero_zero(multiset(A)),aa(B,multiset(A),F,I2)) )
=> subset_mset(A,zero_zero(multiset(A)),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,I4)) ) ) ) ).
% subset_mset.sum_pos
tff(fact_4427_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A4: set(B),F: fun(B,multiset(A)),G: fun(B,multiset(A))] :
( finite_finite2(B,A4)
=> ( ( A4 != bot_bot(set(B)) )
=> ( ! [X3: B] :
( pp(member(B,X3,A4))
=> subset_mset(A,aa(B,multiset(A),F,X3),aa(B,multiset(A),G,X3)) )
=> subset_mset(A,groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),F,A4),groups3894954378712506084id_sum(multiset(A),B,plus_plus(multiset(A)),zero_zero(multiset(A)),G,A4)) ) ) ) ).
% subset_mset.sum_strict_mono
tff(fact_4428_ofilter__ordLess,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),field2(A,R)))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4)))),R),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% ofilter_ordLess
tff(fact_4429_ofilter__subset__ordLess,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B5: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A4)
=> ( order_ofilter(A,R,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A4),B5))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,B5,aTP_Lamp_ma(set(A),fun(A,set(A)),B5)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).
% ofilter_subset_ordLess
tff(fact_4430_ofilter__subset__ordLeq,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B5: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A4)
=> ( order_ofilter(A,R,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,B5,aTP_Lamp_ma(set(A),fun(A,set(A)),B5)))),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).
% ofilter_subset_ordLeq
tff(fact_4431_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType,Xs: list(A),F: fun(A,nat)] :
( ( ( Xs = nil(A) )
=> ( size_list(A,F,Xs) = zero_zero(nat) ) )
& ( ( Xs != nil(A) )
=> ( size_list(A,F,Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,hd(A,Xs))),size_list(A,F,tl(A,Xs)))) ) ) ) ).
% Nitpick.size_list_simp(1)
tff(fact_4432_length__tl,axiom,
! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),tl(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).
% length_tl
tff(fact_4433_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list(A),X2: A] :
( ( L != nil(A) )
=> ( ( ( X2 = hd(A,L) )
| pp(member(A,X2,aa(list(A),set(A),set2(A),tl(A,L)))) )
<=> pp(member(A,X2,aa(list(A),set(A),set2(A),L))) ) ) ).
% in_hd_or_tl_conv
tff(fact_4434_tl__replicate,axiom,
! [A: $tType,N: nat,X2: A] : tl(A,replicate(A,N,X2)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X2) ).
% tl_replicate
tff(fact_4435_rev__split__conv,axiom,
! [A: $tType,L: list(A)] :
( ( L != nil(A) )
=> ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),rev(A,tl(A,L))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,L)),nil(A))) = rev(A,L) ) ) ).
% rev_split_conv
tff(fact_4436_ordLess__irreflexive,axiom,
! [A: $tType,R: set(product_prod(A,A))] : ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R),bNF_We4044943003108391690rdLess(A,A))) ).
% ordLess_irreflexive
tff(fact_4437_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_We4044943003108391690rdLess(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_We4044943003108391690rdLess(A,C))) ) ) ).
% ordLess_transitive
tff(fact_4438_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_Wellorder_ordLeq(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_Wellorder_ordLeq(A,C))) ) ) ).
% ordLeq_transitive
tff(fact_4439_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> ~ pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordLeq(B,A))) ) ).
% not_ordLess_ordLeq
tff(fact_4440_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B))) ) ).
% ordLess_imp_ordLeq
tff(fact_4441_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_We4044943003108391690rdLess(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_We4044943003108391690rdLess(A,C))) ) ) ).
% ordLeq_ordLess_trans
tff(fact_4442_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_Wellorder_ordLeq(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_We4044943003108391690rdLess(A,C))) ) ) ).
% ordLess_ordLeq_trans
tff(fact_4443_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( ~ pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_We4044943003108391690rdLess(B,A)))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).
% not_ordLess_iff_ordLeq
tff(fact_4444_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( ~ pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordLeq(B,A)))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ).
% not_ordLeq_iff_ordLess
tff(fact_4445_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
| pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).
% ordLess_or_ordLeq
tff(fact_4446_in__set__tlD,axiom,
! [A: $tType,X2: A,Xs: list(A)] :
( pp(member(A,X2,aa(list(A),set(A),set2(A),tl(A,Xs))))
=> pp(member(A,X2,aa(list(A),set(A),set2(A),Xs))) ) ).
% in_set_tlD
tff(fact_4447_tl__obtain__elem,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( ( tl(A,Xs) = nil(A) )
=> ~ ! [E3: A] : Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) ) ) ).
% tl_obtain_elem
tff(fact_4448_not__hd__in__tl,axiom,
! [A: $tType,X2: A,Xs: list(A)] :
( ( X2 != hd(A,Xs) )
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> pp(member(A,X2,aa(list(A),set(A),set2(A),tl(A,Xs)))) ) ) ).
% not_hd_in_tl
tff(fact_4449_tl__last,axiom,
! [A: $tType,Xs: list(A)] :
( ( tl(A,Xs) != nil(A) )
=> ( last(A,Xs) = last(A,tl(A,Xs)) ) ) ).
% tl_last
tff(fact_4450_rev__butlast__is__tl__rev,axiom,
! [A: $tType,L: list(A)] : rev(A,butlast(A,L)) = tl(A,rev(A,L)) ).
% rev_butlast_is_tl_rev
tff(fact_4451_tl__subset,axiom,
! [A: $tType,Xs: list(A),A4: set(A)] :
( ( Xs != nil(A) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),tl(A,Xs))),A4)) ) ) ).
% tl_subset
tff(fact_4452_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( Xs != nil(A) )
=> ( aa(nat,A,nth(A,tl(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).
% Misc.nth_tl
tff(fact_4453_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),P: fun(B,fun(A,bool))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(B,fun(A,bool),P,aa(nat,B,nth(B,Ys),N3)),aa(nat,A,nth(A,Xs),N3))) )
=> ! [N9: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N9),aa(list(A),nat,size_size(list(A)),tl(A,Xs))))
=> pp(aa(A,bool,aa(B,fun(A,bool),P,aa(nat,B,nth(B,tl(B,Ys)),N9)),aa(nat,A,nth(A,tl(A,Xs)),N9))) ) ) ) ).
% list_take_induct_tl2
tff(fact_4454_exists__minim__Well__order,axiom,
! [A: $tType,R4: set(set(product_prod(A,A)))] :
( ( R4 != bot_bot(set(set(product_prod(A,A)))) )
=> ( ! [X3: set(product_prod(A,A))] :
( pp(member(set(product_prod(A,A)),X3,R4))
=> order_well_order_on(A,field2(A,X3),X3) )
=> ? [X3: set(product_prod(A,A))] :
( pp(member(set(product_prod(A,A)),X3,R4))
& ! [Xa3: set(product_prod(A,A))] :
( pp(member(set(product_prod(A,A)),Xa3,R4))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).
% exists_minim_Well_order
tff(fact_4455_ordLeq__total,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
| pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).
% ordLeq_total
tff(fact_4456_ordLeq__reflexive,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R),bNF_Wellorder_ordLeq(A,A))) ) ).
% ordLeq_reflexive
tff(fact_4457_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> ( order_well_order_on(A,field2(A,R),R)
& order_well_order_on(B,field2(B,R3),R3) ) ) ).
% ordLeq_Well_order_simp
tff(fact_4458_tl__take,axiom,
! [A: $tType,N: nat,Xs: list(A)] : tl(A,take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),tl(A,Xs)) ).
% tl_take
tff(fact_4459_distinct__hd__tl,axiom,
! [A: $tType,Xs: list(A),X2: A] :
( distinct(A,Xs)
=> ( ( X2 = hd(A,Xs) )
=> ~ pp(member(A,X2,aa(list(A),set(A),set2(A),tl(A,Xs)))) ) ) ).
% distinct_hd_tl
tff(fact_4460_butlast__rev__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( butlast(A,rev(A,Xs)) = rev(A,tl(A,Xs)) ) ) ).
% butlast_rev_tl
tff(fact_4461_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( finite_finite2(A,field2(A,R))
=> ( ~ finite_finite2(B,field2(B,R3))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).
% finite_ordLess_infinite
tff(fact_4462_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
<=> ! [X4: A] :
( pp(member(A,X4,field2(A,R)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,order_underS(A,R,X4),aa(A,fun(A,set(A)),aTP_Lamp_ss(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X4)))),R3),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
tff(fact_4463_underS__Restr__ordLess,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( order_well_order_on(A,field2(A,R),R)
=> ( ( field2(A,R) != bot_bot(set(A)) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,order_underS(A,R,A3),aa(A,fun(A,set(A)),aTP_Lamp_ss(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),A3)))),R),bNF_We4044943003108391690rdLess(A,A))) ) ) ).
% underS_Restr_ordLess
tff(fact_4464_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Ys))) ) ).
% horner_sum_append
tff(fact_4465_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),aa(num,num,bit0,N)) = code_Neg(bitM(N)) ).
% Code_Numeral.sub_code(4)
tff(fact_4466_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,X2: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs))) ) ).
% horner_sum_simps(2)
tff(fact_4467_underS__E,axiom,
! [A: $tType,I: A,R4: set(product_prod(A,A)),J: A] :
( pp(member(A,I,order_underS(A,R4,J)))
=> ( ( I != J )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R4)) ) ) ).
% underS_E
tff(fact_4468_underS__I,axiom,
! [A: $tType,I: A,J: A,R4: set(product_prod(A,A))] :
( ( I != J )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J),R4))
=> pp(member(A,I,order_underS(A,R4,J))) ) ) ).
% underS_I
tff(fact_4469_underS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_underS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_st(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% underS_def
tff(fact_4470_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] : aa(code_integer,code_integer,code_dup,code_Neg(N)) = code_Neg(aa(num,num,bit0,N)) ).
% Code_Numeral.dup_code(3)
tff(fact_4471_less__integer__code_I3_J,axiom,
! [L: num] : ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),code_Neg(L))) ).
% less_integer_code(3)
tff(fact_4472_less__integer__code_I7_J,axiom,
! [K: num] : pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),code_Neg(K)),zero_zero(code_integer))) ).
% less_integer_code(7)
tff(fact_4473_plus__integer__code_I6_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_Neg(M)),code_Neg(N)) = code_Neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% plus_integer_code(6)
tff(fact_4474_less__eq__integer__code_I7_J,axiom,
! [K: num] : pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),code_Neg(K)),zero_zero(code_integer))) ).
% less_eq_integer_code(7)
tff(fact_4475_less__eq__integer__code_I3_J,axiom,
! [L: num] : ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),code_Neg(L))) ).
% less_eq_integer_code(3)
tff(fact_4476_less__integer__code_I9_J,axiom,
! [K: num,L: num] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),code_Neg(K)),code_Neg(L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),L),K)) ) ).
% less_integer_code(9)
tff(fact_4477_less__eq__integer__code_I9_J,axiom,
! [K: num,L: num] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),code_Neg(K)),code_Neg(L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),L),K)) ) ).
% less_eq_integer_code(9)
tff(fact_4478_minus__integer__code_I6_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_Neg(M)),code_Neg(N)) = aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,N),M) ).
% minus_integer_code(6)
tff(fact_4479_divmod__abs__code_I2_J,axiom,
! [K: num,L: num] : code_divmod_abs(code_Neg(K),code_Neg(L)) = unique8689654367752047608divmod(code_integer,K,L) ).
% divmod_abs_code(2)
tff(fact_4480_horner__sum__foldr,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_su(fun(B,A),fun(A,fun(B,fun(A,A))),F),A3),Xs),zero_zero(A)) ) ).
% horner_sum_foldr
tff(fact_4481_underS__incl__iff,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( order_679001287576687338der_on(A,field2(A,R),R)
=> ( pp(member(A,A3,field2(A,R)))
=> ( pp(member(A,B2,field2(A,R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2)))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)) ) ) ) ) ).
% underS_incl_iff
tff(fact_4482_Code__Numeral_Osub__code_I5_J,axiom,
! [N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),aa(num,num,bit1,N)) = code_Neg(aa(num,num,bit0,N)) ).
% Code_Numeral.sub_code(5)
tff(fact_4483_ordLess__iff__ordIso__Restr,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_We4044943003108391690rdLess(B,A)))
<=> ? [X4: A] :
( pp(member(A,X4,field2(A,R)))
& pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,order_underS(A,R,X4),aa(A,fun(A,set(A)),aTP_Lamp_ss(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X4)))),bNF_Wellorder_ordIso(B,A))) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
tff(fact_4484_sum__list__update,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [K: nat,Xs: list(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K,X2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),X2)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).
% sum_list_update
tff(fact_4485_length__product__lists,axiom,
! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xss)),one_one(nat)) ).
% length_product_lists
tff(fact_4486_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( comm_semiring_0(B)
& comm_semiring_0(A) )
=> ! [A4: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B)))
=> pp(aa(fun(fun(D,B),fun(B,fun(list(D),B))),bool,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),bool),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B5,A4),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A4,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B5),A4))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).
% horner_sum_transfer
tff(fact_4487_sum__list_OCons,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [X2: A,Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ).
% sum_list.Cons
tff(fact_4488_sum__list__append,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),aa(list(A),A,groups8242544230860333062m_list(A),Ys)) ) ).
% sum_list_append
tff(fact_4489_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_Wellorder_ordIso(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_Wellorder_ordIso(A,C))) ) ) ).
% ordIso_transitive
tff(fact_4490_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordIso(B,A))) ) ).
% ordIso_symmetric
tff(fact_4491_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& monoid_add(A) )
=> ! [A4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B)))
=> pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A4),A4),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).
% sum_list_transfer
tff(fact_4492_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),L: list(A),L4: list(B),Q: fun(list(A),fun(list(B),bool))] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),L),L4))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
=> ( ! [X3: A,X5: B,Ls2: list(A),Ls3: list(B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),X5))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Ls2),Ls3))
=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,Ls2),Ls3))
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ls2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X5),Ls3))) ) ) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,L),L4)) ) ) ) ).
% Misc.list_all2_induct
tff(fact_4493_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
<=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
& pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordLeq(B,A))) ) ) ).
% ordIso_iff_ordLeq
tff(fact_4494_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B))) ) ).
% ordIso_imp_ordLeq
tff(fact_4495_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_Wellorder_ordLeq(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_Wellorder_ordLeq(A,C))) ) ) ).
% ordIso_ordLeq_trans
tff(fact_4496_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_Wellorder_ordIso(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_Wellorder_ordLeq(A,C))) ) ) ).
% ordLeq_ordIso_trans
tff(fact_4497_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B))) ) ).
% not_ordLess_ordIso
tff(fact_4498_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_We4044943003108391690rdLess(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_We4044943003108391690rdLess(A,C))) ) ) ).
% ordIso_ordLess_trans
tff(fact_4499_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R8: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R3),R8),bNF_Wellorder_ordIso(B,C)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C))),aa(set(product_prod(A,A)),fun(set(product_prod(C,C)),product_prod(set(product_prod(A,A)),set(product_prod(C,C)))),product_Pair(set(product_prod(A,A)),set(product_prod(C,C))),R),R8),bNF_We4044943003108391690rdLess(A,C))) ) ) ).
% ordLess_ordIso_trans
tff(fact_4500_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bz(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G),Xs))) ) ).
% sum_list_addf
tff(fact_4501_finite__well__order__on__ordIso,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
( finite_finite2(A,A4)
=> ( order_well_order_on(A,A4,R)
=> ( order_well_order_on(A,A4,R3)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R3),bNF_Wellorder_ordIso(A,A))) ) ) ) ).
% finite_well_order_on_ordIso
tff(fact_4502_ordIso__reflexive,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R),bNF_Wellorder_ordIso(A,A))) ) ).
% ordIso_reflexive
tff(fact_4503_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),uminus_uminus(A)),F)),Xs)) ) ).
% uminus_sum_list_map
tff(fact_4504_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
<=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
| pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B))) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
tff(fact_4505_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ).
% sum_list.eq_foldr
tff(fact_4506_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R),bNF_Wellorder_ordLeq(A,B)))
<=> ? [P6: set(product_prod(B,B))] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),field2(B,P6)),field2(B,R)))
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),P6),bNF_Wellorder_ordIso(A,B)))
& pp(member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P6),R),bNF_Wellorder_ordLeq(B,B))) ) ) ).
% internalize_ordLeq
tff(fact_4507_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),R),bNF_We4044943003108391690rdLess(A,B)))
<=> ? [P6: set(product_prod(B,B))] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),field2(B,P6)),field2(B,R)))
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R3),P6),bNF_Wellorder_ordIso(A,B)))
& pp(member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P6),R),bNF_We4044943003108391690rdLess(B,B))) ) ) ).
% internalize_ordLess
tff(fact_4508_card__length__sum__list__rec,axiom,
! [M: nat,N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
=> ( finite_card(list(nat),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_sv(nat,fun(nat,fun(list(nat),bool)),M),N5))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(list(nat),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_sw(nat,fun(nat,fun(list(nat),bool)),M),N5)))),finite_card(list(nat),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_sx(nat,fun(nat,fun(list(nat),bool)),M),N5)))) ) ) ).
% card_length_sum_list_rec
tff(fact_4509_size__list__conv__sum__list,axiom,
! [B: $tType,F: fun(B,nat),Xs: list(B)] : size_list(B,F,Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).
% size_list_conv_sum_list
tff(fact_4510_card__length__sum__list,axiom,
! [M: nat,N5: nat] : finite_card(list(nat),aa(fun(list(nat),bool),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),bool),aTP_Lamp_sv(nat,fun(nat,fun(list(nat),bool)),M),N5))) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),M)),one_one(nat))),N5) ).
% card_length_sum_list
tff(fact_4511_sum__list__Suc,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,aTP_Lamp_ie(fun(A,nat),fun(A,nat),F)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).
% sum_list_Suc
tff(fact_4512_select__weight__def,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))),aTP_Lamp_sy(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).
% select_weight_def
tff(fact_4513_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_mult(B)
& monoid_mult(A) )
=> ! [A4: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A4,one_one(A)),one_one(B)))
=> ( pp(aa(fun(B,fun(B,B)),bool,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B)))
=> pp(aa(fun(list(B),B),bool,aa(fun(list(A),A),fun(fun(list(B),B),bool),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A4),A4),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B))) ) ) ) ).
% prod_list_transfer
tff(fact_4514_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
( order_well_order_on(A,field2(A,R),R)
=> ( inj_on(A,B,F,field2(A,R))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_We2720479622203943262_image(A,B,R,F)),bNF_Wellorder_ordIso(A,B))) ) ) ).
% dir_image_ordIso
tff(fact_4515_underS__incr,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( trans(A,R)
=> ( antisym(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2))) ) ) ) ).
% underS_incr
tff(fact_4516_prod__list_ONil,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aa(list(A),A,groups5270119922927024881d_list(A),nil(A)) = one_one(A) ) ) ).
% prod_list.Nil
tff(fact_4517_antisymD,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( antisym(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),R))
=> ( A3 = B2 ) ) ) ) ).
% antisymD
tff(fact_4518_antisymI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X3: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X3),R))
=> ( X3 = Y4 ) ) )
=> antisym(A,R) ) ).
% antisymI
tff(fact_4519_antisym__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( antisym(A,R)
<=> ! [X4: A,Y3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R))
=> ( X4 = Y3 ) ) ) ) ).
% antisym_def
tff(fact_4520_dir__image__def,axiom,
! [A2: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,A2)] : bNF_We2720479622203943262_image(A,A2,R,F) = aa(fun(product_prod(A2,A2),bool),set(product_prod(A2,A2)),collect(product_prod(A2,A2)),aa(fun(A,A2),fun(product_prod(A2,A2),bool),aTP_Lamp_sz(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),R),F)) ).
% dir_image_def
tff(fact_4521_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = aa(A,A,foldr(A,A,times_times(A),Xs),one_one(A)) ) ).
% prod_list.eq_foldr
tff(fact_4522_pick__same,axiom,
! [A: $tType,L: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pick(A,aa(list(A),list(product_prod(code_natural,A)),map(A,product_prod(code_natural,A),aa(code_natural,fun(A,product_prod(code_natural,A)),product_Pair(code_natural,A),one_one(code_natural))),Xs),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,A,nth(A,Xs),L) ) ) ).
% pick_same
tff(fact_4523_flip__pred,axiom,
! [A: $tType,B: $tType,A4: set(product_prod(A,B)),R4: fun(B,fun(A,bool))] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),conversep(B,A,R4)))))
=> pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(product_prod(A,B)),set(product_prod(B,A)),image2(product_prod(A,B),product_prod(B,A),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_it(A,fun(B,product_prod(B,A))))),A4)),aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),R4)))) ) ).
% flip_pred
tff(fact_4524_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X2: A,Y: B] : aa(product_prod(A,B),set(A),basic_fsts(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)) = aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))) ).
% prod_set_simps(1)
tff(fact_4525_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] : basic_snds(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)) = aa(set(B),set(B),insert2(B,Y),bot_bot(set(B))) ).
% prod_set_simps(2)
tff(fact_4526_natural__of__nat__of__nat,axiom,
code_natural_of_nat = semiring_1_of_nat(code_natural) ).
% natural_of_nat_of_nat
tff(fact_4527_division__segment__natural_Oabs__eq,axiom,
! [X2: nat] : aa(code_natural,code_natural,euclid7384307370059645450egment(code_natural),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,euclid7384307370059645450egment(nat),X2)) ).
% division_segment_natural.abs_eq
tff(fact_4528_set__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se5668285175392031749et_bit(code_natural),Xa),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),Xa),X2)) ).
% set_bit_natural.abs_eq
tff(fact_4529_flip__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se8732182000553998342ip_bit(code_natural),Xa),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),Xa),X2)) ).
% flip_bit_natural.abs_eq
tff(fact_4530_unset__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2638667681897837118et_bit(code_natural),Xa),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),Xa),X2)) ).
% unset_bit_natural.abs_eq
tff(fact_4531_euclidean__size__natural_Oabs__eq,axiom,
! [X2: nat] : aa(code_natural,nat,euclid6346220572633701492n_size(code_natural),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,nat,euclid6346220572633701492n_size(nat),X2) ).
% euclidean_size_natural.abs_eq
tff(fact_4532_bit__natural_Oabs__eq,axiom,
! [X2: nat] : aa(code_natural,fun(nat,bool),bit_se5641148757651400278ts_bit(code_natural),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),X2) ).
% bit_natural.abs_eq
tff(fact_4533_natural__of__nat__cases,axiom,
! [X2: code_natural] :
~ ! [Y4: nat] :
( ( X2 = aa(nat,code_natural,code_natural_of_nat,Y4) )
=> ~ pp(member(nat,Y4,top_top(set(nat)))) ) ).
% natural_of_nat_cases
tff(fact_4534_natural__of__nat__induct,axiom,
! [P: fun(code_natural,bool),X2: code_natural] :
( ! [Y4: nat] :
( pp(member(nat,Y4,top_top(set(nat))))
=> pp(aa(code_natural,bool,P,aa(nat,code_natural,code_natural_of_nat,Y4))) )
=> pp(aa(code_natural,bool,P,X2)) ) ).
% natural_of_nat_induct
tff(fact_4535_natural__of__nat__inject,axiom,
! [X2: nat,Y: nat] :
( pp(member(nat,X2,top_top(set(nat))))
=> ( pp(member(nat,Y,top_top(set(nat))))
=> ( ( aa(nat,code_natural,code_natural_of_nat,X2) = aa(nat,code_natural,code_natural_of_nat,Y) )
<=> ( X2 = Y ) ) ) ) ).
% natural_of_nat_inject
tff(fact_4536_take__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2584673776208193580ke_bit(code_natural),Xa),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Xa),X2)) ).
% take_bit_natural.abs_eq
tff(fact_4537_and__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se5824344872417868541ns_and(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Xa),X2)) ).
% and_natural.abs_eq
tff(fact_4538_xor__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se5824344971392196577ns_xor(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Xa),X2)) ).
% xor_natural.abs_eq
tff(fact_4539_or__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se1065995026697491101ons_or(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Xa),X2)) ).
% or_natural.abs_eq
tff(fact_4540_mask__natural_Oabs__eq,axiom,
! [X2: nat] : aa(nat,code_natural,bit_se2239418461657761734s_mask(code_natural),X2) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,bit_se2239418461657761734s_mask(nat),X2)) ).
% mask_natural.abs_eq
tff(fact_4541_push__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4730199178511100633sh_bit(code_natural),Xa),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Xa),X2)) ).
% push_bit_natural.abs_eq
tff(fact_4542_drop__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4197421643247451524op_bit(code_natural),Xa),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Xa),X2)) ).
% drop_bit_natural.abs_eq
tff(fact_4543_zero__natural__def,axiom,
zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).
% zero_natural_def
tff(fact_4544_one__natural__def,axiom,
one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).
% one_natural_def
tff(fact_4545_less__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa),X2)) ) ).
% less_natural.abs_eq
tff(fact_4546_plus__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),X2)) ).
% plus_natural.abs_eq
tff(fact_4547_minus__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),X2)) ).
% minus_natural.abs_eq
tff(fact_4548_less__eq__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X2)) ) ).
% less_eq_natural.abs_eq
tff(fact_4549_times__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa),X2)) ).
% times_natural.abs_eq
tff(fact_4550_divide__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),X2)) ).
% divide_natural.abs_eq
tff(fact_4551_mask__natural__def,axiom,
bit_se2239418461657761734s_mask(code_natural) = aa(fun(nat,nat),fun(nat,code_natural),map_fun(nat,nat,nat,code_natural,id(nat),code_natural_of_nat),bit_se2239418461657761734s_mask(nat)) ).
% mask_natural_def
tff(fact_4552_modulo__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),X2)) ).
% modulo_natural.abs_eq
tff(fact_4553_select__def,axiom,
! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(nat,code_natural,code_natural_of_nat,aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_ta(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).
% select_def
tff(fact_4554_Nat_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_natural,code_Nat,aa(int,code_integer,code_integer_of_int,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,X2)) ).
% Nat.abs_eq
tff(fact_4555_Nat__def,axiom,
code_Nat = aa(fun(int,nat),fun(code_integer,code_natural),map_fun(code_integer,int,nat,code_natural,code_int_of_integer,code_natural_of_nat),nat2) ).
% Nat_def
tff(fact_4556_natural__of__integer_Oabs__eq,axiom,
! [X2: int] : aa(code_integer,code_natural,code_n4118661773612635043nteger,aa(int,code_integer,code_integer_of_int,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,X2)) ).
% natural_of_integer.abs_eq
tff(fact_4557_natural__of__nat__of__natural__inverse,axiom,
! [N: code_natural] : aa(nat,code_natural,code_natural_of_nat,aa(code_natural,nat,code_nat_of_natural,N)) = N ).
% natural_of_nat_of_natural_inverse
tff(fact_4558_nat__of__natural__of__nat__inverse,axiom,
! [N: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,code_natural_of_nat,N)) = N ).
% nat_of_natural_of_nat_inverse
tff(fact_4559_nat__of__natural__inverse,axiom,
! [X2: code_natural] : aa(nat,code_natural,code_natural_of_nat,aa(code_natural,nat,code_nat_of_natural,X2)) = X2 ).
% nat_of_natural_inverse
tff(fact_4560_nat__of__natural__min,axiom,
! [K: code_natural,L: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,ord_min(code_natural,K),L)) = aa(nat,nat,ord_min(nat,aa(code_natural,nat,code_nat_of_natural,K)),aa(code_natural,nat,code_nat_of_natural,L)) ).
% nat_of_natural_min
tff(fact_4561_nat__of__natural__max,axiom,
! [K: code_natural,L: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,ord_max(code_natural,K),L)) = aa(nat,nat,ord_max(nat,aa(code_natural,nat,code_nat_of_natural,K)),aa(code_natural,nat,code_nat_of_natural,L)) ).
% nat_of_natural_max
tff(fact_4562_of__nat__of__natural,axiom,
! [N: code_natural] : aa(nat,code_natural,semiring_1_of_nat(code_natural),aa(code_natural,nat,code_nat_of_natural,N)) = N ).
% of_nat_of_natural
tff(fact_4563_nat__of__natural__of__nat,axiom,
! [N: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,semiring_1_of_nat(code_natural),N)) = N ).
% nat_of_natural_of_nat
tff(fact_4564_zero__natural_Orep__eq,axiom,
aa(code_natural,nat,code_nat_of_natural,zero_zero(code_natural)) = zero_zero(nat) ).
% zero_natural.rep_eq
tff(fact_4565_one__natural_Orep__eq,axiom,
aa(code_natural,nat,code_nat_of_natural,one_one(code_natural)) = one_one(nat) ).
% one_natural.rep_eq
tff(fact_4566_plus__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% plus_natural.rep_eq
tff(fact_4567_minus__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% minus_natural.rep_eq
tff(fact_4568_nat__of__natural__numeral,axiom,
! [K: num] : aa(code_natural,nat,code_nat_of_natural,aa(num,code_natural,numeral_numeral(code_natural),K)) = aa(num,nat,numeral_numeral(nat),K) ).
% nat_of_natural_numeral
tff(fact_4569_times__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% times_natural.rep_eq
tff(fact_4570_divide__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% divide_natural.rep_eq
tff(fact_4571_modulo__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% modulo_natural.rep_eq
tff(fact_4572_division__segment__natural_Orep__eq,axiom,
! [X2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,euclid7384307370059645450egment(code_natural),X2)) = aa(nat,nat,euclid7384307370059645450egment(nat),aa(code_natural,nat,code_nat_of_natural,X2)) ).
% division_segment_natural.rep_eq
tff(fact_4573_euclidean__size__natural_Orep__eq,axiom,
! [X2: code_natural] : aa(code_natural,nat,euclid6346220572633701492n_size(code_natural),X2) = aa(nat,nat,euclid6346220572633701492n_size(nat),aa(code_natural,nat,code_nat_of_natural,X2)) ).
% euclidean_size_natural.rep_eq
tff(fact_4574_unset__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2638667681897837118et_bit(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),X2),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% unset_bit_natural.rep_eq
tff(fact_4575_flip__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se8732182000553998342ip_bit(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),X2),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% flip_bit_natural.rep_eq
tff(fact_4576_set__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se5668285175392031749et_bit(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),X2),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% set_bit_natural.rep_eq
tff(fact_4577_Nat_Orep__eq,axiom,
! [X2: code_integer] : aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_Nat,X2)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X2)) ).
% Nat.rep_eq
tff(fact_4578_push__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4730199178511100633sh_bit(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),X2),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% push_bit_natural.rep_eq
tff(fact_4579_mask__natural_Orep__eq,axiom,
! [X2: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,bit_se2239418461657761734s_mask(code_natural),X2)) = aa(nat,nat,bit_se2239418461657761734s_mask(nat),X2) ).
% mask_natural.rep_eq
tff(fact_4580_and__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se5824344872417868541ns_and(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% and_natural.rep_eq
tff(fact_4581_bit__natural_Orep__eq,axiom,
! [X2: code_natural] : aa(code_natural,fun(nat,bool),bit_se5641148757651400278ts_bit(code_natural),X2) = aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(code_natural,nat,code_nat_of_natural,X2)) ).
% bit_natural.rep_eq
tff(fact_4582_nat__of__natural__inject,axiom,
! [X2: code_natural,Y: code_natural] :
( ( aa(code_natural,nat,code_nat_of_natural,X2) = aa(code_natural,nat,code_nat_of_natural,Y) )
<=> ( X2 = Y ) ) ).
% nat_of_natural_inject
tff(fact_4583_natural__eqI,axiom,
! [M: code_natural,N: code_natural] :
( ( aa(code_natural,nat,code_nat_of_natural,M) = aa(code_natural,nat,code_nat_of_natural,N) )
=> ( M = N ) ) ).
% natural_eqI
tff(fact_4584_natural__eq__iff,axiom,
! [M: code_natural,N: code_natural] :
( ( M = N )
<=> ( aa(code_natural,nat,code_nat_of_natural,M) = aa(code_natural,nat,code_nat_of_natural,N) ) ) ).
% natural_eq_iff
tff(fact_4585_natural__cases,axiom,
! [M: code_natural] :
~ ! [N3: nat] : M != aa(nat,code_natural,semiring_1_of_nat(code_natural),N3) ).
% natural_cases
tff(fact_4586_nat__of__natural,axiom,
! [X2: code_natural] : pp(member(nat,aa(code_natural,nat,code_nat_of_natural,X2),top_top(set(nat)))) ).
% nat_of_natural
tff(fact_4587_nat__of__natural__cases,axiom,
! [Y: nat] :
( pp(member(nat,Y,top_top(set(nat))))
=> ~ ! [X3: code_natural] : Y != aa(code_natural,nat,code_nat_of_natural,X3) ) ).
% nat_of_natural_cases
tff(fact_4588_nat__of__natural__induct,axiom,
! [Y: nat,P: fun(nat,bool)] :
( pp(member(nat,Y,top_top(set(nat))))
=> ( ! [X3: code_natural] : pp(aa(nat,bool,P,aa(code_natural,nat,code_nat_of_natural,X3)))
=> pp(aa(nat,bool,P,Y)) ) ) ).
% nat_of_natural_induct
tff(fact_4589_take__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2584673776208193580ke_bit(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),X2),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% take_bit_natural.rep_eq
tff(fact_4590_or__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se1065995026697491101ons_or(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% or_natural.rep_eq
tff(fact_4591_xor__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),bit_se5824344971392196577ns_xor(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% xor_natural.rep_eq
tff(fact_4592_drop__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4197421643247451524op_bit(code_natural),X2),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),X2),aa(code_natural,nat,code_nat_of_natural,Xa)) ).
% drop_bit_natural.rep_eq
tff(fact_4593_natural__of__integer_Orep__eq,axiom,
! [X2: code_integer] : aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_n4118661773612635043nteger,X2)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X2)) ).
% natural_of_integer.rep_eq
tff(fact_4594_less__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),X2),Xa))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa))) ) ).
% less_natural.rep_eq
tff(fact_4595_natural__of__nat__inverse,axiom,
! [Y: nat] :
( pp(member(nat,Y,top_top(set(nat))))
=> ( aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,code_natural_of_nat,Y)) = Y ) ) ).
% natural_of_nat_inverse
tff(fact_4596_type__definition__natural,axiom,
type_definition(code_natural,nat,code_nat_of_natural,code_natural_of_nat,top_top(set(nat))) ).
% type_definition_natural
tff(fact_4597_or__natural__def,axiom,
bit_se1065995026697491101ons_or(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se1065995026697491101ons_or(nat)) ).
% or_natural_def
tff(fact_4598_xor__natural__def,axiom,
bit_se5824344971392196577ns_xor(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se5824344971392196577ns_xor(nat)) ).
% xor_natural_def
tff(fact_4599_and__natural__def,axiom,
bit_se5824344872417868541ns_and(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se5824344872417868541ns_and(nat)) ).
% and_natural_def
tff(fact_4600_less__eq__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X2),Xa))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(code_natural,nat,code_nat_of_natural,X2)),aa(code_natural,nat,code_nat_of_natural,Xa))) ) ).
% less_eq_natural.rep_eq
tff(fact_4601_division__segment__natural__def,axiom,
euclid7384307370059645450egment(code_natural) = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),euclid7384307370059645450egment(nat)) ).
% division_segment_natural_def
tff(fact_4602_euclidean__size__natural__def,axiom,
euclid6346220572633701492n_size(code_natural) = aa(fun(nat,nat),fun(code_natural,nat),map_fun(code_natural,nat,nat,nat,code_nat_of_natural,id(nat)),euclid6346220572633701492n_size(nat)) ).
% euclidean_size_natural_def
tff(fact_4603_plus__natural__def,axiom,
plus_plus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),plus_plus(nat)) ).
% plus_natural_def
tff(fact_4604_minus__natural__def,axiom,
minus_minus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),minus_minus(nat)) ).
% minus_natural_def
tff(fact_4605_bit__natural__def,axiom,
bit_se5641148757651400278ts_bit(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(nat,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(nat,bool),code_nat_of_natural,id(fun(nat,bool))),bit_se5641148757651400278ts_bit(nat)) ).
% bit_natural_def
tff(fact_4606_less__natural__def,axiom,
ord_less(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(code_natural,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(code_natural,bool),code_nat_of_natural,map_fun(code_natural,nat,bool,bool,code_nat_of_natural,id(bool))),ord_less(nat)) ).
% less_natural_def
tff(fact_4607_times__natural__def,axiom,
times_times(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),times_times(nat)) ).
% times_natural_def
tff(fact_4608_divide__natural__def,axiom,
divide_divide(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),divide_divide(nat)) ).
% divide_natural_def
tff(fact_4609_take__bit__natural__def,axiom,
bit_se2584673776208193580ke_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se2584673776208193580ke_bit(nat)) ).
% take_bit_natural_def
tff(fact_4610_push__bit__natural__def,axiom,
bit_se4730199178511100633sh_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se4730199178511100633sh_bit(nat)) ).
% push_bit_natural_def
tff(fact_4611_less__eq__natural__def,axiom,
ord_less_eq(code_natural) = aa(fun(nat,fun(nat,bool)),fun(code_natural,fun(code_natural,bool)),map_fun(code_natural,nat,fun(nat,bool),fun(code_natural,bool),code_nat_of_natural,map_fun(code_natural,nat,bool,bool,code_nat_of_natural,id(bool))),ord_less_eq(nat)) ).
% less_eq_natural_def
tff(fact_4612_modulo__natural__def,axiom,
modulo_modulo(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),modulo_modulo(nat)) ).
% modulo_natural_def
tff(fact_4613_drop__bit__natural__def,axiom,
bit_se4197421643247451524op_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se4197421643247451524op_bit(nat)) ).
% drop_bit_natural_def
tff(fact_4614_unset__bit__natural__def,axiom,
bit_se2638667681897837118et_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se2638667681897837118et_bit(nat)) ).
% unset_bit_natural_def
tff(fact_4615_flip__bit__natural__def,axiom,
bit_se8732182000553998342ip_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se8732182000553998342ip_bit(nat)) ).
% flip_bit_natural_def
tff(fact_4616_set__bit__natural__def,axiom,
bit_se5668285175392031749et_bit(code_natural) = aa(fun(nat,fun(nat,nat)),fun(nat,fun(code_natural,code_natural)),map_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),id(nat),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),bit_se5668285175392031749et_bit(nat)) ).
% set_bit_natural_def
tff(fact_4617_natural__decr,axiom,
! [N: code_natural] :
( ( N != zero_zero(code_natural) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,N)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,N))) ) ).
% natural_decr
tff(fact_4618_natural__of__integer__def,axiom,
code_n4118661773612635043nteger = aa(fun(int,nat),fun(code_integer,code_natural),map_fun(code_integer,int,nat,code_natural,code_int_of_integer,code_natural_of_nat),nat2) ).
% natural_of_integer_def
tff(fact_4619_integer__of__natural__of__integer,axiom,
! [K: code_integer] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(code_integer,code_natural,code_n4118661773612635043nteger,K)) = aa(code_integer,code_integer,ord_max(code_integer,zero_zero(code_integer)),K) ).
% integer_of_natural_of_integer
tff(fact_4620_integer__of__natural_Orep__eq,axiom,
! [X2: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,X2)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,X2)) ).
% integer_of_natural.rep_eq
tff(fact_4621_int__of__integer__of__natural,axiom,
! [N: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,N)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,N)) ).
% int_of_integer_of_natural
tff(fact_4622_integer__of__natural__of__nat,axiom,
! [N: nat] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,semiring_1_of_nat(code_natural),N)) = aa(nat,code_integer,semiring_1_of_nat(code_integer),N) ).
% integer_of_natural_of_nat
tff(fact_4623_natural__of__integer__of__natural,axiom,
! [N: code_natural] : aa(code_integer,code_natural,code_n4118661773612635043nteger,aa(code_natural,code_integer,code_i5400310926305786745atural,N)) = N ).
% natural_of_integer_of_natural
tff(fact_4624_integer__of__natural__def,axiom,
code_i5400310926305786745atural = aa(fun(nat,int),fun(code_natural,code_integer),map_fun(code_natural,nat,int,code_integer,code_nat_of_natural,code_integer_of_int),semiring_1_of_nat(int)) ).
% integer_of_natural_def
tff(fact_4625_integer__of__natural_Oabs__eq,axiom,
! [X2: nat] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,code_natural_of_nat,X2)) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X2)) ).
% integer_of_natural.abs_eq
tff(fact_4626_Code__Numeral_Osize__nat__def,axiom,
code_size_nat = code_nat_of_natural ).
% Code_Numeral.size_nat_def
tff(fact_4627_Code__Numeral_OSuc__def,axiom,
code_Suc = aa(fun(nat,nat),fun(code_natural,code_natural),map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat),suc) ).
% Code_Numeral.Suc_def
tff(fact_4628_sum__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A4: multiset(B),B5: multiset(B),G: fun(B,A)] :
( ( inter_mset(B,A4,B5) = zero_zero(multiset(B)) )
=> ( aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),union_mset(B,A4,B5))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),A4))),aa(multiset(A),A,comm_m7189776963980413722m_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,G),B5))) ) ) ) ).
% sum_mset.union_disjoint
tff(fact_4629_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( aa(code_natural,code_natural,code_Suc,Natural) = aa(code_natural,code_natural,code_Suc,Natural2) )
<=> ( Natural = Natural2 ) ) ).
% natural.inject
tff(fact_4630_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: fun(code_natural,fun(T,T)),Natural: code_natural] : aa(code_natural,T,code_rec_natural(T,F1,F22),aa(code_natural,code_natural,code_Suc,Natural)) = aa(T,T,aa(code_natural,fun(T,T),F22,Natural),aa(code_natural,T,code_rec_natural(T,F1,F22),Natural)) ).
% natural.simps(7)
tff(fact_4631_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: fun(code_natural,T),Natural: code_natural] : aa(code_natural,T,code_case_natural(T,F1,F22),aa(code_natural,code_natural,code_Suc,Natural)) = aa(code_natural,T,F22,Natural) ).
% natural.simps(5)
tff(fact_4632_Suc_Orep__eq,axiom,
! [X2: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,code_Suc,X2)) = aa(nat,nat,suc,aa(code_natural,nat,code_nat_of_natural,X2)) ).
% Suc.rep_eq
tff(fact_4633_union__sup__distrib__right,axiom,
! [A: $tType,C5: multiset(A),A4: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),union_mset(A,A4,B5)) = union_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B5)) ).
% union_sup_distrib_right
tff(fact_4634_sup__union__distrib__left,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A),C5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),union_mset(A,A4,B5)),C5) = union_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),C5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C5)) ).
% sup_union_distrib_left
tff(fact_4635_natural_Oexhaust,axiom,
! [Y: code_natural] :
( ( Y != zero_zero(code_natural) )
=> ~ ! [Natural3: code_natural] : Y != aa(code_natural,code_natural,code_Suc,Natural3) ) ).
% natural.exhaust
tff(fact_4636_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] : zero_zero(code_natural) != aa(code_natural,code_natural,code_Suc,Natural2) ).
% natural.distinct(1)
tff(fact_4637_natural_Odistinct_I2_J,axiom,
! [Natural2: code_natural] : aa(code_natural,code_natural,code_Suc,Natural2) != zero_zero(code_natural) ).
% natural.distinct(2)
tff(fact_4638_union__mset__def,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : union_mset(A,A4,B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),A4)) ).
% union_mset_def
tff(fact_4639_union__diff__inter__eq__sup,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),inter_mset(A,A4,B5)) = union_mset(A,A4,B5) ).
% union_diff_inter_eq_sup
tff(fact_4640_union__diff__sup__eq__inter,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)),union_mset(A,A4,B5)) = inter_mset(A,A4,B5) ).
% union_diff_sup_eq_inter
tff(fact_4641_size__Un__Int,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),A4)),aa(multiset(A),nat,size_size(multiset(A)),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),union_mset(A,A4,B5))),aa(multiset(A),nat,size_size(multiset(A)),inter_mset(A,A4,B5))) ).
% size_Un_Int
tff(fact_4642_Suc_Oabs__eq,axiom,
! [X2: nat] : aa(code_natural,code_natural,code_Suc,aa(nat,code_natural,code_natural_of_nat,X2)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,suc,X2)) ).
% Suc.abs_eq
tff(fact_4643_Suc__natural__minus__one,axiom,
! [N: code_natural] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),aa(code_natural,code_natural,code_Suc,N)),one_one(code_natural)) = N ).
% Suc_natural_minus_one
tff(fact_4644_size__Un__disjoint,axiom,
! [A: $tType,A4: multiset(A),B5: multiset(A)] :
( ( inter_mset(A,A4,B5) = zero_zero(multiset(A)) )
=> ( aa(multiset(A),nat,size_size(multiset(A)),union_mset(A,A4,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),A4)),aa(multiset(A),nat,size_size(multiset(A)),B5)) ) ) ).
% size_Un_disjoint
tff(fact_4645_random__aux__set_Ocases,axiom,
! [X2: product_prod(code_natural,code_natural)] :
( ! [J3: code_natural] : X2 != aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),zero_zero(code_natural)),J3)
=> ~ ! [I2: code_natural,J3: code_natural] : X2 != aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),aa(code_natural,code_natural,code_Suc,I2)),J3) ) ).
% random_aux_set.cases
tff(fact_4646_Eps__Opt__def,axiom,
! [A: $tType,P: fun(A,bool)] :
( ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
=> ( eps_Opt(A,P) = aa(A,option(A),some(A),fChoice(A,P)) ) )
& ( ~ ? [X_13: A] : pp(aa(A,bool,P,X_13))
=> ( eps_Opt(A,P) = none(A) ) ) ) ).
% Eps_Opt_def
tff(fact_4647_acyclic__insert,axiom,
! [A: $tType,Y: A,X2: A,R: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2)),R))
<=> ( transitive_acyclic(A,R)
& ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R))) ) ) ).
% acyclic_insert
tff(fact_4648_acyclic__empty,axiom,
! [A: $tType] : transitive_acyclic(A,bot_bot(set(product_prod(A,A)))) ).
% acyclic_empty
tff(fact_4649_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ne(A,fun(B,fun(A,fun(B,bool))),X2),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y) ).
% Eps_case_prod_eq
tff(fact_4650_some__insert__self,axiom,
! [A: $tType,S2: set(A)] :
( ( S2 != bot_bot(set(A)) )
=> ( aa(set(A),set(A),insert2(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),S2))),S2) = S2 ) ) ).
% some_insert_self
tff(fact_4651_some__elem,axiom,
! [A: $tType,S2: set(A)] :
( ( S2 != bot_bot(set(A)) )
=> pp(member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),S2)),S2)) ) ).
% some_elem
tff(fact_4652_cyclic__subset,axiom,
! [A: $tType,R4: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,R4)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),S2))
=> ~ transitive_acyclic(A,S2) ) ) ).
% cyclic_subset
tff(fact_4653_acyclic__union_I1_J,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A4),B5))
=> transitive_acyclic(A,A4) ) ).
% acyclic_union(1)
tff(fact_4654_acyclic__union_I2_J,axiom,
! [A: $tType,A4: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A4),B5))
=> transitive_acyclic(A,B5) ) ).
% acyclic_union(2)
tff(fact_4655_inv__on__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),X2: B] : aa(B,A,inv_on(A,B,F,A4),X2) = fChoice(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_nd(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F),A4),X2)) ).
% inv_on_def
tff(fact_4656_split__paired__Eps,axiom,
! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_tb(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P))) ).
% split_paired_Eps
tff(fact_4657_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [R: set(product_prod(B,B)),F: fun(B,A)] :
( ! [A5: B,B3: B] :
( pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B3),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,B3)),aa(B,A,F,A5))) )
=> transitive_acyclic(B,R) ) ) ).
% acyclicI_order
tff(fact_4658_subset__mset_OacyclicI__order,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,B)),F: fun(B,multiset(A))] :
( ! [A5: B,B3: B] :
( pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B3),R))
=> subset_mset(A,aa(B,multiset(A),F,B3),aa(B,multiset(A),F,A5)) )
=> transitive_acyclic(B,R) ) ).
% subset_mset.acyclicI_order
tff(fact_4659_acyclicI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,R)))
=> transitive_acyclic(A,R) ) ).
% acyclicI
tff(fact_4660_acyclic__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( transitive_acyclic(A,R)
<=> ! [X4: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4),transitive_trancl(A,R))) ) ).
% acyclic_def
tff(fact_4661_cyclicE,axiom,
! [A: $tType,G: set(product_prod(A,A))] :
( ~ transitive_acyclic(A,G)
=> ~ ! [X3: A] : ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3),transitive_trancl(A,G))) ) ).
% cyclicE
tff(fact_4662_some__theI,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] :
( ? [A14: A,X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),P,A14),X_12))
=> ( ! [B12: B,B23: B] :
( ? [A5: A] : pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B12))
=> ( ? [A5: A] : pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B23))
=> ( B12 = B23 ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,fChoice(A,aTP_Lamp_tc(fun(A,fun(B,bool)),fun(A,bool),P))),the(B,aTP_Lamp_td(fun(A,fun(B,bool)),fun(B,bool),P)))) ) ) ).
% some_theI
tff(fact_4663_acyclic__insert__cyclic,axiom,
! [A: $tType,G: set(product_prod(A,A)),X2: A,Y: A] :
( transitive_acyclic(A,G)
=> ( ~ transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),G))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2),transitive_rtrancl(A,G))) ) ) ).
% acyclic_insert_cyclic
tff(fact_4664_fun__of__rel__def,axiom,
! [A: $tType,B: $tType,R4: set(product_prod(B,A)),X2: B] : fun_of_rel(B,A,R4,X2) = fChoice(A,aa(B,fun(A,bool),aTP_Lamp_au(set(product_prod(B,A)),fun(B,fun(A,bool)),R4),X2)) ).
% fun_of_rel_def
tff(fact_4665_irrefl__tranclI,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A] :
( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),converse(A,A,R)),transitive_rtrancl(A,R)) = bot_bot(set(product_prod(A,A))) )
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),transitive_trancl(A,R))) ) ).
% irrefl_tranclI
tff(fact_4666_quicksort_Opelims,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: list(A),Y: list(A)] :
( ( linorder_quicksort(A,X2) = Y )
=> ( pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),X2))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = nil(A) )
=> ~ pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),nil(A))) ) )
=> ~ ! [X3: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
=> ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),linorder_quicksort(A,filter(A,aTP_Lamp_te(A,fun(A,bool),X3),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),linorder_quicksort(A,filter(A,aa(A,fun(A,bool),ord_less_eq(A),X3),Xs2)))) )
=> ~ pp(aa(list(A),bool,accp(list(A),linord6200660962353139674rt_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) ) ) ) ) ) ).
% quicksort.pelims
tff(fact_4667_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),converse(B,A,R)))
<=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),R)) ) ).
% converse_iff
tff(fact_4668_pair__set__inverse,axiom,
! [A: $tType,B: $tType,P: fun(B,fun(A,bool))] : converse(B,A,aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_tf(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P))) ).
% pair_set_inverse
tff(fact_4669_below__Id__inv,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),converse(A,A,R4)),id2(A)))
<=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),id2(A))) ) ).
% below_Id_inv
tff(fact_4670_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R))
=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),converse(A,B,R))) ) ).
% converseI
tff(fact_4671_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod(A,B),R: set(product_prod(B,A))] :
( pp(member(product_prod(A,B),Yx,converse(B,A,R)))
=> ~ ! [X3: B,Y4: A] :
( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y4),X3) )
=> ~ pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X3),Y4),R)) ) ) ).
% converseE
tff(fact_4672_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),converse(B,A,R)))
=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3),R)) ) ).
% converseD
tff(fact_4673_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set(product_prod(A,B))] :
( pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22),converse(A,B,R)))
<=> ? [A7: A,B7: B] :
( ( A1 = B7 )
& ( A22 = A7 )
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7),R)) ) ) ).
% converse.simps
tff(fact_4674_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set(product_prod(A,B))] :
( pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22),converse(A,B,R)))
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A22),A1),R)) ) ).
% converse.cases
tff(fact_4675_in__listrel1__converse,axiom,
! [A: $tType,X2: list(A),Y: list(A),R: set(product_prod(A,A))] :
( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),listrel1(A,converse(A,A,R))))
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y),converse(list(A),list(A),listrel1(A,R)))) ) ).
% in_listrel1_converse
tff(fact_4676_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X2: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2),transitive_rtrancl(A,R)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,converse(A,A,R)))) ) ).
% rtrancl_converseI
tff(fact_4677_rtrancl__converseD,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,converse(A,A,R))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2),transitive_rtrancl(A,R))) ) ).
% rtrancl_converseD
tff(fact_4678_trancl__converseI,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),converse(A,A,transitive_trancl(A,R))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,converse(A,A,R)))) ) ).
% trancl_converseI
tff(fact_4679_trancl__converseD,axiom,
! [A: $tType,X2: A,Y: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_trancl(A,converse(A,A,R))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),converse(A,A,transitive_trancl(A,R)))) ) ).
% trancl_converseD
tff(fact_4680_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( finite_finite2(product_prod(A,A),R4)
=> ( wf(A,converse(A,A,R4))
<=> wf(A,R4) ) ) ).
% finite_wf_eq_wf_converse
tff(fact_4681_converse__unfold,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A))] : converse(B,A,R) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_tg(set(product_prod(B,A)),fun(A,fun(B,bool)),R))) ).
% converse_unfold
tff(fact_4682_conversep__converse__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X: B,Xa3: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X),Xa3))
<=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Xa3),converse(A,B,R))) ) ).
% conversep_converse_eq
tff(fact_4683_converse__def,axiom,
! [B: $tType,A: $tType,X: set(product_prod(A,B))] : converse(A,B,X) = aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),X)))) ).
% converse_def
tff(fact_4684_is__singleton__altdef,axiom,
! [A: $tType,A4: set(A)] :
( is_singleton(A,A4)
<=> ( finite_card(A,A4) = one_one(nat) ) ) ).
% is_singleton_altdef
tff(fact_4685_Code__Numeral_Osub__code_I2_J,axiom,
! [M: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit0,M)),one2) = aa(num,code_integer,code_Pos,bitM(M)) ).
% Code_Numeral.sub_code(2)
tff(fact_4686_su__rel__fun__def,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B)] :
( su_rel_fun(A,B,F5,F)
<=> ( ! [A8: A,B9: B,B10: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B9),F5))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B10),F5))
=> ( B9 = B10 ) ) )
& ! [A8: A,P5: bool] :
( ! [B9: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B9),F5))
=> pp(P5) )
=> pp(P5) )
& ! [A8: A] : aa(A,B,F,A8) = the(B,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),F5),A8)) ) ) ).
% su_rel_fun_def
tff(fact_4687_Code__Numeral_OPos__def,axiom,
code_Pos = numeral_numeral(code_integer) ).
% Code_Numeral.Pos_def
tff(fact_4688_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B),A4: A] :
( su_rel_fun(A,B,F5,F)
=> ~ ! [B8: B] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B8),F5)) ) ).
% su_rel_fun.surjective
tff(fact_4689_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F5: set(product_prod(A,B)),F: fun(A,B),A4: A,B5: B,B11: B] :
( su_rel_fun(A,B,F5,F)
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5),F5))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B11),F5))
=> ( B5 = B11 ) ) ) ) ).
% su_rel_fun.unique
tff(fact_4690_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B),A4: A,B5: B] :
( su_rel_fun(A,B,F5,F)
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5),F5))
=> ( B5 = aa(A,B,F,A4) ) ) ) ).
% su_rel_fun.repr2
tff(fact_4691_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B),A4: A] :
( su_rel_fun(A,B,F5,F)
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),aa(A,B,F,A4)),F5)) ) ).
% su_rel_fun.repr1
tff(fact_4692_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B),A4: A,B5: B] :
( su_rel_fun(A,B,F5,F)
=> ( ( aa(A,B,F,A4) = B5 )
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B5),F5)) ) ) ).
% su_rel_fun.repr
tff(fact_4693_Pos__fold_I2_J,axiom,
! [K: num] : aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,K)) = aa(num,code_integer,code_Pos,aa(num,num,bit0,K)) ).
% Pos_fold(2)
tff(fact_4694_Pos__fold_I1_J,axiom,
aa(num,code_integer,numeral_numeral(code_integer),one2) = aa(num,code_integer,code_Pos,one2) ).
% Pos_fold(1)
tff(fact_4695_Pos__fold_I3_J,axiom,
! [K: num] : aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,K)) = aa(num,code_integer,code_Pos,aa(num,num,bit1,K)) ).
% Pos_fold(3)
tff(fact_4696_one__integer__code,axiom,
one_one(code_integer) = aa(num,code_integer,code_Pos,one2) ).
% one_integer_code
tff(fact_4697_Code__Numeral_Odup__code_I2_J,axiom,
! [N: num] : aa(code_integer,code_integer,code_dup,aa(num,code_integer,code_Pos,N)) = aa(num,code_integer,code_Pos,aa(num,num,bit0,N)) ).
% Code_Numeral.dup_code(2)
tff(fact_4698_less__integer__code_I4_J,axiom,
! [K: num] : ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(num,code_integer,code_Pos,K)),zero_zero(code_integer))) ).
% less_integer_code(4)
tff(fact_4699_less__integer__code_I2_J,axiom,
! [L: num] : pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),aa(num,code_integer,code_Pos,L))) ).
% less_integer_code(2)
tff(fact_4700_plus__integer__code_I3_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(num,code_integer,code_Pos,M)),aa(num,code_integer,code_Pos,N)) = aa(num,code_integer,code_Pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% plus_integer_code(3)
tff(fact_4701_less__eq__integer__code_I2_J,axiom,
! [L: num] : pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),aa(num,code_integer,code_Pos,L))) ).
% less_eq_integer_code(2)
tff(fact_4702_less__eq__integer__code_I4_J,axiom,
! [K: num] : ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,code_Pos,K)),zero_zero(code_integer))) ).
% less_eq_integer_code(4)
tff(fact_4703_less__integer__code_I8_J,axiom,
! [K: num,L: num] : pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),code_Neg(K)),aa(num,code_integer,code_Pos,L))) ).
% less_integer_code(8)
tff(fact_4704_less__integer__code_I6_J,axiom,
! [K: num,L: num] : ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(num,code_integer,code_Pos,K)),code_Neg(L))) ).
% less_integer_code(6)
tff(fact_4705_uminus__integer__code_I3_J,axiom,
! [M: num] : aa(code_integer,code_integer,uminus_uminus(code_integer),code_Neg(M)) = aa(num,code_integer,code_Pos,M) ).
% uminus_integer_code(3)
tff(fact_4706_uminus__integer__code_I2_J,axiom,
! [M: num] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,code_Pos,M)) = code_Neg(M) ).
% uminus_integer_code(2)
tff(fact_4707_Code__Numeral_ONeg__def,axiom,
! [N: num] : code_Neg(N) = aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,code_Pos,N)) ).
% Code_Numeral.Neg_def
tff(fact_4708_less__eq__integer__code_I8_J,axiom,
! [K: num,L: num] : pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),code_Neg(K)),aa(num,code_integer,code_Pos,L))) ).
% less_eq_integer_code(8)
tff(fact_4709_less__eq__integer__code_I6_J,axiom,
! [K: num,L: num] : ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,code_Pos,K)),code_Neg(L))) ).
% less_eq_integer_code(6)
tff(fact_4710_times__integer__code_I3_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,code_Pos,M)),aa(num,code_integer,code_Pos,N)) = aa(num,code_integer,code_Pos,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_integer_code(3)
tff(fact_4711_less__integer__code_I5_J,axiom,
! [K: num,L: num] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),aa(num,code_integer,code_Pos,K)),aa(num,code_integer,code_Pos,L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),K),L)) ) ).
% less_integer_code(5)
tff(fact_4712_less__eq__integer__code_I5_J,axiom,
! [K: num,L: num] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,code_Pos,K)),aa(num,code_integer,code_Pos,L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),K),L)) ) ).
% less_eq_integer_code(5)
tff(fact_4713_minus__integer__code_I3_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(num,code_integer,code_Pos,M)),aa(num,code_integer,code_Pos,N)) = aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N) ).
% minus_integer_code(3)
tff(fact_4714_divmod__abs__code_I1_J,axiom,
! [K: num,L: num] : code_divmod_abs(aa(num,code_integer,code_Pos,K),aa(num,code_integer,code_Pos,L)) = unique8689654367752047608divmod(code_integer,K,L) ).
% divmod_abs_code(1)
tff(fact_4715_minus__integer__code_I4_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(num,code_integer,code_Pos,M)),code_Neg(N)) = aa(num,code_integer,code_Pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% minus_integer_code(4)
tff(fact_4716_minus__integer__code_I5_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),code_Neg(M)),aa(num,code_integer,code_Pos,N)) = code_Neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% minus_integer_code(5)
tff(fact_4717_times__integer__code_I4_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,code_Pos,M)),code_Neg(N)) = code_Neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_integer_code(4)
tff(fact_4718_times__integer__code_I5_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Neg(M)),aa(num,code_integer,code_Pos,N)) = code_Neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_integer_code(5)
tff(fact_4719_times__integer__code_I6_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_Neg(M)),code_Neg(N)) = aa(num,code_integer,code_Pos,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_integer_code(6)
tff(fact_4720_plus__integer__code_I4_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(num,code_integer,code_Pos,M)),code_Neg(N)) = aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,M),N) ).
% plus_integer_code(4)
tff(fact_4721_plus__integer__code_I5_J,axiom,
! [M: num,N: num] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_Neg(M)),aa(num,code_integer,code_Pos,N)) = aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,N),M) ).
% plus_integer_code(5)
tff(fact_4722_divmod__abs__code_I3_J,axiom,
! [K: num,L: num] : code_divmod_abs(code_Neg(K),aa(num,code_integer,code_Pos,L)) = unique8689654367752047608divmod(code_integer,K,L) ).
% divmod_abs_code(3)
tff(fact_4723_divmod__abs__code_I4_J,axiom,
! [K: num,L: num] : code_divmod_abs(aa(num,code_integer,code_Pos,K),code_Neg(L)) = unique8689654367752047608divmod(code_integer,K,L) ).
% divmod_abs_code(4)
tff(fact_4724_Code__Numeral_Osub__code_I3_J,axiom,
! [M: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,aa(num,num,bit1,M)),one2) = aa(num,code_integer,code_Pos,aa(num,num,bit0,M)) ).
% Code_Numeral.sub_code(3)
tff(fact_4725_su__rel__fun_Of__def,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B),A4: A] :
( su_rel_fun(A,B,F5,F)
=> ( aa(A,B,F,A4) = the(B,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),F5),A4)) ) ) ).
% su_rel_fun.f_def
tff(fact_4726_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F5: set(product_prod(A,B)),F: fun(A,B)] :
( ! [A10: A,B8: B,B13: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B8),F5))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B13),F5))
=> ( B8 = B13 ) ) )
=> ( ! [A10: A,P4: bool] :
( ! [B14: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A10),B14),F5))
=> pp(P4) )
=> pp(P4) )
=> ( ! [A10: A] : aa(A,B,F,A10) = the(B,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),F5),A10))
=> su_rel_fun(A,B,F5,F) ) ) ) ).
% su_rel_fun.intro
tff(fact_4727_aboveS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_aboveS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_th(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% aboveS_def
tff(fact_4728_relInvImage__def,axiom,
! [B: $tType,A: $tType,A4: set(A),R4: set(product_prod(B,B)),F: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A4,R4,F) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_ti(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A4),R4),F)) ).
% relInvImage_def
tff(fact_4729_prod__list__def,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( groups5270119922927024881d_list(A) = groups_monoid_F(A,times_times(A),one_one(A)) ) ) ).
% prod_list_def
tff(fact_4730_sum__list__def,axiom,
! [A: $tType] :
( monoid_add(A)
=> ( groups8242544230860333062m_list(A) = groups_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).
% sum_list_def
tff(fact_4731_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [X2: B,Xs: list(B),F: fun(B,A)] :
( pp(member(B,X2,aa(list(B),set(B),set2(B),Xs)))
=> ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X2)),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),remove1(B,X2,Xs)))) ) ) ) ).
% sum_list_map_remove1
tff(fact_4732_normalize__stable,axiom,
! [Q3: int,P3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3))
=> ( algebr8660921524188924756oprime(int,P3,Q3)
=> ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) ) ) ) ).
% normalize_stable
tff(fact_4733_distinct__count__atmost__1,axiom,
! [A: $tType,X2: list(A)] :
( distinct(A,X2)
<=> ! [A7: A] :
( ( pp(member(A,A7,aa(list(A),set(A),set2(A),X2)))
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset(A,X2)),A7) = one_one(nat) ) )
& ( ~ pp(member(A,A7,aa(list(A),set(A),set2(A),X2)))
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset(A,X2)),A7) = zero_zero(nat) ) ) ) ) ).
% distinct_count_atmost_1
tff(fact_4734_coprime__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,A3)
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% coprime_self
tff(fact_4735_mset__empty__count,axiom,
! [A: $tType,M2: multiset(A)] :
( ! [P6: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),P6) = zero_zero(nat)
<=> ( M2 = zero_zero(multiset(A)) ) ) ).
% mset_empty_count
tff(fact_4736_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).
% coprime_imp_gcd_eq_1
tff(fact_4737_count__union,axiom,
! [A: $tType,M2: multiset(A),N5: multiset(A),A3: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),A3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N5),A3)) ).
% count_union
tff(fact_4738_coprime__0__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,zero_zero(A))
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% coprime_0_right_iff
tff(fact_4739_coprime__0__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,zero_zero(A),A3)
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% coprime_0_left_iff
tff(fact_4740_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
<=> ( pp(dvd_dvd(A,C2,one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_right_iff
tff(fact_4741_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
<=> ( pp(dvd_dvd(A,C2,one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_left_iff
tff(fact_4742_is__unit__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2),one_one(A)))
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_gcd
tff(fact_4743_count__greater__eq__one__iff,axiom,
! [A: $tType,M2: multiset(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),X2)))
<=> pp(member(A,X2,set_mset(A,M2))) ) ).
% count_greater_eq_one_iff
tff(fact_4744_count__mset__set_I1_J,axiom,
! [A: $tType,A4: set(A),X2: A] :
( finite_finite2(A,A4)
=> ( pp(member(A,X2,A4))
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,A4)),X2) = one_one(nat) ) ) ) ).
% count_mset_set(1)
tff(fact_4745_coprime__1__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).
% coprime_1_left
tff(fact_4746_coprime__1__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).
% coprime_1_right
tff(fact_4747_coprime__add__one__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),A3) ) ).
% coprime_add_one_left
tff(fact_4748_coprime__add__one__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).
% coprime_add_one_right
tff(fact_4749_coprime__doff__one__right,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).
% coprime_doff_one_right
tff(fact_4750_coprime__diff__one__left,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A)),A3) ) ).
% coprime_diff_one_left
tff(fact_4751_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_right_imp_coprime
tff(fact_4752_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_left_imp_coprime
tff(fact_4753_coprime__common__divisor,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( pp(dvd_dvd(A,C2,A3))
=> ( pp(dvd_dvd(A,C2,B2))
=> pp(dvd_dvd(A,C2,one_one(A))) ) ) ) ) ).
% coprime_common_divisor
tff(fact_4754_coprime__absorb__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [Y: A,X2: A] :
( pp(dvd_dvd(A,Y,X2))
=> ( algebr8660921524188924756oprime(A,X2,Y)
<=> pp(dvd_dvd(A,Y,one_one(A))) ) ) ) ).
% coprime_absorb_right
tff(fact_4755_coprime__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,D3: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C2,D3)
=> ( ! [E3: A] :
( ~ pp(dvd_dvd(A,E3,one_one(A)))
=> ( pp(dvd_dvd(A,E3,A3))
=> ( pp(dvd_dvd(A,E3,B2))
=> pp(dvd_dvd(A,E3,C2)) ) ) )
=> ( ! [E3: A] :
( ~ pp(dvd_dvd(A,E3,one_one(A)))
=> ( pp(dvd_dvd(A,E3,A3))
=> ( pp(dvd_dvd(A,E3,B2))
=> pp(dvd_dvd(A,E3,D3)) ) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% coprime_imp_coprime
tff(fact_4756_coprime__absorb__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X2: A,Y: A] :
( pp(dvd_dvd(A,X2,Y))
=> ( algebr8660921524188924756oprime(A,X2,Y)
<=> pp(dvd_dvd(A,X2,one_one(A))) ) ) ) ).
% coprime_absorb_left
tff(fact_4757_not__coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(dvd_dvd(A,C2,A3))
=> ( pp(dvd_dvd(A,C2,B2))
=> ( ~ pp(dvd_dvd(A,C2,one_one(A)))
=> ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% not_coprimeI
tff(fact_4758_not__coprimeE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ~ algebr8660921524188924756oprime(A,A3,B2)
=> ~ ! [C3: A] :
( pp(dvd_dvd(A,C3,A3))
=> ( pp(dvd_dvd(A,C3,B2))
=> pp(dvd_dvd(A,C3,one_one(A))) ) ) ) ) ).
% not_coprimeE
tff(fact_4759_coprime__def,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ! [C4: A] :
( pp(dvd_dvd(A,C4,A3))
=> ( pp(dvd_dvd(A,C4,B2))
=> pp(dvd_dvd(A,C4,one_one(A))) ) ) ) ) ).
% coprime_def
tff(fact_4760_coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ! [C3: A] :
( pp(dvd_dvd(A,C3,A3))
=> ( pp(dvd_dvd(A,C3,B2))
=> pp(dvd_dvd(A,C3,one_one(A))) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprimeI
tff(fact_4761_plus__multiset_Orep__eq,axiom,
! [A: $tType,X2: multiset(A),Xa: multiset(A),X: A] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Xa)),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),X2),X)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),Xa),X)) ).
% plus_multiset.rep_eq
tff(fact_4762_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).
% coprime_iff_gcd_eq_1
tff(fact_4763_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% gcd_eq_1_imp_coprime
tff(fact_4764_quotient__of__coprime,axiom,
! [R: rat,P3: int,Q3: int] :
( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
=> algebr8660921524188924756oprime(int,P3,Q3) ) ).
% quotient_of_coprime
tff(fact_4765_normalize__coprime,axiom,
! [R: product_prod(int,int),P3: int,Q3: int] :
( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
=> algebr8660921524188924756oprime(int,P3,Q3) ) ).
% normalize_coprime
tff(fact_4766_invertible__coprime,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2) = one_one(A) )
=> algebr8660921524188924756oprime(A,A3,C2) ) ) ).
% invertible_coprime
tff(fact_4767_count__ne__remove,axiom,
! [A: $tType,X2: A,T2: A,S2: multiset(A)] :
( ( X2 != T2 )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),S2),X2) = aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),S2),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),T2),zero_zero(multiset(A))))),X2) ) ) ).
% count_ne_remove
tff(fact_4768_count__in__diffI,axiom,
! [A: $tType,N5: multiset(A),X2: A,M2: multiset(A)] :
( ! [N3: nat] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N5),X2) != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),X2))
=> pp(member(A,X2,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),N5)))) ) ).
% count_in_diffI
tff(fact_4769_count__single,axiom,
! [A: $tType,B2: A,A3: A] :
( ( ( B2 = A3 )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))),A3) = one_one(nat) ) )
& ( ( B2 != A3 )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),B2),zero_zero(multiset(A)))),A3) = zero_zero(nat) ) ) ) ).
% count_single
tff(fact_4770_in__diff__countE,axiom,
! [A: $tType,X2: A,M2: multiset(A),N5: multiset(A)] :
( pp(member(A,X2,set_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M2),N5))))
=> ~ ! [N3: nat] : aa(A,nat,aa(multiset(A),fun(A,nat),count(A),M2),X2) != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,N3)),aa(A,nat,aa(multiset(A),fun(A,nat),count(A),N5),X2)) ) ).
% in_diff_countE
tff(fact_4771_remove1__tl,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( remove1(A,hd(A,Xs),Xs) = tl(A,Xs) ) ) ).
% remove1_tl
tff(fact_4772_count__mset__set__finite__iff,axiom,
! [A: $tType,S2: set(A),A3: A] :
( finite_finite2(A,S2)
=> ( ( pp(member(A,A3,S2))
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S2)),A3) = one_one(nat) ) )
& ( ~ pp(member(A,A3,S2))
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,S2)),A3) = zero_zero(nat) ) ) ) ) ).
% count_mset_set_finite_iff
tff(fact_4773_count__mset__set_H,axiom,
! [A: $tType,A4: set(A),X2: A] :
( ( ( finite_finite2(A,A4)
& pp(member(A,X2,A4)) )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,A4)),X2) = one_one(nat) ) )
& ( ~ ( finite_finite2(A,A4)
& pp(member(A,X2,A4)) )
=> ( aa(A,nat,aa(multiset(A),fun(A,nat),count(A),mset_set(A,A4)),X2) = zero_zero(nat) ) ) ) ).
% count_mset_set'
tff(fact_4774_length__remove1,axiom,
! [A: $tType,X2: A,Xs: list(A)] :
( ( pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),remove1(A,X2,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) )
& ( ~ pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),remove1(A,X2,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).
% length_remove1
tff(fact_4775_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_tj(B,fun(A,fun(B,A)),Y),C2)),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),Y)) ) ).
% prod_mset_delta'
tff(fact_4776_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Y: B,C2: A,A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aa(A,fun(B,A),aTP_Lamp_tk(B,fun(A,fun(B,A)),Y),C2)),A4)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(B,nat,aa(multiset(B),fun(B,nat),count(B),A4),Y)) ) ).
% prod_mset_delta
tff(fact_4777_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult(A)
& ord(B) )
=> ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ).
% prod.atLeastLessThan_reindex
tff(fact_4778_bij__betw__id,axiom,
! [A: $tType,A4: set(A)] : bij_betw(A,A,id(A),A4,A4) ).
% bij_betw_id
tff(fact_4779_bij__betw__add,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,A4: set(A),B5: set(A)] :
( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A3),A4,B5)
<=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),A4) = B5 ) ) ) ).
% bij_betw_add
tff(fact_4780_prod__mset__empty,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),zero_zero(multiset(A))) = one_one(A) ) ) ).
% prod_mset_empty
tff(fact_4781_prod__mset_Ounion,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: multiset(A),N5: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M2)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),N5)) ) ).
% prod_mset.union
tff(fact_4782_prod__mset__Un,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(A),B5: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4)),aa(multiset(A),A,comm_m9189036328036947845d_mset(A),B5)) ) ).
% prod_mset_Un
tff(fact_4783_bij__betw__of__nat,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N5: set(nat),A4: set(A)] :
( bij_betw(nat,A,semiring_1_of_nat(A),N5,A4)
<=> ( aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),N5) = A4 ) ) ) ).
% bij_betw_of_nat
tff(fact_4784_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(B)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),aa(multiset(B),multiset(A),image_mset(B,A,aTP_Lamp_fl(B,A)),A4)) = one_one(A) ) ).
% prod_mset.neutral_const
tff(fact_4785_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> ( aa(set(A),set(B),image2(A,B,F),A4) = B5 ) ) ).
% bij_betw_imp_surj_on
tff(fact_4786_f__the__inv__into__f__bij__betw,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B),X2: B] :
( bij_betw(A,B,F,A4,B5)
=> ( ( bij_betw(A,B,F,A4,B5)
=> pp(member(B,X2,B5)) )
=> ( aa(A,B,F,aa(B,A,the_inv_into(A,B,A4,F),X2)) = X2 ) ) ) ).
% f_the_inv_into_f_bij_betw
tff(fact_4787_bij__betw__the__inv__into,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> bij_betw(B,A,the_inv_into(A,B,A4,F),B5,A4) ) ).
% bij_betw_the_inv_into
tff(fact_4788_bij__betw__id__iff,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( bij_betw(A,A,id(A),A4,B5)
<=> ( A4 = B5 ) ) ).
% bij_betw_id_iff
tff(fact_4789_bij__betwE,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> ! [X: A] :
( pp(member(A,X,A4))
=> pp(member(B,aa(A,B,F,X),B5)) ) ) ).
% bij_betwE
tff(fact_4790_bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> ? [G3: fun(B,A)] : bij_betw(B,A,G3,B5,A4) ) ).
% bij_betw_inv
tff(fact_4791_bij__betw__cong,axiom,
! [A: $tType,B: $tType,A4: set(A),F: fun(A,B),G: fun(A,B),A15: set(B)] :
( ! [A5: A] :
( pp(member(A,A5,A4))
=> ( aa(A,B,F,A5) = aa(A,B,G,A5) ) )
=> ( bij_betw(A,B,F,A4,A15)
<=> bij_betw(A,B,G,A4,A15) ) ) ).
% bij_betw_cong
tff(fact_4792_bij__betw__apply,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B),A3: A] :
( bij_betw(A,B,F,A4,B5)
=> ( pp(member(A,A3,A4))
=> pp(member(B,aa(A,B,F,A3),B5)) ) ) ).
% bij_betw_apply
tff(fact_4793_bij__betw__iff__bijections,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
<=> ? [G5: fun(B,A)] :
( ! [X4: A] :
( pp(member(A,X4,A4))
=> ( pp(member(B,aa(A,B,F,X4),B5))
& ( aa(B,A,G5,aa(A,B,F,X4)) = X4 ) ) )
& ! [X4: B] :
( pp(member(B,X4,B5))
=> ( pp(member(A,aa(B,A,G5,X4),A4))
& ( aa(A,B,F,aa(B,A,G5,X4)) = X4 ) ) ) ) ) ).
% bij_betw_iff_bijections
tff(fact_4794_involuntory__imp__bij,axiom,
! [A: $tType,F: fun(A,A)] :
( ! [X3: A] : aa(A,A,F,aa(A,A,F,X3)) = X3
=> bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).
% involuntory_imp_bij
tff(fact_4795_bij__pointE,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Y: B] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ~ ! [X3: A] :
( ( Y = aa(A,B,F,X3) )
=> ~ ! [X13: A] :
( ( Y = aa(A,B,F,X13) )
=> ( X13 = X3 ) ) ) ) ).
% bij_pointE
tff(fact_4796_bij__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
<=> ! [X4: B] :
? [Xa2: A] :
( ( aa(A,B,F,Xa2) = X4 )
& ! [Y3: A] :
( ( aa(A,B,F,Y3) = X4 )
=> ( Y3 = Xa2 ) ) ) ) ).
% bij_iff
tff(fact_4797_bij__betw__imp__inj__on,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> inj_on(A,B,F,A4) ) ).
% bij_betw_imp_inj_on
tff(fact_4798_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),A4: set(A),B5: set(B),G: fun(B,C),C5: set(C)] :
( bij_betw(A,B,F,A4,B5)
=> ( bij_betw(B,C,G,B5,C5)
=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),A4,C5) ) ) ).
% bij_betw_trans
tff(fact_4799_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),A4: set(A),A15: set(B),F2: fun(B,C),A16: set(C)] :
( bij_betw(A,B,F,A4,A15)
=> ( bij_betw(B,C,F2,A15,A16)
<=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F),A4,A16) ) ) ).
% bij_betw_comp_iff
tff(fact_4800_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( bij_betw(A,B,F,A4,bot_bot(set(B)))
=> ( A4 = bot_bot(set(A)) ) ) ).
% bij_betw_empty2
tff(fact_4801_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(B)] :
( bij_betw(A,B,F,bot_bot(set(A)),A4)
=> ( A4 = bot_bot(set(B)) ) ) ).
% bij_betw_empty1
tff(fact_4802_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A4: set(A),F2: fun(B,A),F: fun(A,B),A15: set(B)] :
( ! [X3: A] :
( pp(member(A,X3,A4))
=> ( aa(B,A,F2,aa(A,B,F,X3)) = X3 ) )
=> ( ! [X3: B] :
( pp(member(B,X3,A15))
=> ( aa(A,B,F,aa(B,A,F2,X3)) = X3 ) )
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F),A4)),A15))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A15)),A4))
=> bij_betw(A,B,F,A4,A15) ) ) ) ) ).
% bij_betw_byWitness
tff(fact_4803_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),A15: set(B),B5: set(A),B11: set(B)] :
( bij_betw(A,B,F,A4,A15)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A4))
=> ( ( aa(set(A),set(B),image2(A,B,F),B5) = B11 )
=> bij_betw(A,B,F,B5,B11) ) ) ) ).
% bij_betw_subset
tff(fact_4804_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A)] :
( bij_betw(A,B,F,A4,top_top(set(B)))
=> ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ).
% bij_betw_imp_surj
tff(fact_4805_bij__is__surj,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ).
% bij_is_surj
tff(fact_4806_bij__uminus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> bij_betw(A,A,uminus_uminus(A),top_top(set(A)),top_top(set(A))) ) ).
% bij_uminus
tff(fact_4807_bij__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( bij_betw(B,C,G,top_top(set(B)),top_top(set(C)))
=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),top_top(set(A)),top_top(set(C))) ) ) ).
% bij_comp
tff(fact_4808_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,A4)
=> bij_betw(A,B,F,A4,aa(set(A),set(B),image2(A,B,F),A4)) ) ).
% inj_on_imp_bij_betw
tff(fact_4809_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( inj_on(A,B,F,A4)
=> ( ( aa(set(A),set(B),image2(A,B,F),A4) = B5 )
=> bij_betw(A,B,F,A4,B5) ) ) ).
% bij_betw_imageI
tff(fact_4810_bij__betw__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
<=> ( inj_on(A,B,F,A4)
& ( aa(set(A),set(B),image2(A,B,F),A4) = B5 ) ) ) ).
% bij_betw_def
tff(fact_4811_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( finite_finite2(A,A4)
=> ( finite_finite2(B,B5)
=> ( ? [F6: fun(A,B)] : bij_betw(A,B,F6,A4,B5)
<=> ( finite_card(A,A4) = finite_card(B,B5) ) ) ) ) ).
% bij_betw_iff_card
tff(fact_4812_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( finite_finite2(A,A4)
=> ( finite_finite2(B,B5)
=> ( ( finite_card(A,A4) = finite_card(B,B5) )
=> ? [H2: fun(A,B)] : bij_betw(A,B,H2,A4,B5) ) ) ) ).
% finite_same_card_bij
tff(fact_4813_bij__is__inj,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> inj_on(A,B,F,top_top(set(A))) ) ).
% bij_is_inj
tff(fact_4814_bij__id,axiom,
! [A: $tType] : bij_betw(A,A,id(A),top_top(set(A)),top_top(set(A))) ).
% bij_id
tff(fact_4815_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X2: nat] :
( algebr8660921524188924756oprime(nat,A3,B2)
=> ( pp(dvd_dvd(nat,X2,A3))
=> ( pp(dvd_dvd(nat,X2,B2))
=> ( X2 = one_one(nat) ) ) ) ) ).
% coprime_common_divisor_nat
tff(fact_4816_prod__mset_Oneutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A4: multiset(A)] :
( ! [X3: A] :
( pp(member(A,X3,set_mset(A,A4)))
=> ( X3 = one_one(A) ) )
=> ( aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4) = one_one(A) ) ) ) ).
% prod_mset.neutral
tff(fact_4817_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F2: fun(A,B),A15: set(A),A16: set(B),F: fun(C,A),A4: set(C)] :
( bij_betw(A,B,F2,A15,A16)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F),A4)),A15))
=> ( bij_betw(C,A,F,A4,A15)
<=> bij_betw(C,B,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F2),F),A4,A16) ) ) ) ).
% bij_betw_comp_iff2
tff(fact_4818_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A4: set(A),F: fun(A,B),A15: set(B)] :
( ~ pp(member(A,B2,A4))
=> ( ~ pp(member(B,aa(A,B,F,B2),A15))
=> ( bij_betw(A,B,F,A4,A15)
=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A15),aa(set(B),set(B),insert2(B,aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw
tff(fact_4819_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A4: set(A),F: fun(A,B),A15: set(B)] :
( ~ pp(member(A,B2,A4))
=> ( ~ pp(member(B,aa(A,B,F,B2),A15))
=> ( bij_betw(A,B,F,A4,A15)
<=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A15),aa(set(B),set(B),insert2(B,aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).
% notIn_Un_bij_betw3
tff(fact_4820_bijI,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
=> bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).
% bijI
tff(fact_4821_bij__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
<=> ( inj_on(A,B,F,top_top(set(A)))
& ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ) ).
% bij_def
tff(fact_4822_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),C5: set(A),B5: set(B),D4: set(B)] :
( bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D4))
=> ( bij_betw(A,B,F,C5,D4)
=> ( ( aa(set(A),set(A),inf_inf(set(A),A4),C5) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),inf_inf(set(B),B5),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,F,A4,B5) ) ) ) ) ).
% bij_betw_partition
tff(fact_4823_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),B5: set(B),C5: set(A),D4: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> ( bij_betw(A,B,F,C5,D4)
=> ( ( aa(set(B),set(B),inf_inf(set(B),B5),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B5),D4)) ) ) ) ).
% bij_betw_combine
tff(fact_4824_o__bij,axiom,
! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),G),F) = id(A) )
=> ( ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),F),G) = id(B) )
=> bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).
% o_bij
tff(fact_4825_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F),A4)) ) ) ).
% bij_image_Compl_eq
tff(fact_4826_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(A,A)),F: fun(A,B),A15: set(B)] :
( order_well_order_on(A,A4,R)
=> ( bij_betw(A,B,F,A4,A15)
=> ? [R9: set(product_prod(B,B))] :
( order_well_order_on(B,A15,R9)
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R9),bNF_Wellorder_ordIso(A,B))) ) ) ) ).
% Well_order_iso_copy
tff(fact_4827_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A4: set(A),C5: set(B),G: fun(A,B),B5: set(A),D4: set(B)] :
( bij_betw(A,B,F,A4,C5)
=> ( bij_betw(A,B,G,B5,D4)
=> ( ( aa(set(A),set(A),inf_inf(set(A),A4),B5) = bot_bot(set(A)) )
=> ( ( aa(set(B),set(B),inf_inf(set(B),C5),D4) = bot_bot(set(B)) )
=> bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_nn(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C5),D4)) ) ) ) ) ).
% bij_betw_disjoint_Un
tff(fact_4828_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A4: multiset(A)] :
( pp(dvd_dvd(A,aa(multiset(A),A,comm_m9189036328036947845d_mset(A),A4),one_one(A)))
<=> ! [X4: A] :
( pp(member(A,X4,set_mset(A,A4)))
=> pp(dvd_dvd(A,X4,one_one(A))) ) ) ) ).
% is_unit_prod_mset_iff
tff(fact_4829_prod__mset_Oeq__fold,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M2: multiset(A)] : aa(multiset(A),A,comm_m9189036328036947845d_mset(A),M2) = fold_mset(A,A,times_times(A),one_one(A),M2) ) ).
% prod_mset.eq_fold
tff(fact_4830_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B5: set(B),A4: set(A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),vimage(A,B,F),B5)),A4))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),image2(A,B,F),A4))) ) ) ).
% vimage_subset_eq
tff(fact_4831_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M2: set(A)] :
( finite_finite2(A,M2)
=> ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(A,M2)),M2) ) ).
% ex_bij_betw_nat_finite
tff(fact_4832_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M2: set(A)] :
( finite_finite2(A,M2)
=> ? [H2: fun(nat,A)] : bij_betw(nat,A,H2,set_or1337092689740270186AtMost(nat,one_one(nat),finite_card(A,M2)),M2) ) ).
% ex_bij_betw_nat_finite_1
tff(fact_4833_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> algebr8660921524188924756oprime(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).
% coprime_diff_one_right_nat
tff(fact_4834_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),N) ) ).
% coprime_diff_one_left_nat
tff(fact_4835_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S5: set(B),T4: set(C),H: fun(B,C),S2: set(B),T5: set(C),G: fun(C,A)] :
( finite_finite2(B,S5)
=> ( finite_finite2(C,T4)
=> ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))
=> ( ! [A5: B] :
( pp(member(B,A5,S5))
=> ( aa(C,A,G,aa(B,C,H,A5)) = one_one(A) ) )
=> ( ! [B3: C] :
( pp(member(C,B3,T4))
=> ( aa(C,A,G,B3) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_tl(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T5) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
tff(fact_4836_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add(A)
& ord(B) )
=> ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ).
% sum.atLeastAtMost_reindex
tff(fact_4837_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add(A)
& ord(B) )
=> ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ).
% sum.atLeastLessThan_reindex
tff(fact_4838_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult(A)
& ord(B) )
=> ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ).
% prod.atLeastAtMost_reindex
tff(fact_4839_antisymp__antisym__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( antisymp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> antisym(A,R) ) ).
% antisymp_antisym_eq
tff(fact_4840_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(A,B),H: fun(A,C)] :
( ( aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F) = G )
=> ( ( H = aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F) )
<=> ( bNF_convol(A,B,C,G,H) = F ) ) ) ).
% convol_expand_snd'
tff(fact_4841_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(A,B)] :
( ( aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F) = G )
=> ( bNF_convol(A,B,C,G,aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F)) = F ) ) ).
% convol_expand_snd
tff(fact_4842_convol__def,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C),X: A] : aa(A,product_prod(B,C),bNF_convol(A,B,C,F,G),X) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,F,X)),aa(A,C,G,X)) ).
% convol_def
tff(fact_4843_bij__swap,axiom,
! [A: $tType,B: $tType] : bij_betw(product_prod(A,B),product_prod(B,A),product_swap(A,B),top_top(set(product_prod(A,B))),top_top(set(product_prod(B,A)))) ).
% bij_swap
tff(fact_4844_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] : aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),bNF_convol(A,B,C,F,G)) = F ).
% fst_convol
tff(fact_4845_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,C),G: fun(A,B)] : aa(fun(A,product_prod(C,B)),fun(A,B),aa(fun(product_prod(C,B),B),fun(fun(A,product_prod(C,B)),fun(A,B)),comp(product_prod(C,B),B,A),product_snd(C,B)),bNF_convol(A,C,B,F,G)) = G ).
% snd_convol
tff(fact_4846_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: fun(D,B),G: fun(D,C),H: fun(A,D)] : aa(fun(A,D),fun(A,product_prod(B,C)),aa(fun(D,product_prod(B,C)),fun(fun(A,D),fun(A,product_prod(B,C))),comp(D,product_prod(B,C),A),bNF_convol(D,B,C,F,G)),H) = bNF_convol(A,B,C,aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),F),H),aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),G),H)) ).
% convol_o
tff(fact_4847_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M2: set(A)] :
( finite_finite2(A,M2)
=> ? [H2: fun(A,nat)] : bij_betw(A,nat,H2,M2,set_or7035219750837199246ssThan(nat,zero_zero(nat),finite_card(A,M2))) ) ).
% ex_bij_betw_finite_nat
tff(fact_4848_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),X2: C] : aa(C,product_prod(A,B),aa(fun(C,product_prod(C,B)),fun(C,product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(fun(C,product_prod(C,B)),fun(C,product_prod(A,B))),comp(product_prod(C,B),product_prod(A,B),C),product_map_prod(C,A,B,B,F,id(B))),bNF_convol(C,C,B,id(C),G)),X2) = aa(C,product_prod(A,B),bNF_convol(C,A,B,aa(fun(C,A),fun(C,A),aa(fun(A,A),fun(fun(C,A),fun(C,A)),comp(A,A,C),id(A)),F),G),X2) ).
% map_prod_o_convol_id
tff(fact_4849_Heap__lub__empty,axiom,
! [A: $tType] : aa(set(heap_Time_Heap(A)),heap_Time_Heap(A),heap_Time_Heap_lub(A),bot_bot(set(heap_Time_Heap(A)))) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_ao(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% Heap_lub_empty
tff(fact_4850_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> semilattice_neutr(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% and.semilattice_neutr_axioms
tff(fact_4851_map__prod__ident,axiom,
! [B: $tType,A: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_ae(A,A),aTP_Lamp_hm(B,B)),X) = X ).
% map_prod_ident
tff(fact_4852_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,A3)),aa(D,B,G,B2)) ).
% map_prod_simp
tff(fact_4853_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X2: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),X2)) = aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X2)) ).
% fst_map_prod
tff(fact_4854_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(C,B),G: fun(D,A),X2: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F,G),X2)) = aa(D,A,G,aa(product_prod(C,D),D,product_snd(C,D),X2)) ).
% snd_map_prod
tff(fact_4855_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R4: set(product_prod(A,B)),F: fun(A,C),G: fun(B,D)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R4))
=> pp(member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F,A3)),aa(B,D,G,B2)),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F,G)),R4))) ) ).
% map_prod_imageI
tff(fact_4856_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,C),G: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),aa(fun(product_prod(C,D),C),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C)),comp(product_prod(C,D),C,product_prod(A,B)),product_fst(C,D)),product_map_prod(A,C,B,D,F,G)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)) ).
% fst_comp_map_prod
tff(fact_4857_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,D),G: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),aa(fun(product_prod(D,C),C),fun(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C)),comp(product_prod(D,C),C,product_prod(A,B)),product_snd(D,C)),product_map_prod(A,D,B,C,F,G)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),aa(fun(B,C),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),C)),comp(B,C,product_prod(A,B)),G),product_snd(A,B)) ).
% snd_comp_map_prod
tff(fact_4858_prod_Omap__comp,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: fun(C,E),G22: fun(D,F4),F1: fun(A,C),F22: fun(B,D),V: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F4),product_map_prod(C,E,D,F4,G1,G22),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),V)) = aa(product_prod(A,B),product_prod(E,F4),product_map_prod(A,E,B,F4,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),G1),F1),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G22),F22)),V) ).
% prod.map_comp
tff(fact_4859_map__prod_Ocompositionality,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: fun(C,E),G: fun(D,F4),H: fun(A,C),I: fun(B,D),Prod: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F4),product_map_prod(C,E,D,F4,F,G),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,H,I),Prod)) = aa(product_prod(A,B),product_prod(E,F4),product_map_prod(A,E,B,F4,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G),I)),Prod) ).
% map_prod.compositionality
tff(fact_4860_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_ae(A,A),aTP_Lamp_hm(B,B)) = id(product_prod(A,B)) ).
% map_prod.identity
tff(fact_4861_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F: fun(C,A),G: fun(D,B),R4: set(product_prod(C,D))] :
( pp(member(product_prod(A,B),C2,aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G)),R4)))
=> ~ ! [X3: C,Y4: D] :
( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X3)),aa(D,B,G,Y4)) )
=> ~ pp(member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X3),Y4),R4)) ) ) ).
% prod_fun_imageE
tff(fact_4862_apfst__def,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,C)] : product_apfst(A,C,B,F) = product_map_prod(A,C,B,B,F,id(B)) ).
% apfst_def
tff(fact_4863_apsnd__def,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F) = product_map_prod(A,A,B,C,id(A),F) ).
% apsnd_def
tff(fact_4864_map__prod__def,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_tm(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F),G)) ).
% map_prod_def
tff(fact_4865_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F4: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F4,D),G22: fun(B,F4)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),aa(fun(E,C),fun(fun(A,E),fun(A,C)),comp(E,C,A),F1),F22),aa(fun(B,F4),fun(B,D),aa(fun(F4,D),fun(fun(B,F4),fun(B,D)),comp(F4,D,B),G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F4)),fun(product_prod(A,B),product_prod(C,D)),aa(fun(product_prod(E,F4),product_prod(C,D)),fun(fun(product_prod(A,B),product_prod(E,F4)),fun(product_prod(A,B),product_prod(C,D))),comp(product_prod(E,F4),product_prod(C,D),product_prod(A,B)),product_map_prod(E,C,F4,D,F1,G1)),product_map_prod(A,E,B,F4,F22,G22)) ).
% map_prod_compose
tff(fact_4866_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F4: $tType,D: $tType,B: $tType,F: fun(C,E),G: fun(D,F4),H: fun(A,C),I: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F4)),aa(fun(product_prod(C,D),product_prod(E,F4)),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F4))),comp(product_prod(C,D),product_prod(E,F4),product_prod(A,B)),product_map_prod(C,E,D,F4,F,G)),product_map_prod(A,C,B,D,H,I)) = product_map_prod(A,E,B,F4,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G),I)) ).
% map_prod.comp
tff(fact_4867_map__prod__surj__on,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(B,A),A4: set(B),A15: set(A),G: fun(D,C),B5: set(D),B11: set(C)] :
( ( aa(set(B),set(A),image2(B,A,F),A4) = A15 )
=> ( ( aa(set(D),set(C),image2(D,C,G),B5) = B11 )
=> ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F,G)),product_Sigma(B,D,A4,aTP_Lamp_tn(set(D),fun(B,set(D)),B5))) = product_Sigma(A,C,A15,aTP_Lamp_to(set(C),fun(A,set(C)),B11)) ) ) ) ).
% map_prod_surj_on
tff(fact_4868_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: fun(D,B),H22: fun(E,C),F: fun(A,D),G: fun(A,E)] : aa(fun(A,product_prod(D,E)),fun(A,product_prod(B,C)),aa(fun(product_prod(D,E),product_prod(B,C)),fun(fun(A,product_prod(D,E)),fun(A,product_prod(B,C))),comp(product_prod(D,E),product_prod(B,C),A),product_map_prod(D,B,E,C,H1,H22)),bNF_convol(A,D,E,F,G)) = bNF_convol(A,B,C,aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),H1),F),aa(fun(A,E),fun(A,C),aa(fun(E,C),fun(fun(A,E),fun(A,C)),comp(E,C,A),H22),G)) ).
% map_prod_o_convol
tff(fact_4869_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(A,B),A4: set(A),G: fun(C,D),B5: set(C)] :
( inj_on(A,B,F,A4)
=> ( inj_on(C,D,G,B5)
=> inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F,G),product_Sigma(A,C,A4,aTP_Lamp_to(set(C),fun(A,set(C)),B5))) ) ) ).
% map_prod_inj_on
tff(fact_4870_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,B),G: fun(C,D)] :
( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
=> ( ( aa(set(C),set(D),image2(C,D,G),top_top(set(C))) = top_top(set(D)) )
=> ( aa(set(product_prod(A,C)),set(product_prod(B,D)),image2(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F,G)),top_top(set(product_prod(A,C)))) = top_top(set(product_prod(B,D))) ) ) ) ).
% map_prod_surj
tff(fact_4871_Zorns__po__lemma,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_7125193373082350890der_on(A,field2(A,R),R)
=> ( ! [C8: set(A)] :
( pp(member(set(A),C8,chains(A,R)))
=> ? [X: A] :
( pp(member(A,X,field2(A,R)))
& ! [Xa4: A] :
( pp(member(A,Xa4,C8))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X),R)) ) ) )
=> ? [X3: A] :
( pp(member(A,X3,field2(A,R)))
& ! [Xa3: A] :
( pp(member(A,Xa3,field2(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa3),R))
=> ( Xa3 = X3 ) ) ) ) ) ) ).
% Zorns_po_lemma
tff(fact_4872_assert__def,axiom,
! [A: $tType,P: fun(A,bool),X2: A] :
( ( pp(aa(A,bool,P,X2))
=> ( heap_Time_assert(A,P,X2) = aa(A,heap_Time_Heap(A),heap_Time_return(A),X2) ) )
& ( ~ pp(aa(A,bool,P,X2))
=> ( heap_Time_assert(A,P,X2) = heap_Time_raise(A,aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,fFalse)),nil(char)))))))) ) ) ) ).
% assert_def
tff(fact_4873_power__int__add__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X2: A,M: int] :
( ( ( X2 != zero_zero(A) )
| ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X2,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X2,M)),X2) ) ) ) ).
% power_int_add_1
tff(fact_4874_power__int__1__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int] : power_int(A,one_one(A),N) = one_one(A) ) ).
% power_int_1_left
tff(fact_4875_power__int__0__right,axiom,
! [B: $tType] :
( ( inverse(B)
& power(B) )
=> ! [X2: B] : power_int(B,X2,zero_zero(int)) = one_one(B) ) ).
% power_int_0_right
tff(fact_4876_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)) = one_one(A) ) ).
% power_int_minus_one_mult_self
tff(fact_4877_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),M)),B2)) = B2 ) ).
% power_int_minus_one_mult_self'
tff(fact_4878_power__int__add__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X2: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X2,aa(num,int,numeral_numeral(int),M))),power_int(A,X2,aa(num,int,numeral_numeral(int),N))) = power_int(A,X2,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).
% power_int_add_numeral
tff(fact_4879_power__int__add__numeral2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X2: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X2,aa(num,int,numeral_numeral(int),M))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X2,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X2,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).
% power_int_add_numeral2
tff(fact_4880_power__int__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X2: A,N: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),X2),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,X2,N)) ) ).
% power_int_one_over
tff(fact_4881_power__int__0__left__If,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int] :
( ( ( M = zero_zero(int) )
=> ( power_int(A,zero_zero(A),M) = one_one(A) ) )
& ( ( M != zero_zero(int) )
=> ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ) ).
% power_int_0_left_If
tff(fact_4882_power__int__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N5: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N5))) ) ) ) ).
% power_int_increasing
tff(fact_4883_power__int__strict__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N5: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,A3,N5))) ) ) ) ).
% power_int_strict_increasing
tff(fact_4884_power__int__minus__one__minus,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),N)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N) ) ).
% power_int_minus_one_minus
tff(fact_4885_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: int,B2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A3),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A3)) ) ).
% power_int_minus_one_diff_commute
tff(fact_4886_power__int__strict__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N5: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N5)),power_int(A,A3,N))) ) ) ) ) ).
% power_int_strict_decreasing
tff(fact_4887_one__le__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X2,N))) ) ) ) ).
% one_le_power_int
tff(fact_4888_one__less__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A3,N))) ) ) ) ).
% one_less_power_int
tff(fact_4889_power__int__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X2: A,M: int,N: int] :
( ( ( X2 != zero_zero(A) )
| ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
=> ( power_int(A,X2,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X2,M)),power_int(A,X2,N)) ) ) ) ).
% power_int_add
tff(fact_4890_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( division_ring(A)
& one(B)
& uminus(B) )
=> ! [X2: C,A3: A,N: int] :
( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X2)
=> ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N)),power_int(A,A3,N)) ) ) ) ).
% power_int_minus_left_distrib
tff(fact_4891_power__int__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N5: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
=> ( ( ( A3 != zero_zero(A) )
| ( N5 != zero_zero(int) )
| ( N = zero_zero(int) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N5)),power_int(A,A3,N))) ) ) ) ) ) ).
% power_int_decreasing
tff(fact_4892_power__int__le__one,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X2,N)),one_one(A))) ) ) ) ) ).
% power_int_le_one
tff(fact_4893_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,M: int,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X2,M)),power_int(A,X2,N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N)) ) ) ) ) ).
% power_int_le_imp_le_exp
tff(fact_4894_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,M: int,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X2,M)),power_int(A,X2,N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N)) ) ) ) ) ).
% power_int_le_imp_less_exp
tff(fact_4895_power__int__add__1_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X2: A,M: int] :
( ( ( X2 != zero_zero(A) )
| ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X2,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),one_one(int))) = aa(A,A,aa(A,fun(A,A),times_times(A),X2),power_int(A,X2,M)) ) ) ) ).
% power_int_add_1'
tff(fact_4896_integer__of__char__code,axiom,
! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B72)),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B62))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B52))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B42))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B32))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B22))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B1))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B0)) ).
% integer_of_char_code
tff(fact_4897_char__of__def,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: A] : unique5772411509450598832har_of(A,N) = char2(aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),N)),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),one_one(nat)),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).
% char_of_def
tff(fact_4898_numeral__unfold__funpow,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,compow(fun(A,A),aa(num,nat,numeral_numeral(nat),K),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).
% numeral_unfold_funpow
tff(fact_4899_relpowp__1,axiom,
! [A: $tType,P: fun(A,fun(A,bool))] : compow(fun(A,fun(A,bool)),one_one(nat),P) = P ).
% relpowp_1
tff(fact_4900_Suc__funpow,axiom,
! [N: nat] : compow(fun(nat,nat),N,suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).
% Suc_funpow
tff(fact_4901_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F: fun(A,A)] : compow(fun(fun(B,A),fun(B,A)),N,aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),F)) = aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),compow(fun(A,A),N,F)) ).
% comp_funpow
tff(fact_4902_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F: fun(A,A)] : compow(fun(A,A),aa(nat,nat,suc,N),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),F),compow(fun(A,A),N,F)) ).
% funpow.simps(2)
tff(fact_4903_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F: fun(A,A)] : compow(fun(A,A),aa(nat,nat,suc,N),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),N,F)),F) ).
% funpow_Suc_right
tff(fact_4904_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F: fun(A,A)] : compow(fun(A,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),M,F)),compow(fun(A,A),N,F)) ).
% funpow_add
tff(fact_4905_of__nat__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,compow(fun(A,A),N,aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).
% of_nat_def
tff(fact_4906_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [K: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A3) = aa(A,A,compow(fun(A,A),aa(num,nat,numeral_numeral(nat),K),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).
% numeral_add_unfold_funpow
tff(fact_4907_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F: fun(A,A),S: A] :
( ( aa(A,A,compow(fun(A,A),N,F),S) = S )
=> ( ! [M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
=> ( aa(A,A,compow(fun(A,A),M4,F),S) != S ) ) )
=> inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_tp(fun(A,A),fun(A,fun(nat,A)),F),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).
% inj_on_funpow_least
tff(fact_4908_card__Un__disjnt,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( finite_finite2(A,A4)
=> ( finite_finite2(A,B5)
=> ( disjnt(A,A4,B5)
=> ( finite_card(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(A,B5)) ) ) ) ) ).
% card_Un_disjnt
tff(fact_4909_plus__multiset__def,axiom,
! [A: $tType] : plus_plus(multiset(A)) = aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(multiset(A),fun(multiset(A),multiset(A))),map_fun(multiset(A),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),count(A),map_fun(multiset(A),fun(A,nat),fun(A,nat),multiset(A),count(A),abs_multiset(A))),aTP_Lamp_tq(fun(A,nat),fun(fun(A,nat),fun(A,nat)))) ).
% plus_multiset_def
tff(fact_4910_mset__subset__add__iff2,axiom,
! [A: $tType,I: nat,J: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,I,U)),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,J,U)),N))
<=> subset_mset(A,M,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),N)) ) ) ).
% mset_subset_add_iff2
tff(fact_4911_repeat__mset__distrib2,axiom,
! [A: $tType,N: nat,A4: multiset(A),B5: multiset(A)] : repeat_mset(A,N,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,N,A4)),repeat_mset(A,N,B5)) ).
% repeat_mset_distrib2
tff(fact_4912_repeat__mset__Suc,axiom,
! [A: $tType,N: nat,M2: multiset(A)] : repeat_mset(A,aa(nat,nat,suc,N),M2) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),repeat_mset(A,N,M2)) ).
% repeat_mset_Suc
tff(fact_4913_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C5: set(A),A4: set(B),B5: set(B)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,C5,aTP_Lamp_jm(set(B),fun(A,set(B)),A4)),product_Sigma(A,B,C5,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))
<=> ( ( C5 = bot_bot(set(A)) )
| disjnt(B,A4,B5) ) ) ).
% disjnt_Times1_iff
tff(fact_4914_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A4: set(A),C5: set(B),B5: set(A)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),C5)),product_Sigma(A,B,B5,aTP_Lamp_jm(set(B),fun(A,set(B)),C5)))
<=> ( ( C5 = bot_bot(set(B)) )
| disjnt(A,A4,B5) ) ) ).
% disjnt_Times2_iff
tff(fact_4915_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set(A),C5: fun(A,set(B)),B5: set(A)] :
( disjnt(product_prod(A,B),product_Sigma(A,B,A4,C5),product_Sigma(A,B,B5,C5))
<=> ( ! [X4: A] :
( pp(member(A,X4,aa(set(A),set(A),inf_inf(set(A),A4),B5)))
=> ( aa(A,set(B),C5,X4) = bot_bot(set(B)) ) )
| disjnt(A,A4,B5) ) ) ).
% disjnt_Sigma_iff
tff(fact_4916_left__add__mult__distrib__mset,axiom,
! [A: $tType,I: nat,U: multiset(A),J: nat,K: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,I,U)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,J,U)),K)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J),U)),K) ).
% left_add_mult_distrib_mset
tff(fact_4917_repeat__mset__distrib,axiom,
! [A: $tType,M: nat,N: nat,A4: multiset(A)] : repeat_mset(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,M,A4)),repeat_mset(A,N,A4)) ).
% repeat_mset_distrib
tff(fact_4918_mset__subseteq__add__iff1,axiom,
! [A: $tType,J: nat,I: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,I,U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,J,U)),N)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J),U)),M)),N)) ) ) ).
% mset_subseteq_add_iff1
tff(fact_4919_mset__subseteq__add__iff2,axiom,
! [A: $tType,I: nat,J: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,I,U)),M)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,J,U)),N)))
<=> pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),N))) ) ) ).
% mset_subseteq_add_iff2
tff(fact_4920_mset__subset__add__iff1,axiom,
! [A: $tType,J: nat,I: nat,U: multiset(A),M: multiset(A),N: multiset(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,I,U)),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,J,U)),N))
<=> subset_mset(A,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),repeat_mset(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J),U)),M),N) ) ) ).
% mset_subset_add_iff1
tff(fact_4921_plus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: fun(A,nat),X2: fun(A,nat)] :
( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_ts(fun(A,nat),bool)),Xa),Xa))
=> ( pp(aa(fun(A,nat),bool,aa(fun(A,nat),fun(fun(A,nat),bool),bNF_eq_onp(fun(A,nat),aTP_Lamp_ts(fun(A,nat),bool)),X2),X2))
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(fun(A,nat),multiset(A),abs_multiset(A),Xa)),aa(fun(A,nat),multiset(A),abs_multiset(A),X2)) = aa(fun(A,nat),multiset(A),abs_multiset(A),aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_tq(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Xa),X2)) ) ) ) ).
% plus_multiset.abs_eq
tff(fact_4922_repeat__mset__distrib__add__mset,axiom,
! [A: $tType,N: nat,A3: A,A4: multiset(A)] : repeat_mset(A,N,aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A3),A4)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),replicate_mset(A,N,A3)),repeat_mset(A,N,A4)) ).
% repeat_mset_distrib_add_mset
tff(fact_4923_o__prs_I2_J,axiom,
! [F4: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H6: $tType,G2: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),R32: fun(E,fun(E,bool)),Abs32: fun(E,F4),Rep32: fun(F4,E)] :
( quotient3(A,B,R12,Abs1,Rep1)
=> ( quotient3(C,D,R23,Abs22,Rep22)
=> ( quotient3(E,F4,R32,Abs32,Rep32)
=> ( aa(fun(fun(G2,H6),fun(fun(A,G2),fun(A,H6))),fun(fun(G2,H6),fun(fun(B,G2),fun(B,H6))),map_fun(fun(G2,H6),fun(G2,H6),fun(fun(A,G2),fun(A,H6)),fun(fun(B,G2),fun(B,H6)),id(fun(G2,H6)),map_fun(fun(B,G2),fun(A,G2),fun(A,H6),fun(B,H6),map_fun(A,B,G2,G2,Abs1,id(G2)),map_fun(B,A,H6,H6,Rep1,id(H6)))),comp(G2,H6,A)) = comp(G2,H6,B) ) ) ) ) ).
% o_prs(2)
tff(fact_4924_abs__o__rep,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A)] :
( quotient3(A,B,R4,Abs,Rep)
=> ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),Abs),Rep) = id(B) ) ) ).
% abs_o_rep
tff(fact_4925_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F4: $tType,D: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),R32: fun(E,fun(E,bool)),Abs32: fun(E,F4),Rep32: fun(F4,E)] :
( quotient3(A,B,R12,Abs1,Rep1)
=> ( quotient3(C,D,R23,Abs22,Rep22)
=> ( quotient3(E,F4,R32,Abs32,Rep32)
=> ( aa(fun(fun(C,E),fun(fun(A,C),fun(A,E))),fun(fun(D,F4),fun(fun(B,D),fun(B,F4))),map_fun(fun(D,F4),fun(C,E),fun(fun(A,C),fun(A,E)),fun(fun(B,D),fun(B,F4)),map_fun(C,D,F4,E,Abs22,Rep32),map_fun(fun(B,D),fun(A,C),fun(A,E),fun(B,F4),map_fun(A,B,D,C,Abs1,Rep22),map_fun(B,A,E,F4,Rep1,Abs32))),comp(C,E,A)) = comp(D,F4,B) ) ) ) ) ).
% o_prs(1)
tff(fact_4926_plus__multiset_Orsp,axiom,
! [A: $tType] : pp(aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),bool),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(fun(A,nat),fun(A,nat)),fun(fun(A,nat),fun(A,nat)),bNF_eq_onp(fun(A,nat),aTP_Lamp_ts(fun(A,nat),bool)),bNF_rel_fun(fun(A,nat),fun(A,nat),fun(A,nat),fun(A,nat),bNF_eq_onp(fun(A,nat),aTP_Lamp_ts(fun(A,nat),bool)),bNF_eq_onp(fun(A,nat),aTP_Lamp_ts(fun(A,nat),bool)))),aTP_Lamp_tq(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),aTP_Lamp_tq(fun(A,nat),fun(fun(A,nat),fun(A,nat))))) ).
% plus_multiset.rsp
tff(fact_4927_or__not__num__neg_Opelims,axiom,
! [X2: num,Xa: num,Y: num] :
( ( bit_or_not_num_neg(X2,Xa) = Y )
=> ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X2),Xa)))
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [M4: num] :
( ( Xa = aa(num,num,bit0,M4) )
=> ( ( Y = aa(num,num,bit1,M4) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,M4)))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [M4: num] :
( ( Xa = aa(num,num,bit1,M4) )
=> ( ( Y = aa(num,num,bit1,M4) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,M4)))) ) ) )
=> ( ! [N3: num] :
( ( X2 = aa(num,num,bit0,N3) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,num,bit0,one2) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N3)),one2))) ) ) )
=> ( ! [N3: num] :
( ( X2 = aa(num,num,bit0,N3) )
=> ! [M4: num] :
( ( Xa = aa(num,num,bit0,M4) )
=> ( ( Y = bitM(bit_or_not_num_neg(N3,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N3)),aa(num,num,bit0,M4)))) ) ) )
=> ( ! [N3: num] :
( ( X2 = aa(num,num,bit0,N3) )
=> ! [M4: num] :
( ( Xa = aa(num,num,bit1,M4) )
=> ( ( Y = aa(num,num,bit0,bit_or_not_num_neg(N3,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,N3)),aa(num,num,bit1,M4)))) ) ) )
=> ( ! [N3: num] :
( ( X2 = aa(num,num,bit1,N3) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N3)),one2))) ) ) )
=> ( ! [N3: num] :
( ( X2 = aa(num,num,bit1,N3) )
=> ! [M4: num] :
( ( Xa = aa(num,num,bit0,M4) )
=> ( ( Y = bitM(bit_or_not_num_neg(N3,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N3)),aa(num,num,bit0,M4)))) ) ) )
=> ~ ! [N3: num] :
( ( X2 = aa(num,num,bit1,N3) )
=> ! [M4: num] :
( ( Xa = aa(num,num,bit1,M4) )
=> ( ( Y = bitM(bit_or_not_num_neg(N3,M4)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N3)),aa(num,num,bit1,M4)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
tff(fact_4928_cr__natural__def,axiom,
! [X: nat,Xa3: code_natural] :
( pp(aa(code_natural,bool,aa(nat,fun(code_natural,bool),code_cr_natural,X),Xa3))
<=> ( X = aa(code_natural,nat,code_nat_of_natural,Xa3) ) ) ).
% cr_natural_def
tff(fact_4929_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).
% inv_o_cancel
tff(fact_4930_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G),hilbert_inv_into(A,B,top_top(set(A)),F))),F) = G ) ) ).
% o_inv_o_cancel
tff(fact_4931_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),L: list(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(list(B),list(A),map(B,A,hilbert_inv_into(A,B,top_top(set(A)),F)),aa(list(A),list(B),map(A,B,F),L)) = L ) ) ).
% inj_map_inv_f
tff(fact_4932_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A4: set(C),X2: B] :
( inj_on(A,B,F,aa(set(C),set(A),image2(C,A,G),A4))
=> ( inj_on(C,A,G,A4)
=> ( pp(member(B,X2,aa(set(A),set(B),image2(A,B,F),aa(set(C),set(A),image2(C,A,G),A4))))
=> ( aa(B,C,hilbert_inv_into(C,B,A4,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)),X2) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),hilbert_inv_into(C,A,A4,G)),hilbert_inv_into(A,B,aa(set(C),set(A),image2(C,A,G),A4),F)),X2) ) ) ) ) ).
% inv_into_comp
tff(fact_4933_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),G) = id(A) )
=> ( ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),G),F) = id(B) )
=> ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).
% inv_unique_comp
tff(fact_4934_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(C,A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( bij_betw(C,A,G,top_top(set(C)),top_top(set(A)))
=> ( hilbert_inv_into(C,B,top_top(set(C)),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)) = aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),hilbert_inv_into(C,A,top_top(set(C)),G)),hilbert_inv_into(A,B,top_top(set(A)),F)) ) ) ) ).
% o_inv_distrib
tff(fact_4935_surj__iff,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
<=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),hilbert_inv_into(B,A,top_top(set(B)),F)) = id(A) ) ) ).
% surj_iff
tff(fact_4936_inj__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
<=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).
% inj_iff
tff(fact_4937_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F: fun(A,A),N: nat] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> ! [X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),N,F)),compow(fun(A,A),N,hilbert_inv_into(A,A,top_top(set(A)),F))),X) = X ) ).
% fn_o_inv_fn_is_id
tff(fact_4938_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F: fun(A,A),N: nat] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> ! [X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),N,hilbert_inv_into(A,A,top_top(set(A)),F))),compow(fun(A,A),N,F)),X) = X ) ).
% inv_fn_o_fn_is_id
tff(fact_4939_bijection_Oinv__comp__right,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),F),hilbert_inv_into(A,A,top_top(set(A)),F)) = id(A) ) ) ).
% bijection.inv_comp_right
tff(fact_4940_bijection_Oinv__comp__left,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),hilbert_inv_into(A,A,top_top(set(A)),F)),F) = id(A) ) ) ).
% bijection.inv_comp_left
tff(fact_4941_finite__reachable__restrictedI,axiom,
! [A: $tType,Q: set(A),I4: set(A),E4: set(product_prod(A,A))] :
( finite_finite2(A,Q)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),I4),Q))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),range2(A,A,E4)),Q))
=> finite_finite2(A,image(A,A,transitive_rtrancl(A,E4),I4)) ) ) ) ).
% finite_reachable_restrictedI
tff(fact_4942_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R: set(product_prod(B,A))] : range2(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R)) = aa(set(A),set(A),insert2(A,B2),range2(B,A,R)) ).
% Range_insert
tff(fact_4943_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
( pp(member(A,A3,range2(B,A,R)))
<=> ? [Y3: B] : pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y3),A3),R)) ) ).
% Range_iff
tff(fact_4944_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A))] :
( pp(member(A,B2,range2(B,A,R)))
=> ~ ! [A5: B] : ~ pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B2),R)) ) ).
% RangeE
tff(fact_4945_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R))
=> pp(member(B,B2,range2(A,B,R))) ) ).
% Range.intros
tff(fact_4946_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
( pp(member(B,A3,range2(A,B,R)))
<=> ? [A7: A,B7: B] :
( ( A3 = B7 )
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7),R)) ) ) ).
% Range.simps
tff(fact_4947_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
( pp(member(B,A3,range2(A,B,R)))
=> ~ ! [A5: A] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),A3),R)) ) ).
% Range.cases
tff(fact_4948_trancl__Image__in__Range,axiom,
! [A: $tType,R4: set(product_prod(A,A)),V5: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,transitive_trancl(A,R4),V5)),range2(A,A,R4))) ).
% trancl_Image_in_Range
tff(fact_4949_Range__rel__restrict,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),range2(A,A,rel_restrict(A,R4,A4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),range2(A,A,R4)),A4))) ).
% Range_rel_restrict
tff(fact_4950_dom__ran__disj__comp,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( ( aa(set(A),set(A),inf_inf(set(A),domain(A,A,R4)),range2(A,A,R4)) = bot_bot(set(A)) )
=> ( relcomp(A,A,A,R4,R4) = bot_bot(set(product_prod(A,A))) ) ) ).
% dom_ran_disj_comp
tff(fact_4951_wf__max,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( wf(A,converse(A,A,R4))
=> ( ( R4 != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M4: A] : ~ pp(member(A,M4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),range2(A,A,R4)),domain(A,A,R4)))) ) ) ).
% wf_max
tff(fact_4952_Rangep__Range__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X: B] :
( pp(aa(B,bool,rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X))
<=> pp(member(B,X,range2(A,B,R))) ) ).
% Rangep_Range_eq
tff(fact_4953_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] : domain(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R)) = aa(set(A),set(A),insert2(A,A3),domain(A,B,R)) ).
% Domain_insert
tff(fact_4954_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(member(A,A3,domain(A,B,R)))
=> ~ ! [B3: B] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R)) ) ).
% Domain.cases
tff(fact_4955_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(member(A,A3,domain(A,B,R)))
<=> ? [A7: A,B7: B] :
( ( A3 = A7 )
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7),R)) ) ) ).
% Domain.simps
tff(fact_4956_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),R))
=> pp(member(A,A3,domain(A,B,R))) ) ).
% Domain.DomainI
tff(fact_4957_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(member(A,A3,domain(A,B,R)))
=> ~ ! [B3: B] : ~ pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),R)) ) ).
% DomainE
tff(fact_4958_Domain__iff,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(member(A,A3,domain(A,B,R)))
<=> ? [Y3: B] : pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),Y3),R)) ) ).
% Domain_iff
tff(fact_4959_Not__Domain__rtrancl,axiom,
! [A: $tType,X2: A,R4: set(product_prod(A,A)),Y: A] :
( ~ pp(member(A,X2,domain(A,A,R4)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R4)))
<=> ( X2 = Y ) ) ) ).
% Not_Domain_rtrancl
tff(fact_4960_Domain__unfold,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] : domain(A,B,R) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_tt(set(product_prod(A,B)),fun(A,bool),R)) ).
% Domain_unfold
tff(fact_4961_for__in__RI,axiom,
! [B: $tType,A: $tType,X2: A,R4: set(product_prod(A,B))] :
( pp(member(A,X2,domain(A,B,R4)))
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),fun_of_rel(A,B,R4,X2)),R4)) ) ).
% for_in_RI
tff(fact_4962_Domain__rel__restrict,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A4: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),domain(A,A,rel_restrict(A,R4,A4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),domain(A,A,R4)),A4))) ).
% Domain_rel_restrict
tff(fact_4963_wf__no__path,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( ( aa(set(A),set(A),inf_inf(set(A),domain(A,A,R4)),range2(A,A,R4)) = bot_bot(set(A)) )
=> wf(A,R4) ) ).
% wf_no_path
tff(fact_4964_wf__min,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( wf(A,R4)
=> ( ( R4 != bot_bot(set(product_prod(A,A))) )
=> ~ ! [M4: A] : ~ pp(member(A,M4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),domain(A,A,R4)),range2(A,A,R4)))) ) ) ).
% wf_min
tff(fact_4965_Range__def,axiom,
! [B: $tType,A: $tType,X: set(product_prod(A,B))] : range2(A,B,X) = aa(fun(B,bool),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),X))) ).
% Range_def
tff(fact_4966_and_Omonoid__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% and.monoid_axioms
tff(fact_4967_cr__integer__def,axiom,
! [X: int,Xa3: code_integer] :
( pp(aa(code_integer,bool,aa(int,fun(code_integer,bool),code_cr_integer,X),Xa3))
<=> ( X = aa(code_integer,int,code_int_of_integer,Xa3) ) ) ).
% cr_integer_def
tff(fact_4968_wo__rel_OisMinim__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A4: set(A),B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( bNF_We4791949203932849705sMinim(A,R,A4,B2)
<=> ( pp(member(A,B2,A4))
& ! [X4: A] :
( pp(member(A,X4,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X4),R)) ) ) ) ) ).
% wo_rel.isMinim_def
tff(fact_4969_monoid_Oleft__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,Z2),A3) = A3 ) ) ).
% monoid.left_neutral
tff(fact_4970_monoid_Oright__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).
% monoid.right_neutral
tff(fact_4971_add_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.monoid_axioms
tff(fact_4972_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> monoid(A,times_times(A),one_one(A)) ) ).
% mult.monoid_axioms
tff(fact_4973_integer_Opcr__cr__eq,axiom,
code_pcr_integer = code_cr_integer ).
% integer.pcr_cr_eq
tff(fact_4974_multp__implies__one__step,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),M2: multiset(A),N5: multiset(A)] :
( transp(A,R)
=> ( multp(A,R,M2,N5)
=> ? [I7: multiset(A),J5: multiset(A)] :
( ( N5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),J5) )
& ? [K9: multiset(A)] :
( ( M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),I7),K9) )
& ( J5 != zero_zero(multiset(A)) )
& ! [X: A] :
( pp(member(A,X,set_mset(A,K9)))
=> ? [Xa4: A] :
( pp(member(A,Xa4,set_mset(A,J5)))
& pp(aa(A,bool,aa(A,fun(A,bool),R,X),Xa4)) ) ) ) ) ) ) ).
% multp_implies_one_step
tff(fact_4975_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),compow(set(product_prod(A,A)),N,R4)))
<=> ? [F6: fun(nat,A)] :
( ( aa(nat,A,F6,zero_zero(nat)) = A3 )
& ( aa(nat,A,F6,N) = B2 )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F6,I3)),aa(nat,A,F6,aa(nat,nat,suc,I3))),R4)) ) ) ) ).
% relpow_fun_conv
tff(fact_4976_irreflp__irrefl__eq,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( irreflp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R4))
<=> irrefl(A,R4) ) ).
% irreflp_irrefl_eq
tff(fact_4977_relpow__1,axiom,
! [A: $tType,R4: set(product_prod(A,A))] : compow(set(product_prod(A,A)),one_one(nat),R4) = R4 ).
% relpow_1
tff(fact_4978_multp__cancel,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X7: multiset(A),Z7: multiset(A),Y7: multiset(A)] :
( transp(A,R)
=> ( irreflp(A,R)
=> ( multp(A,R,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X7),Z7),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y7),Z7))
<=> multp(A,R,X7,Y7) ) ) ) ).
% multp_cancel
tff(fact_4979_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R4: set(product_prod(A,A)),X: A,Y5: A,Z5: A] :
( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y5),compow(set(product_prod(A,A)),N,R4)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z5),R4)) )
=> ? [W3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),W3),R4))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W3),Z5),compow(set(product_prod(A,A)),N,R4))) ) ) ).
% relpow_Suc_D2'
tff(fact_4980_relpow__0__E,axiom,
! [A: $tType,X2: A,Y: A,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),compow(set(product_prod(A,A)),zero_zero(nat),R4)))
=> ( X2 = Y ) ) ).
% relpow_0_E
tff(fact_4981_relpow__0__I,axiom,
! [A: $tType,X2: A,R4: set(product_prod(A,A))] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),compow(set(product_prod(A,A)),zero_zero(nat),R4))) ).
% relpow_0_I
tff(fact_4982_relpow__Suc__E,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R4)))
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),compow(set(product_prod(A,A)),N,R4)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),R4)) ) ) ).
% relpow_Suc_E
tff(fact_4983_relpow__Suc__I,axiom,
! [A: $tType,X2: A,Y: A,N: nat,R4: set(product_prod(A,A)),Z2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),compow(set(product_prod(A,A)),N,R4)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R4))) ) ) ).
% relpow_Suc_I
tff(fact_4984_relpow__Suc__D2,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R4)))
=> ? [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),R4))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),compow(set(product_prod(A,A)),N,R4))) ) ) ).
% relpow_Suc_D2
tff(fact_4985_relpow__Suc__E2,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R4)))
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),compow(set(product_prod(A,A)),N,R4))) ) ) ).
% relpow_Suc_E2
tff(fact_4986_relpow__Suc__I2,axiom,
! [A: $tType,X2: A,Y: A,R4: set(product_prod(A,A)),Z2: A,N: nat] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R4))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),compow(set(product_prod(A,A)),N,R4)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),aa(nat,nat,suc,N),R4))) ) ) ).
% relpow_Suc_I2
tff(fact_4987_relpow__add,axiom,
! [A: $tType,M: nat,N: nat,R4: set(product_prod(A,A))] : compow(set(product_prod(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),R4) = relcomp(A,A,A,compow(set(product_prod(A,A)),M,R4),compow(set(product_prod(A,A)),N,R4)) ).
% relpow_add
tff(fact_4988_transp__trans__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( transp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> trans(A,R) ) ).
% transp_trans_eq
tff(fact_4989_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R4: set(product_prod(A,A)),X: A,Xa3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),compow(fun(A,fun(A,bool)),N,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R4)),X),Xa3))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa3),compow(set(product_prod(A,A)),N,R4))) ) ).
% relpowp_relpow_eq
tff(fact_4990_relpow__E2,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),N,R4)))
=> ( ( ( N = zero_zero(nat) )
=> ( X2 != Z2 ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N = aa(nat,nat,suc,M4) )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),compow(set(product_prod(A,A)),M4,R4))) ) ) ) ) ).
% relpow_E2
tff(fact_4991_relpow__E,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R4: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),compow(set(product_prod(A,A)),N,R4)))
=> ( ( ( N = zero_zero(nat) )
=> ( X2 != Z2 ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N = aa(nat,nat,suc,M4) )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y4),compow(set(product_prod(A,A)),M4,R4)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z2),R4)) ) ) ) ) ).
% relpow_E
tff(fact_4992_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list(A),I4: set(nat)] : nths(A,drop(A,N,Xs),I4) = nths(A,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I4)) ).
% nths_drop
tff(fact_4993_nths__append,axiom,
! [A: $tType,L: list(A),L4: list(A),A4: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L4),A4) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A4)),nths(A,L4,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_tu(list(A),fun(set(nat),fun(nat,bool)),L),A4)))) ).
% nths_append
tff(fact_4994_asymp__asym__eq,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( asymp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R4))
<=> asym(A,R4) ) ).
% asymp_asym_eq
tff(fact_4995_asym_Ocases,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( asym(A,A3)
=> ! [A14: A,B6: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A14),B6),A3))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B6),A14),A3)) ) ) ).
% asym.cases
tff(fact_4996_asym_Osimps,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( asym(A,A3)
<=> ? [R10: set(product_prod(A,A))] :
( ( A3 = R10 )
& ! [X4: A,Xa2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R10))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R10)) ) ) ) ).
% asym.simps
tff(fact_4997_asym_Ointros,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( ! [A5: A,B3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B3),R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A5),R4)) )
=> asym(A,R4) ) ).
% asym.intros
tff(fact_4998_asymD,axiom,
! [A: $tType,R4: set(product_prod(A,A)),X2: A,Y: A] :
( asym(A,R4)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2),R4)) ) ) ).
% asymD
tff(fact_4999_asym__iff,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( asym(A,R4)
<=> ! [X4: A,Y3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),R4))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X4),R4)) ) ) ).
% asym_iff
tff(fact_5000_lexord__asymmetric,axiom,
! [A: $tType,R4: set(product_prod(A,A)),A3: list(A),B2: list(A)] :
( asym(A,R4)
=> ( pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A3),B2),lexord(A,R4)))
=> ~ pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),B2),A3),lexord(A,R4))) ) ) ).
% lexord_asymmetric
tff(fact_5001_ordLess__iff,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
<=> ( order_well_order_on(A,field2(A,R),R)
& order_well_order_on(B,field2(B,R3),R3)
& ~ ? [X_1: fun(B,A)] : bNF_Wellorder_embed(B,A,R3,R,X_1) ) ) ).
% ordLess_iff
tff(fact_5002_option_Oin__rel,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),A3),B2))
<=> ? [Z4: option(product_prod(A,B))] :
( pp(member(option(product_prod(A,B)),Z4,aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_tv(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R4))))
& ( aa(option(product_prod(A,B)),option(A),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)),Z4) = A3 )
& ( aa(option(product_prod(A,B)),option(B),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)),Z4) = B2 ) ) ) ).
% option.in_rel
tff(fact_5003_bit__double__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A3)),N))
<=> ( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
& ( N != zero_zero(nat) )
& bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ).
% bit_double_iff
tff(fact_5004_rel__option__None2,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X2: option(A)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),X2),none(B)))
<=> ( X2 = none(A) ) ) ).
% rel_option_None2
tff(fact_5005_rel__option__None1,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X2: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),none(A)),X2))
<=> ( X2 = none(B) ) ) ).
% rel_option_None1
tff(fact_5006_bit__minus__1__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),one_one(A))),N))
<=> bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).
% bit_minus_1_iff
tff(fact_5007_rel__option__iff,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: option(A),Y: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X2),Y))
<=> pp(aa(product_prod(option(A),option(B)),bool,aa(fun(option(A),fun(option(B),bool)),fun(product_prod(option(A),option(B)),bool),product_case_prod(option(A),option(B),bool),aTP_Lamp_ty(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),R4)),aa(option(B),product_prod(option(A),option(B)),aa(option(A),fun(option(B),product_prod(option(A),option(B))),product_Pair(option(A),option(B)),X2),Y))) ) ).
% rel_option_iff
tff(fact_5008_option_Orel__mono,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R4),Ra2))
=> pp(aa(fun(option(A),fun(option(B),bool)),bool,aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),bool),ord_less_eq(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2))) ) ).
% option.rel_mono
tff(fact_5009_option_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X22: A,Y2: B] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(A,option(A),some(A),X22)),aa(B,option(B),some(B),Y2)))
<=> pp(aa(B,bool,aa(A,fun(B,bool),R4,X22),Y2)) ) ).
% option.rel_inject(2)
tff(fact_5010_option_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X22: A,Y2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,X22),Y2))
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(A,option(A),some(A),X22)),aa(B,option(B),some(B),Y2))) ) ).
% option.rel_intros(2)
tff(fact_5011_option__rel__Some1,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),X2: A,Y: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),aa(A,option(A),some(A),X2)),Y))
<=> ? [Y10: B] :
( ( Y = aa(B,option(B),some(B),Y10) )
& pp(aa(B,bool,aa(A,fun(B,bool),A4,X2),Y10)) ) ) ).
% option_rel_Some1
tff(fact_5012_option__rel__Some2,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool)),X2: option(A),Y: B] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),X2),aa(B,option(B),some(B),Y)))
<=> ? [X14: A] :
( ( X2 = aa(A,option(A),some(A),X14) )
& pp(aa(B,bool,aa(A,fun(B,bool),A4,X14),Y)) ) ) ).
% option_rel_Some2
tff(fact_5013_option_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),none(A)),none(B))) ).
% option.ctr_transfer(1)
tff(fact_5014_rel__option__inf,axiom,
! [B: $tType,A: $tType,A4: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool))] : aa(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),inf_inf(fun(option(A),fun(option(B),bool)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),B5)) = aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),inf_inf(fun(A,fun(B,bool)),A4),B5)) ).
% rel_option_inf
tff(fact_5015_option_Orel__refl,axiom,
! [B: $tType,Ra2: fun(B,fun(B,bool)),X2: option(B)] :
( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra2,X3),X3))
=> pp(aa(option(B),bool,aa(option(B),fun(option(B),bool),aa(fun(B,fun(B,bool)),fun(option(B),fun(option(B),bool)),rel_option(B,B),Ra2),X2),X2)) ) ).
% option.rel_refl
tff(fact_5016_option_Orel__eq,axiom,
! [A: $tType] : aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),fequal(A)) = fequal(option(A)) ).
% option.rel_eq
tff(fact_5017_option_Orel__cong,axiom,
! [A: $tType,B: $tType,X2: option(A),Ya: option(A),Y: option(B),Xa: option(B),R4: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),Ya)))
=> ( pp(member(B,Yb,aa(option(B),set(B),set_option(B),Xa)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z3),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z3),Yb)) ) ) )
=> ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X2),Y))
<=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2),Ya),Xa)) ) ) ) ) ).
% option.rel_cong
tff(fact_5018_option_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: option(A),Y: option(B),Ra2: fun(A,fun(B,bool))] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X2),Y))
=> ( ! [Z3: A,Yb: B] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),X2)))
=> ( pp(member(B,Yb,aa(option(B),set(B),set_option(B),Y)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z3),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z3),Yb)) ) ) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2),X2),Y)) ) ) ).
% option.rel_mono_strong
tff(fact_5019_option_Orel__refl__strong,axiom,
! [A: $tType,X2: option(A),Ra2: fun(A,fun(A,bool))] :
( ! [Z3: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra2,Z3),Z3)) )
=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),Ra2),X2),X2)) ) ).
% option.rel_refl_strong
tff(fact_5020_rel__option__reflI,axiom,
! [A: $tType,Y: option(A),P: fun(A,fun(A,bool))] :
( ! [X3: A] :
( pp(member(A,X3,aa(option(A),set(A),set_option(A),Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,X3),X3)) )
=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),P),Y),Y)) ) ).
% rel_option_reflI
tff(fact_5021_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X2: option(A),Y: option(B)] :
( pp(aa(option(B),bool,aa(option(C),fun(option(B),bool),aa(fun(C,fun(B,bool)),fun(option(C),fun(option(B),bool)),rel_option(C,B),Sb),aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),I),X2)),Y))
<=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_js(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X2),Y)) ) ).
% option.rel_map(1)
tff(fact_5022_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X2: option(A),G: fun(B,C),Y: option(B)] :
( pp(aa(option(C),bool,aa(option(A),fun(option(C),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),X2),aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G),Y)))
<=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_jt(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X2),Y)) ) ).
% option.rel_map(2)
tff(fact_5023_option_Orel__flip,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(B),B2: option(A)] :
( pp(aa(option(A),bool,aa(option(B),fun(option(A),bool),aa(fun(B,fun(A,bool)),fun(option(B),fun(option(A),bool)),rel_option(B,A),conversep(A,B,R4)),A3),B2))
<=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),B2),A3)) ) ).
% option.rel_flip
tff(fact_5024_option_Orel__conversep,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(option(B),fun(option(A),bool)),rel_option(B,A),conversep(A,B,R4)) = conversep(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) ).
% option.rel_conversep
tff(fact_5025_option_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(B,option(B)),bool,aa(fun(A,option(A)),fun(fun(B,option(B)),bool),bNF_rel_fun(A,B,option(A),option(B),R4,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)),some(A)),some(B))) ).
% option.ctr_transfer(2)
tff(fact_5026_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),fequal(bool)),aTP_Lamp_tz(option(A),bool)),aTP_Lamp_ua(option(B),bool))) ).
% option.disc_transfer(2)
tff(fact_5027_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),fequal(bool)),aTP_Lamp_ub(option(A),bool)),aTP_Lamp_uc(option(B),bool))) ).
% option.disc_transfer(1)
tff(fact_5028_option__bind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] : pp(aa(fun(option(B),fun(fun(B,option(D)),option(D))),bool,aa(fun(option(A),fun(fun(A,option(C)),option(C))),fun(fun(option(B),fun(fun(B,option(D)),option(D))),bool),bNF_rel_fun(option(A),option(B),fun(fun(A,option(C)),option(C)),fun(fun(B,option(D)),option(D)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),bNF_rel_fun(fun(A,option(C)),fun(B,option(D)),option(C),option(D),bNF_rel_fun(A,B,option(C),option(D),A4,aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B5)),aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B5))),bind(A,C)),bind(B,D))) ).
% option_bind_transfer
tff(fact_5029_option_Orel__transp,axiom,
! [A: $tType,R4: fun(A,fun(A,bool))] :
( transp(A,R4)
=> transp(option(A),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R4)) ) ).
% option.rel_transp
tff(fact_5030_option_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),Y2: A] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(A,option(A),some(A),Y2)),none(B))) ).
% option.rel_distinct(2)
tff(fact_5031_option_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),Y2: B] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),none(A)),aa(B,option(B),some(B),Y2))) ).
% option.rel_distinct(1)
tff(fact_5032_option_Orel__cases,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),A3),B2))
=> ( ( ( A3 = none(A) )
=> ( B2 != none(B) ) )
=> ~ ! [X3: A] :
( ( A3 = aa(A,option(A),some(A),X3) )
=> ! [Y4: B] :
( ( B2 = aa(B,option(B),some(B),Y4) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),R4,X3),Y4)) ) ) ) ) ).
% option.rel_cases
tff(fact_5033_option_Orel__induct,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: option(A),Y: option(B),Q: fun(option(A),fun(option(B),bool))] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X2),Y))
=> ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,none(A)),none(B)))
=> ( ! [A23: A,B23: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),R4,A23),B23))
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,aa(A,option(A),some(A),A23)),aa(B,option(B),some(B),B23))) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,X2),Y)) ) ) ) ).
% option.rel_induct
tff(fact_5034_option_Omap__transfer,axiom,
! [A: $tType,B: $tType,F4: $tType,E: $tType,Rb2: fun(A,fun(E,bool)),Sd: fun(B,fun(F4,bool))] : pp(aa(fun(fun(E,F4),fun(option(E),option(F4))),bool,aa(fun(fun(A,B),fun(option(A),option(B))),fun(fun(fun(E,F4),fun(option(E),option(F4))),bool),bNF_rel_fun(fun(A,B),fun(E,F4),fun(option(A),option(B)),fun(option(E),option(F4)),bNF_rel_fun(A,E,B,F4,Rb2,Sd),bNF_rel_fun(option(A),option(E),option(B),option(F4),aa(fun(A,fun(E,bool)),fun(option(A),fun(option(E),bool)),rel_option(A,E),Rb2),aa(fun(B,fun(F4,bool)),fun(option(B),fun(option(F4),bool)),rel_option(B,F4),Sd))),map_option(A,B)),map_option(E,F4))) ).
% option.map_transfer
tff(fact_5035_option_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: fun(C,fun(D,bool)),R4: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S2,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R4,S2),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),S2))),case_option(C,A)),case_option(D,B))) ).
% option.case_transfer
tff(fact_5036_option_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(option(A),fun(option(B),bool)),fun(option(C),fun(option(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(option(A),option(C),fun(option(B),bool),fun(option(D),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),bNF_rel_fun(option(B),option(D),bool,bool,aa(fun(B,fun(D,bool)),fun(option(B),fun(option(D),bool)),rel_option(B,D),Sc),fequal(bool)))),rel_option(A,B)),rel_option(C,D))) ).
% option.rel_transfer
tff(fact_5037_bit__minus__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),A3)),N))
<=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
& ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N)) ) ) ) ).
% bit_minus_iff
tff(fact_5038_option_Orel__sel,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),A3),B2))
<=> ( ( ( A3 = none(A) )
<=> ( B2 = none(B) ) )
& ( ( A3 != none(A) )
=> ( ( B2 != none(B) )
=> pp(aa(B,bool,aa(A,fun(B,bool),R4,aa(option(A),A,the2(A),A3)),aa(option(B),B,the2(B),B2))) ) ) ) ) ).
% option.rel_sel
tff(fact_5039_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> ~ ? [X_12: fun(B,A)] : bNF_Wellorder_embed(B,A,R3,R,X_12) ) ).
% ordLess_not_embed
tff(fact_5040_option_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: fun(C,fun(D,bool)),R4: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S2,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R4,S2),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),S2))),rec_option(C,A)),rec_option(D,B))) ).
% option.rec_transfer
tff(fact_5041_bit__2__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))
<=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
& ( N = one_one(nat) ) ) ) ) ).
% bit_2_iff
tff(fact_5042_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,B)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_We4044943003108391690rdLess(A,B)))
=> ( bNF_Wellorder_embed(A,B,R1,R22,F)
=> ( aa(set(A),set(B),image2(A,B,F),field2(A,R1)) != field2(B,R22) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
tff(fact_5043_bit__mask__sub__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A))),N))
<=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).
% bit_mask_sub_iff
tff(fact_5044_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C)),F13: fun(A,C),F23: fun(B,C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_We4044943003108391690rdLess(A,B)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R22),R33),bNF_We4044943003108391690rdLess(B,C)))
=> ( bNF_Wellorder_embed(A,C,R1,R33,F13)
=> ( bNF_Wellorder_embed(B,C,R22,R33,F23)
=> pp(member(product_prod(set(C),set(C)),aa(set(C),product_prod(set(C),set(C)),aa(set(C),fun(set(C),product_prod(set(C),set(C))),product_Pair(set(C),set(C)),aa(set(A),set(C),image2(A,C,F13),field2(A,R1))),aa(set(B),set(C),image2(B,C,F23),field2(B,R22))),bNF_We413866401316099525erIncl(C,R33))) ) ) ) ) ).
% embed_ordLess_ofilterIncl
tff(fact_5045_comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( bNF_Wellorder_embed(A,B,R,R3,F)
=> ( bNF_Wellorder_embed(B,C,R3,R8,F2)
=> bNF_Wellorder_embed(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ).
% comp_embed
tff(fact_5046_override__on__insert,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B),X2: A,X7: set(A)] : override_on(A,B,F,G,aa(set(A),set(A),insert2(A,X2),X7)) = fun_upd(A,B,override_on(A,B,F,G,X7),X2,aa(A,B,G,X2)) ).
% override_on_insert
tff(fact_5047_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A3: A,A4: set(A),F: fun(A,B),G: fun(A,B)] :
( pp(member(A,A3,A4))
=> ( aa(A,B,override_on(A,B,F,G,A4),A3) = aa(A,B,G,A3) ) ) ).
% override_on_apply_in
tff(fact_5048_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A3: A,A4: set(A),F: fun(A,B),G: fun(A,B)] :
( ~ pp(member(A,A3,A4))
=> ( aa(A,B,override_on(A,B,F,G,A4),A3) = aa(A,B,F,A3) ) ) ).
% override_on_apply_notin
tff(fact_5049_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] : override_on(A,B,F,G,bot_bot(set(A))) = F ).
% override_on_emptyset
tff(fact_5050_override__on__def,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B),G: fun(A,B),X: A] :
( ( pp(member(A,X,A4))
=> ( aa(A,B,override_on(A,B,F,G,A4),X) = aa(A,B,G,X) ) )
& ( ~ pp(member(A,X,A4))
=> ( aa(A,B,override_on(A,B,F,G,A4),X) = aa(A,B,F,X) ) ) ) ).
% override_on_def
tff(fact_5051_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B),X2: A,X7: set(A)] : override_on(A,B,F,G,aa(set(A),set(A),insert2(A,X2),X7)) = override_on(A,B,fun_upd(A,B,F,X2,aa(A,B,G,X2)),G,X7) ).
% override_on_insert'
tff(fact_5052_embedS__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( bNF_Wellorder_embedS(A,B,R,R3,F)
=> ( bNF_Wellorder_embed(B,C,R3,R8,F2)
=> bNF_Wellorder_embedS(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ) ) ).
% embedS_comp_embed
tff(fact_5053_embed__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( bNF_Wellorder_embed(A,B,R,R3,F)
=> ( bNF_Wellorder_embedS(B,C,R3,R8,F2)
=> bNF_Wellorder_embedS(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ) ) ).
% embed_comp_embedS
tff(fact_5054_pred__option__parametric,axiom,
! [A: $tType,B: $tType,A4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,A4,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A4),fequal(bool))),pred_option(A)),pred_option(B))) ).
% pred_option_parametric
tff(fact_5055_option_Opred__inject_I2_J,axiom,
! [A: $tType,P: fun(A,bool),A3: A] :
( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),aa(A,option(A),some(A),A3)))
<=> pp(aa(A,bool,P,A3)) ) ).
% option.pred_inject(2)
tff(fact_5056_option_Opred__mono,axiom,
! [A: $tType,P: fun(A,bool),Pa: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Pa))
=> pp(aa(fun(option(A),bool),bool,aa(fun(option(A),bool),fun(fun(option(A),bool),bool),ord_less_eq(fun(option(A),bool)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),P)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa))) ) ).
% option.pred_mono
tff(fact_5057_option_Opred__inject_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),none(A))) ).
% option.pred_inject(1)
tff(fact_5058_option_Opred__True,axiom,
! [A: $tType,X: option(A)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aTP_Lamp_af(A,bool)),X)) ).
% option.pred_True
tff(fact_5059_option_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,bool),X2: option(A),Pa: fun(A,bool)] :
( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X2))
=> ( ! [Z3: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),X2)))
=> ( pp(aa(A,bool,P,Z3))
=> pp(aa(A,bool,Pa,Z3)) ) )
=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),X2)) ) ) ).
% option.pred_mono_strong
tff(fact_5060_option_Opred__cong,axiom,
! [A: $tType,X2: option(A),Ya: option(A),P: fun(A,bool),Pa: fun(A,bool)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(member(A,Z3,aa(option(A),set(A),set_option(A),Ya)))
=> ( pp(aa(A,bool,P,Z3))
<=> pp(aa(A,bool,Pa,Z3)) ) )
=> ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X2))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),Ya)) ) ) ) ).
% option.pred_cong
tff(fact_5061_option_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X2: option(A),Ya: option(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ud(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
=> ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Ya) ) ) ) ).
% option.map_cong_pred
tff(fact_5062_option_Opred__set,axiom,
! [A: $tType,P: fun(A,bool),X: option(A)] :
( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X))
<=> ! [Xa2: A] :
( pp(member(A,Xa2,aa(option(A),set(A),set_option(A),X)))
=> pp(aa(A,bool,P,Xa2)) ) ) ).
% option.pred_set
tff(fact_5063_option_Opred__rel,axiom,
! [A: $tType,P: fun(A,bool),X2: option(A)] :
( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X2))
<=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),bNF_eq_onp(A,P)),X2),X2)) ) ).
% option.pred_rel
tff(fact_5064_option_Orel__eq__onp,axiom,
! [A: $tType,P: fun(A,bool)] : aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),bNF_eq_onp(A,P)) = bNF_eq_onp(option(A),aa(fun(A,bool),fun(option(A),bool),pred_option(A),P)) ).
% option.rel_eq_onp
tff(fact_5065_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X2: option(A)] :
( pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),pred_option(B),Q),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2)))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X2)) ) ).
% option.pred_map
tff(fact_5066_option_Opred__transfer,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,R4,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),fequal(bool))),pred_option(A)),pred_option(B))) ).
% option.pred_transfer
tff(fact_5067_iso__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_well_order_on(B,field2(B,R3),R3)
=> ( bNF_Wellorder_iso(A,B,R,R3,F)
=> ( bNF_Wellorder_embedS(B,C,R3,R8,F2)
=> bNF_Wellorder_embedS(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ) ) ).
% iso_comp_embedS
tff(fact_5068_cut__def,axiom,
! [A: $tType,B: $tType,X2: A,R4: set(product_prod(A,A)),F: fun(A,B),X: A] :
( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X2),R4))
=> ( aa(A,B,cut(A,B,F,R4,X2),X) = aa(A,B,F,X) ) )
& ( ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X2),R4))
=> ( aa(A,B,cut(A,B,F,R4,X2),X) = undefined(B) ) ) ) ).
% cut_def
tff(fact_5069_above__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_above(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% above_def
tff(fact_5070_comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( bNF_Wellorder_iso(A,B,R,R3,F)
=> ( bNF_Wellorder_iso(B,C,R3,R8,F2)
=> bNF_Wellorder_iso(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ).
% comp_iso
tff(fact_5071_iso__forward,axiom,
! [A: $tType,B: $tType,X2: A,Y: A,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( bNF_Wellorder_iso(A,B,R,R3,F)
=> pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X2)),aa(A,B,F,Y)),R3)) ) ) ).
% iso_forward
tff(fact_5072_cut__apply,axiom,
! [B: $tType,A: $tType,X2: A,A3: A,R4: set(product_prod(A,A)),F: fun(A,B)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),A3),R4))
=> ( aa(A,B,cut(A,B,F,R4,A3),X2) = aa(A,B,F,X2) ) ) ).
% cut_apply
tff(fact_5073_cuts__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R4: set(product_prod(A,A)),X2: A,G: fun(A,B)] :
( ( cut(A,B,F,R4,X2) = cut(A,B,G,R4,X2) )
<=> ! [Y3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X2),R4))
=> ( aa(A,B,F,Y3) = aa(A,B,G,Y3) ) ) ) ).
% cuts_eq
tff(fact_5074_iso__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( bNF_Wellorder_iso(A,B,R,R3,F)
=> ( bNF_Wellorder_embed(B,C,R3,R8,F2)
=> bNF_Wellorder_embed(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ).
% iso_comp_embed
tff(fact_5075_embed__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( bNF_Wellorder_embed(A,B,R,R3,F)
=> ( bNF_Wellorder_iso(B,C,R3,R8,F2)
=> bNF_Wellorder_embed(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ).
% embed_comp_iso
tff(fact_5076_embedS__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R8: set(product_prod(C,C)),F2: fun(B,C)] :
( bNF_Wellorder_embedS(A,B,R,R3,F)
=> ( bNF_Wellorder_iso(B,C,R3,R8,F2)
=> bNF_Wellorder_embedS(A,C,R,R8,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F2),F)) ) ) ).
% embedS_comp_iso
tff(fact_5077_iso__iff2,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B)] :
( bNF_Wellorder_iso(A,B,R,R3,F)
<=> ( bij_betw(A,B,F,field2(A,R),field2(B,R3))
& ! [X4: A] :
( pp(member(A,X4,field2(A,R)))
=> ! [Xa2: A] :
( pp(member(A,Xa2,field2(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R))
<=> pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X4)),aa(A,B,F,Xa2)),R3)) ) ) ) ) ) ).
% iso_iff2
tff(fact_5078_iso__backward,axiom,
! [A: $tType,B: $tType,X2: A,Y: A,R3: set(product_prod(A,A)),R: set(product_prod(B,B)),F: fun(B,A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R3))
=> ( bNF_Wellorder_iso(B,A,R,R3,F)
=> pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,hilbert_inv_into(B,A,field2(B,R),F),X2)),aa(A,B,hilbert_inv_into(B,A,field2(B,R),F),Y)),R)) ) ) ).
% iso_backward
tff(fact_5079_lists__of__len__fin1,axiom,
! [A: $tType,P: set(A),N: nat] :
( finite_finite2(A,P)
=> finite_finite2(list(A),aa(set(list(A)),set(list(A)),inf_inf(set(list(A)),lists(A,P)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_ue(nat,fun(list(A),bool),N)))) ) ).
% lists_of_len_fin1
tff(fact_5080_lists__of__len__fin2,axiom,
! [A: $tType,P: set(A),N: nat] :
( finite_finite2(A,P)
=> finite_finite2(list(A),aa(set(list(A)),set(list(A)),inf_inf(set(list(A)),lists(A,P)),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_uf(nat,fun(list(A),bool),N)))) ) ).
% lists_of_len_fin2
tff(fact_5081_Random__Pred_Ounion__def,axiom,
! [A: $tType,R12: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),R23: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),X: product_prod(code_natural,code_natural)] : random_union(A,R12,R23,X) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_uh(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),R23)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R12,X)) ).
% Random_Pred.union_def
tff(fact_5082_lists__image__witness,axiom,
! [A: $tType,B: $tType,X2: list(A),F: fun(B,A),Q: set(B)] :
( pp(member(list(A),X2,lists(A,aa(set(B),set(A),image2(B,A,F),Q))))
=> ~ ! [Xo2: list(B)] :
( pp(member(list(B),Xo2,lists(B,Q)))
=> ( X2 != aa(list(B),list(A),map(B,A,F),Xo2) ) ) ) ).
% lists_image_witness
tff(fact_5083_Random__Pred_Oempty__def,axiom,
! [A: $tType] : random_empty(A) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),bot_bot(pred(A))) ).
% Random_Pred.empty_def
tff(fact_5084_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType,R4: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),F: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),X: product_prod(code_natural,code_natural)] : random_bind(A,B,R4,F,X) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aTP_Lamp_uk(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),F)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R4,X)) ).
% Random_Pred.bind_def
tff(fact_5085_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ( ( finite_finite2(A,A4)
& finite_finite2(B,B5) )
=> ( finite_card(sum_sum(A,B),sum_Plus(A,B,A4,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(B,B5)) ) )
& ( ~ ( finite_finite2(A,A4)
& finite_finite2(B,B5) )
=> ( finite_card(sum_sum(A,B),sum_Plus(A,B,A4,B5)) = zero_zero(nat) ) ) ) ).
% card_Plus_conv_if
tff(fact_5086_card__Plus,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( finite_finite2(A,A4)
=> ( finite_finite2(B,B5)
=> ( finite_card(sum_sum(A,B),sum_Plus(A,B,A4,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(A,A4)),finite_card(B,B5)) ) ) ) ).
% card_Plus
tff(fact_5087_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4) = relcompp(option(A),option(product_prod(A,B)),option(B),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_tv(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)))),bNF_Grp(option(product_prod(A,B)),option(B),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_tv(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)))) ).
% option.rel_compp_Grp
tff(fact_5088_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),unit_f5069060285200089521factor(A,B2)) ) ).
% mult_one_div_unit_factor
tff(fact_5089_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),unit_f5069060285200089521factor(A,B2)) ) ) ) ).
% unit_factor_mult_unit_left
tff(fact_5090_unit__factor__simps_I2_J,axiom,
! [N: nat] : unit_f5069060285200089521factor(nat,aa(nat,nat,suc,N)) = one_one(nat) ).
% unit_factor_simps(2)
tff(fact_5091_unit__factor__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).
% unit_factor_1
tff(fact_5092_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% inv_unit_factor_eq_0_iff
tff(fact_5093_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,B2)),A3) ) ) ) ).
% unit_factor_mult_unit_right
tff(fact_5094_option_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),relcompp(A,B,C,R4,S2)) = relcompp(option(A),option(B),option(C),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(fun(B,fun(C,bool)),fun(option(B),fun(option(C),bool)),rel_option(B,C),S2)) ).
% option.rel_compp
tff(fact_5095_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(B,fun(B,bool)),Abs22: fun(B,C),Rep22: fun(C,B),R24: fun(A,fun(A,bool))] :
( quotient3(A,B,R12,Abs1,Rep1)
=> ( quotient3(B,C,R23,Abs22,Rep22)
=> ( ! [X3: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R24,X3),Y4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R12,X3),X3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R12,Y4),Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),R23,aa(A,B,Abs1,X3)),aa(A,B,Abs1,Y4))) ) ) )
=> ( ! [X3: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),R23,X3),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),R24,aa(B,A,Rep1,X3)),aa(B,A,Rep1,Y4))) )
=> quotient3(A,C,relcompp(A,A,A,R12,relcompp(A,A,A,R24,R12)),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22)) ) ) ) ) ).
% OOO_quotient3
tff(fact_5096_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),Abs22: fun(B,C),Rep22: fun(C,B)] :
( quotient3(A,B,R12,Abs1,Rep1)
=> ( quotient3(B,C,fequal(B),Abs22,Rep22)
=> quotient3(A,C,relcompp(A,A,A,R12,relcompp(A,A,A,fequal(A),R12)),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22)) ) ) ).
% OOO_eq_quotient3
tff(fact_5097_relpowp__add,axiom,
! [A: $tType,M: nat,N: nat,P: fun(A,fun(A,bool))] : compow(fun(A,fun(A,bool)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),P) = relcompp(A,A,A,compow(fun(A,fun(A,bool)),M,P),compow(fun(A,fun(A,bool)),N,P)) ).
% relpowp_add
tff(fact_5098_relcompp__relcomp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,C)),X: A,Xa3: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R),aTP_Lamp_ul(set(product_prod(B,C)),fun(B,fun(C,bool)),S)),X),Xa3))
<=> pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Xa3),relcomp(A,B,C,R,S))) ) ).
% relcompp_relcomp_eq
tff(fact_5099_is__unit__unit__factor,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).
% is_unit_unit_factor
tff(fact_5100_unit__factor__nat__def,axiom,
! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( unit_f5069060285200089521factor(nat,N) = zero_zero(nat) ) )
& ( ( N != zero_zero(nat) )
=> ( unit_f5069060285200089521factor(nat,N) = one_one(nat) ) ) ) ).
% unit_factor_nat_def
tff(fact_5101_option_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),bNF_Grp(A,B,A4,F)) = bNF_Grp(option(A),option(B),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_um(set(A),fun(option(A),bool),A4)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F)) ).
% option.rel_Grp
tff(fact_5102_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : bNF_rel_fun(D,D,A,B,fequal(D),bNF_Grp(A,B,A4,F)) = bNF_Grp(fun(D,A),fun(D,B),aa(fun(fun(D,A),bool),set(fun(D,A)),collect(fun(D,A)),aTP_Lamp_un(set(A),fun(fun(D,A),bool),A4)),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F)) ).
% fun.rel_Grp
tff(fact_5103_relcomp__def,axiom,
! [A: $tType,C: $tType,B: $tType,X: set(product_prod(A,B)),Xa3: set(product_prod(B,C))] : relcomp(A,B,C,X,Xa3) = aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),relcompp(A,B,C,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),X),aTP_Lamp_ul(set(product_prod(B,C)),fun(B,fun(C,bool)),Xa3)))) ).
% relcomp_def
tff(fact_5104_unit__factor__is__unit,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> pp(dvd_dvd(A,unit_f5069060285200089521factor(A,A3),one_one(A))) ) ) ).
% unit_factor_is_unit
tff(fact_5105_unit__factor__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = zero_zero(A) ) )
& ( ~ ( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = one_one(A) ) ) ) ) ).
% unit_factor_gcd
tff(fact_5106_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: fun(A,B),Abs: fun(B,A),G: fun(D,C),P: fun(C,bool),H: fun(C,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(C,fun(A,bool)),fun(D,fun(B,bool)),bNF_vimage2p(D,C,B,A,bool,G,Abs),bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),P),H)) = bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),aa(fun(C,bool),fun(D,bool),aTP_Lamp_uo(fun(D,C),fun(fun(C,bool),fun(D,bool)),G),P)),aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),Rep),H)),G)) ) ) ).
% type_copy_vimage2p_Grp_Abs
tff(fact_5107_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),F: fun(C,D),P: fun(D,bool),H: fun(D,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(D,fun(B,bool)),fun(C,fun(A,bool)),bNF_vimage2p(C,D,A,B,bool,F,Rep),bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),P),H)) = bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),aa(fun(D,bool),fun(C,bool),aTP_Lamp_up(fun(C,D),fun(fun(D,bool),fun(C,bool)),F),P)),aa(fun(C,D),fun(C,A),aa(fun(D,A),fun(fun(C,D),fun(C,A)),comp(D,A,C),aa(fun(D,B),fun(D,A),aa(fun(B,A),fun(fun(D,B),fun(D,A)),comp(B,A,D),Abs),H)),F)) ) ) ).
% type_copy_vimage2p_Grp_Rep
tff(fact_5108_Nat_Otransfer,axiom,
pp(aa(fun(code_integer,code_natural),bool,aa(fun(int,nat),fun(fun(code_integer,code_natural),bool),bNF_rel_fun(int,code_integer,nat,code_natural,code_pcr_integer,code_pcr_natural),nat2),code_Nat)) ).
% Nat.transfer
tff(fact_5109_pcr__natural__def,axiom,
code_pcr_natural = relcompp(nat,nat,code_natural,fequal(nat),code_cr_natural) ).
% pcr_natural_def
tff(fact_5110_natural_Orep__transfer,axiom,
pp(aa(fun(code_natural,nat),bool,aa(fun(nat,nat),fun(fun(code_natural,nat),bool),bNF_rel_fun(nat,code_natural,nat,nat,code_pcr_natural,fequal(nat)),aTP_Lamp_bx(nat,nat)),code_nat_of_natural)) ).
% natural.rep_transfer
tff(fact_5111_natural_Oid__abs__transfer,axiom,
pp(aa(fun(nat,code_natural),bool,aa(fun(nat,nat),fun(fun(nat,code_natural),bool),bNF_rel_fun(nat,nat,nat,code_natural,fequal(nat),code_pcr_natural),aTP_Lamp_bx(nat,nat)),code_natural_of_nat)) ).
% natural.id_abs_transfer
tff(fact_5112_natural_Opcr__cr__eq,axiom,
code_pcr_natural = code_cr_natural ).
% natural.pcr_cr_eq
tff(fact_5113_zero__natural_Otransfer,axiom,
pp(aa(code_natural,bool,aa(nat,fun(code_natural,bool),code_pcr_natural,zero_zero(nat)),zero_zero(code_natural))) ).
% zero_natural.transfer
tff(fact_5114_one__natural_Otransfer,axiom,
pp(aa(code_natural,bool,aa(nat,fun(code_natural,bool),code_pcr_natural,one_one(nat)),one_one(code_natural))) ).
% one_natural.transfer
tff(fact_5115_less__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(code_natural,fun(code_natural,bool)),bool),bNF_rel_fun(nat,code_natural,fun(nat,bool),fun(code_natural,bool),code_pcr_natural,bNF_rel_fun(nat,code_natural,bool,bool,code_pcr_natural,fequal(bool))),ord_less(nat)),ord_less(code_natural))) ).
% less_natural.transfer
tff(fact_5116_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F4: $tType,A: $tType,C: $tType,B: $tType,G2: $tType,F1: fun(F4,A),F22: fun(D,F4),G1: fun(G2,B),G22: fun(E,G2)] : bNF_vimage2p(D,A,E,B,C,aa(fun(D,F4),fun(D,A),aa(fun(F4,A),fun(fun(D,F4),fun(D,A)),comp(F4,A,D),F1),F22),aa(fun(E,G2),fun(E,B),aa(fun(G2,B),fun(fun(E,G2),fun(E,B)),comp(G2,B,E),G1),G22)) = aa(fun(fun(A,fun(B,C)),fun(F4,fun(G2,C))),fun(fun(A,fun(B,C)),fun(D,fun(E,C))),aa(fun(fun(F4,fun(G2,C)),fun(D,fun(E,C))),fun(fun(fun(A,fun(B,C)),fun(F4,fun(G2,C))),fun(fun(A,fun(B,C)),fun(D,fun(E,C)))),comp(fun(F4,fun(G2,C)),fun(D,fun(E,C)),fun(A,fun(B,C))),bNF_vimage2p(D,F4,E,G2,C,F22,G22)),bNF_vimage2p(F4,A,G2,B,C,F1,G1)) ).
% vimage2p_comp
tff(fact_5117_plus__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),plus_plus(nat)),plus_plus(code_natural))) ).
% plus_natural.transfer
tff(fact_5118_Suc_Otransfer,axiom,
pp(aa(fun(code_natural,code_natural),bool,aa(fun(nat,nat),fun(fun(code_natural,code_natural),bool),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural),suc),code_Suc)) ).
% Suc.transfer
tff(fact_5119_minus__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),minus_minus(nat)),minus_minus(code_natural))) ).
% minus_natural.transfer
tff(fact_5120_pcr__integer__def,axiom,
code_pcr_integer = relcompp(int,int,code_integer,fequal(int),code_cr_integer) ).
% pcr_integer_def
tff(fact_5121_less__eq__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(code_natural,fun(code_natural,bool)),bool),bNF_rel_fun(nat,code_natural,fun(nat,bool),fun(code_natural,bool),code_pcr_natural,bNF_rel_fun(nat,code_natural,bool,bool,code_pcr_natural,fequal(bool))),ord_less_eq(nat)),ord_less_eq(code_natural))) ).
% less_eq_natural.transfer
tff(fact_5122_times__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),times_times(nat)),times_times(code_natural))) ).
% times_natural.transfer
tff(fact_5123_divide__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),divide_divide(nat)),divide_divide(code_natural))) ).
% divide_natural.transfer
tff(fact_5124_modulo__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),modulo_modulo(nat)),modulo_modulo(code_natural))) ).
% modulo_natural.transfer
tff(fact_5125_take__bit__natural_Otransfer,axiom,
pp(aa(fun(nat,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),fequal(nat),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se2584673776208193580ke_bit(nat)),bit_se2584673776208193580ke_bit(code_natural))) ).
% take_bit_natural.transfer
tff(fact_5126_xor__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se5824344971392196577ns_xor(nat)),bit_se5824344971392196577ns_xor(code_natural))) ).
% xor_natural.transfer
tff(fact_5127_or__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se1065995026697491101ons_or(nat)),bit_se1065995026697491101ons_or(code_natural))) ).
% or_natural.transfer
tff(fact_5128_and__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(code_natural,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,code_natural,fun(nat,nat),fun(code_natural,code_natural),code_pcr_natural,bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se5824344872417868541ns_and(nat)),bit_se5824344872417868541ns_and(code_natural))) ).
% and_natural.transfer
tff(fact_5129_bit__natural_Otransfer,axiom,
pp(aa(fun(code_natural,fun(nat,bool)),bool,aa(fun(nat,fun(nat,bool)),fun(fun(code_natural,fun(nat,bool)),bool),bNF_rel_fun(nat,code_natural,fun(nat,bool),fun(nat,bool),code_pcr_natural,fequal(fun(nat,bool))),bit_se5641148757651400278ts_bit(nat)),bit_se5641148757651400278ts_bit(code_natural))) ).
% bit_natural.transfer
tff(fact_5130_push__bit__natural_Otransfer,axiom,
pp(aa(fun(nat,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),fequal(nat),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se4730199178511100633sh_bit(nat)),bit_se4730199178511100633sh_bit(code_natural))) ).
% push_bit_natural.transfer
tff(fact_5131_mask__natural_Otransfer,axiom,
pp(aa(fun(nat,code_natural),bool,aa(fun(nat,nat),fun(fun(nat,code_natural),bool),bNF_rel_fun(nat,nat,nat,code_natural,fequal(nat),code_pcr_natural),bit_se2239418461657761734s_mask(nat)),bit_se2239418461657761734s_mask(code_natural))) ).
% mask_natural.transfer
tff(fact_5132_drop__bit__natural_Otransfer,axiom,
pp(aa(fun(nat,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),fequal(nat),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se4197421643247451524op_bit(nat)),bit_se4197421643247451524op_bit(code_natural))) ).
% drop_bit_natural.transfer
tff(fact_5133_division__segment__natural_Otransfer,axiom,
pp(aa(fun(code_natural,code_natural),bool,aa(fun(nat,nat),fun(fun(code_natural,code_natural),bool),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural),euclid7384307370059645450egment(nat)),euclid7384307370059645450egment(code_natural))) ).
% division_segment_natural.transfer
tff(fact_5134_flip__bit__natural_Otransfer,axiom,
pp(aa(fun(nat,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),fequal(nat),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se8732182000553998342ip_bit(nat)),bit_se8732182000553998342ip_bit(code_natural))) ).
% flip_bit_natural.transfer
tff(fact_5135_set__bit__natural_Otransfer,axiom,
pp(aa(fun(nat,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),fequal(nat),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se5668285175392031749et_bit(nat)),bit_se5668285175392031749et_bit(code_natural))) ).
% set_bit_natural.transfer
tff(fact_5136_unset__bit__natural_Otransfer,axiom,
pp(aa(fun(nat,fun(code_natural,code_natural)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(code_natural,code_natural)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(code_natural,code_natural),fequal(nat),bNF_rel_fun(nat,code_natural,nat,code_natural,code_pcr_natural,code_pcr_natural)),bit_se2638667681897837118et_bit(nat)),bit_se2638667681897837118et_bit(code_natural))) ).
% unset_bit_natural.transfer
tff(fact_5137_euclidean__size__natural_Otransfer,axiom,
pp(aa(fun(code_natural,nat),bool,aa(fun(nat,nat),fun(fun(code_natural,nat),bool),bNF_rel_fun(nat,code_natural,nat,nat,code_pcr_natural,fequal(nat)),euclid6346220572633701492n_size(nat)),euclid6346220572633701492n_size(code_natural))) ).
% euclidean_size_natural.transfer
tff(fact_5138_integer__of__natural_Otransfer,axiom,
pp(aa(fun(code_natural,code_integer),bool,aa(fun(nat,int),fun(fun(code_natural,code_integer),bool),bNF_rel_fun(nat,code_natural,int,code_integer,code_pcr_natural,code_pcr_integer),semiring_1_of_nat(int)),code_i5400310926305786745atural)) ).
% integer_of_natural.transfer
tff(fact_5139_natural__of__integer_Otransfer,axiom,
pp(aa(fun(code_integer,code_natural),bool,aa(fun(int,nat),fun(fun(code_integer,code_natural),bool),bNF_rel_fun(int,code_integer,nat,code_natural,code_pcr_integer,code_pcr_natural),nat2),code_n4118661773612635043nteger)) ).
% natural_of_integer.transfer
tff(fact_5140_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : bNF_rel_fun(D,D,A,B,fequal(D),R4) = relcompp(fun(D,A),fun(D,product_prod(A,B)),fun(D,B),conversep(fun(D,product_prod(A,B)),fun(D,A),bNF_Grp(fun(D,product_prod(A,B)),fun(D,A),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_ji(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),A),fun(fun(D,product_prod(A,B)),fun(D,A)),comp(product_prod(A,B),A,D),product_fst(A,B)))),bNF_Grp(fun(D,product_prod(A,B)),fun(D,B),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_ji(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),B),fun(fun(D,product_prod(A,B)),fun(D,B)),comp(product_prod(A,B),B,D),product_snd(A,B)))) ).
% fun.rel_compp_Grp
tff(fact_5141_plus__num__def,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N) = aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(M)),nat_of_num(N))) ).
% plus_num_def
tff(fact_5142_Lcm__no__units,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] : gcd_Lcm(A,A4) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A4),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_uq(A,bool)))) ) ).
% Lcm_no_units
tff(fact_5143_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( remdups_adj(A,Xs) = Ys )
<=> ? [F6: fun(nat,nat)] :
( order_mono(nat,nat,F6)
& ( aa(set(nat),set(nat),image2(nat,nat,F6),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),aa(nat,nat,F6,I3)) ) )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) )
<=> ( aa(nat,nat,F6,I3) = aa(nat,nat,F6,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).
% remdups_adj_altdef
tff(fact_5144_Lcm__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_empty
tff(fact_5145_Lcm__1__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( gcd_Lcm(A,A4) = one_one(A) )
<=> ! [X4: A] :
( pp(member(A,X4,A4))
=> pp(dvd_dvd(A,X4,one_one(A))) ) ) ) ).
% Lcm_1_iff
tff(fact_5146_mono__add,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A3)) ) ).
% mono_add
tff(fact_5147_Lcm__nat__empty,axiom,
gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).
% Lcm_nat_empty
tff(fact_5148_nat__of__num_Osimps_I2_J,axiom,
! [X2: num] : nat_of_num(aa(num,num,bit0,X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(X2)),nat_of_num(X2)) ).
% nat_of_num.simps(2)
tff(fact_5149_nat__of__num__code_I1_J,axiom,
nat_of_num(one2) = one_one(nat) ).
% nat_of_num_code(1)
tff(fact_5150_nat__of__num__add,axiom,
! [X2: num,Y: num] : nat_of_num(aa(num,num,aa(num,fun(num,num),plus_plus(num),X2),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(X2)),nat_of_num(Y)) ).
% nat_of_num_add
tff(fact_5151_nat__of__num__code_I2_J,axiom,
! [N: num] : nat_of_num(aa(num,num,bit0,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(N)),nat_of_num(N)) ).
% nat_of_num_code(2)
tff(fact_5152_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [F: fun(A,B),M: A,N: A,M7: B,N2: B] :
( order_mono(A,B,F)
=> ( ( aa(set(A),set(B),image2(A,B,F),set_or7035219750837199246ssThan(A,M,N)) = set_or7035219750837199246ssThan(B,M7,N2) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
=> ( aa(A,B,F,M) = M7 ) ) ) ) ) ).
% mono_image_least
tff(fact_5153_nat__of__num_Osimps_I3_J,axiom,
! [X2: num] : nat_of_num(aa(num,num,bit1,X2)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(X2)),nat_of_num(X2))) ).
% nat_of_num.simps(3)
tff(fact_5154_unit__factor__Lcm,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( ( gcd_Lcm(A,A4) = zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = zero_zero(A) ) )
& ( ( gcd_Lcm(A,A4) != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A4)) = one_one(A) ) ) ) ) ).
% unit_factor_Lcm
tff(fact_5155_nat__of__num__code_I3_J,axiom,
! [N: num] : nat_of_num(aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(N)),nat_of_num(N))) ).
% nat_of_num_code(3)
tff(fact_5156_remdups__adj_Opelims,axiom,
! [A: $tType,X2: list(A),Y: list(A)] :
( ( remdups_adj(A,X2) = Y )
=> ( pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),X2))
=> ( ( ( X2 = nil(A) )
=> ( ( Y = nil(A) )
=> ~ pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),nil(A))) ) )
=> ( ! [X3: A] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ~ pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) ) )
=> ~ ! [X3: A,Y4: A,Xs2: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) )
=> ( ( ( ( X3 = Y4 )
=> ( Y = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)) ) )
& ( ( X3 != Y4 )
=> ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2))) ) ) )
=> ~ pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)))) ) ) ) ) ) ) ).
% remdups_adj.pelims
tff(fact_5157_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [R: set(product_prod(A,A)),As3: fun(A,B)] :
( bNF_Ca3754400796208372196lChain(A,B,R,As3)
<=> ! [I3: A,J2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I3),J2),R))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As3,I3)),aa(A,B,As3,J2))) ) ) ) ).
% relChain_def
tff(fact_5158_unit__factor__Gcd,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( ( ( gcd_Gcd(A,A4) = zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = zero_zero(A) ) )
& ( ( gcd_Gcd(A,A4) != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A4)) = one_one(A) ) ) ) ) ).
% unit_factor_Gcd
tff(fact_5159_Gcd__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).
% Gcd_UNIV
tff(fact_5160_Gcd__1,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A4: set(A)] :
( pp(member(A,one_one(A),A4))
=> ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ).
% Gcd_1
tff(fact_5161_Gcd__nat__eq__one,axiom,
! [N5: set(nat)] :
( pp(member(nat,one_one(nat),N5))
=> ( gcd_Gcd(nat,N5) = one_one(nat) ) ) ).
% Gcd_nat_eq_one
tff(fact_5162_Gcd__eq__1__I,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,A4: set(A)] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( pp(member(A,A3,A4))
=> ( gcd_Gcd(A,A4) = one_one(A) ) ) ) ) ).
% Gcd_eq_1_I
tff(fact_5163_cofinal__def,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( bNF_Ca7293521722713021262ofinal(A,A4,R)
<=> ! [X4: A] :
( pp(member(A,X4,field2(A,R)))
=> ? [Xa2: A] :
( pp(member(A,Xa2,A4))
& ( X4 != Xa2 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),R)) ) ) ) ).
% cofinal_def
tff(fact_5164_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B5: set(A),I4: set(B),A4: fun(B,set(C))] :
( ~ finite_finite2(A,B5)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(B,A)))
=> ( ! [X3: B] :
( pp(member(B,X3,I4))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(C,A))) )
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I4)))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(C,A))) ) ) ) ).
% card_of_UNION_ordLeq_infinite
tff(fact_5165_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ~ ? [F6: fun(A,B)] :
( inj_on(A,B,F6,A4)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),A4)),B5)) )
<=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_We4044943003108391690rdLess(B,A))) ) ).
% card_of_ordLess
tff(fact_5166_card__of__Times3,axiom,
! [A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4)))),bNF_Wellorder_ordLeq(A,product_prod(A,A)))) ).
% card_of_Times3
tff(fact_5167_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B5: fun(A,set(C))] :
( ! [X3: A] :
( pp(member(A,X3,I4))
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A4,X3))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),B5,X3))),bNF_Wellorder_ordLeq(B,C))) )
=> pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I4,A4))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,I4,B5))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).
% card_of_Sigma_mono1
tff(fact_5168_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,A4,aTP_Lamp_to(set(C),fun(A,set(C)),C5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,B5,aTP_Lamp_mz(set(C),fun(B,set(C)),C5)))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).
% card_of_Times_mono1
tff(fact_5169_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set(A),B5: set(B),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,C5,aTP_Lamp_ur(set(A),fun(C,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,C5,aTP_Lamp_us(set(B),fun(C,set(B)),B5)))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).
% card_of_Times_mono2
tff(fact_5170_infinite__iff__card__of__nat,axiom,
! [A: $tType,A4: set(A)] :
( ~ finite_finite2(A,A4)
<=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(nat,A))) ) ).
% infinite_iff_card_of_nat
tff(fact_5171_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A4: fun(B,set(A)),I4: set(B)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,I4,A4))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ).
% card_of_UNION_Sigma
tff(fact_5172_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] : pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).
% card_of_Times_commute
tff(fact_5173_card__of__Times__same__infinite,axiom,
! [A: $tType,A4: set(A)] :
( ~ finite_finite2(A,A4)
=> pp(member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A4,aTP_Lamp_ma(set(A),fun(A,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).
% card_of_Times_same_infinite
tff(fact_5174_card__of__Times1,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordLeq(B,product_prod(B,A)))) ) ).
% card_of_Times1
tff(fact_5175_card__of__Times2,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ).
% card_of_Times2
tff(fact_5176_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(bool,top_top(set(bool)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A)))))),bNF_Wellorder_ordIso(bool,A))) ) ).
% card_of_bool
tff(fact_5177_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,sum_sum(A,B)))) ).
% card_of_Plus_empty1
tff(fact_5178_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A4))),bNF_Wellorder_ordIso(A,sum_sum(B,A)))) ).
% card_of_Plus_empty2
tff(fact_5179_card__of__Plus__Times__bool,axiom,
! [A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),fun(set(product_prod(product_prod(A,bool),product_prod(A,bool))),product_prod(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool))))),product_Pair(set(product_prod(sum_sum(A,A),sum_sum(A,A))),set(product_prod(product_prod(A,bool),product_prod(A,bool)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A4,A4))),bNF_Ca6860139660246222851ard_of(product_prod(A,bool),product_Sigma(A,bool,A4,aTP_Lamp_ut(A,set(bool))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,bool)))) ).
% card_of_Plus_Times_bool
tff(fact_5180_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] : pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B5,A4))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A)))) ).
% card_of_Plus_commute
tff(fact_5181_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set(A),B5: set(B),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C5,A4))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C5,B5))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).
% card_of_Plus_cong2
tff(fact_5182_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A4,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B5,C5))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).
% card_of_Plus_cong1
tff(fact_5183_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] : pp(member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Ca6860139660246222851ard_of(sum_sum(sum_sum(A,B),C),sum_Plus(sum_sum(A,B),C,sum_Plus(A,B,A4,B5),C5))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),A4,sum_Plus(B,C,B5,C5)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).
% card_of_Plus_assoc
tff(fact_5184_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C),D4: set(D)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C5)),bNF_Ca6860139660246222851ard_of(D,D4)),bNF_Wellorder_ordIso(C,D)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A4,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B5,D4))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).
% card_of_Plus_cong
tff(fact_5185_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] : pp(member(product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),fun(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),product_prod(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))))),product_Pair(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C))))),bNF_Ca6860139660246222851ard_of(product_prod(A,sum_sum(B,C)),product_Sigma(A,sum_sum(B,C),A4,aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_uu(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),B5),C5)))),bNF_Ca6860139660246222851ard_of(sum_sum(product_prod(A,B),product_prod(A,C)),sum_Plus(product_prod(A,B),product_prod(A,C),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)),product_Sigma(A,C,A4,aTP_Lamp_to(set(C),fun(A,set(C)),C5))))),bNF_Wellorder_ordIso(product_prod(A,sum_sum(B,C)),sum_sum(product_prod(A,B),product_prod(A,C))))) ).
% card_of_Times_Plus_distrib
tff(fact_5186_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set(A),B5: set(B),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,C5,A4))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C5,B5))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).
% card_of_Plus_mono2
tff(fact_5187_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A4,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B5,C5))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).
% card_of_Plus_mono1
tff(fact_5188_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C),D4: set(D)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),bNF_Ca6860139660246222851ard_of(C,C5)),bNF_Ca6860139660246222851ard_of(D,D4)),bNF_Wellorder_ordLeq(C,D)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,A4,C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B5,D4))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).
% card_of_Plus_mono
tff(fact_5189_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B5: set(A),A4: set(B)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A4,B5))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ).
% card_of_Plus2
tff(fact_5190_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ).
% card_of_Plus1
tff(fact_5191_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A4: set(A),B5: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A4,B5))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).
% card_of_Un_Plus_ordLeq
tff(fact_5192_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
& pp(member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B5,A4))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ).
% card_of_Plus_infinite
tff(fact_5193_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ).
% card_of_Plus_infinite1
tff(fact_5194_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B5,A4))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ).
% card_of_Plus_infinite2
tff(fact_5195_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C5: set(A),A4: set(B),B5: set(C)] :
( ~ finite_finite2(A,C5)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Ca6860139660246222851ard_of(A,C5)),bNF_We4044943003108391690rdLess(B,A)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),bNF_Ca6860139660246222851ard_of(A,C5)),bNF_We4044943003108391690rdLess(C,A)))
=> pp(member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A4,B5))),bNF_Ca6860139660246222851ard_of(A,C5)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ).
% card_of_Plus_ordLess_infinite
tff(fact_5196_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B5: set(A),I4: set(B),A4: fun(B,set(C))] :
( ~ finite_finite2(A,B5)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(B,A)))
=> ( ! [X3: B] :
( pp(member(B,X3,I4))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(C,A))) )
=> pp(member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I4,A4))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
tff(fact_5197_card__of__refl,axiom,
! [A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(A,A))) ).
% card_of_refl
tff(fact_5198_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set(A),B5: set(B)] :
( ( ( A1 != A22 )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A))))),A4)) )
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).
% card_of_Plus_Times_aux
tff(fact_5199_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( ( B5 != bot_bot(set(B)) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordIso(A,product_prod(B,A)))) ) ) ) ).
% card_of_Times_infinite_simps(4)
tff(fact_5200_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( ( B5 != bot_bot(set(B)) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ).
% card_of_Times_infinite_simps(3)
tff(fact_5201_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( ( B5 != bot_bot(set(B)) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordIso(A,product_prod(A,B)))) ) ) ) ).
% card_of_Times_infinite_simps(2)
tff(fact_5202_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( ( B5 != bot_bot(set(B)) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).
% card_of_Times_infinite_simps(1)
tff(fact_5203_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( ~ finite_finite2(A,A4)
=> ( ( B5 != bot_bot(set(B)) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,B),A)))
& pp(member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B5,aTP_Lamp_jr(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ).
% card_of_Times_infinite
tff(fact_5204_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set(A),B1: B,B22: B,B5: set(B)] :
( ( ( A1 != A22 )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A))))),A4)) )
=> ( ( ( B1 != B22 )
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),insert2(B,B1),aa(set(B),set(B),insert2(B,B22),bot_bot(set(B))))),B5)) )
=> pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).
% card_of_Plus_Times
tff(fact_5205_card__of__image,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A4: set(B)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(B),set(A),image2(B,A,F),A4))),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordLeq(A,B))) ).
% card_of_image
tff(fact_5206_card__of__empty,axiom,
! [B: $tType,A: $tType,A4: set(B)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordLeq(A,B))) ).
% card_of_empty
tff(fact_5207_card__of__empty3,axiom,
! [B: $tType,A: $tType,A4: set(A)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordLeq(A,B)))
=> ( A4 = bot_bot(set(A)) ) ) ).
% card_of_empty3
tff(fact_5208_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> ( finite_finite2(B,B5)
=> finite_finite2(A,A4) ) ) ).
% card_of_ordLeq_finite
tff(fact_5209_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B)))
=> ( ~ finite_finite2(A,A4)
=> ~ finite_finite2(B,B5) ) ) ).
% card_of_ordLeq_infinite
tff(fact_5210_card__of__mono1,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A4),B5))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordLeq(A,A))) ) ).
% card_of_mono1
tff(fact_5211_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B))) ).
% card_of_empty_ordIso
tff(fact_5212_card__of__empty2,axiom,
! [B: $tType,A: $tType,A4: set(A)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B)))
=> ( A4 = bot_bot(set(A)) ) ) ).
% card_of_empty2
tff(fact_5213_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A4: set(A),B5: set(B)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordIso(A,B)))
=> ( finite_finite2(A,A4)
<=> finite_finite2(B,B5) ) ) ).
% card_of_ordIso_finite
tff(fact_5214_card__of__mono2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_Ca6860139660246222851ard_of(B,field2(B,R3))),bNF_Wellorder_ordLeq(A,B))) ) ).
% card_of_mono2
tff(fact_5215_card__of__cong,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_Ca6860139660246222851ard_of(B,field2(B,R3))),bNF_Wellorder_ordIso(A,B))) ) ).
% card_of_cong
tff(fact_5216_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( inj_on(A,B,F,A4)
=> ( ! [A5: A] :
( pp(member(A,A5,A4))
=> pp(member(B,aa(A,B,F,A5),B5)) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B))) ) ) ).
% card_of_ordLeqI
tff(fact_5217_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ? [F3: fun(B,A),B8: set(B)] : bij_betw(B,A,F3,B8,A4) ) ).
% ex_bij_betw
tff(fact_5218_card__of__least,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( order_well_order_on(A,A4,R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,A))) ) ).
% card_of_least
tff(fact_5219_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R3),bNF_We4044943003108391690rdLess(A,B))) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_5220_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A),B5: set(B)] :
( bij_betw(A,B,F,A4,B5)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordIso(A,B))) ) ).
% card_of_ordIsoI
tff(fact_5221_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ? [F6: fun(A,B)] : bij_betw(A,B,F6,A4,B5)
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordIso(A,B))) ) ).
% card_of_ordIso
tff(fact_5222_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S2: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X2: D] :
( ! [Y4: A] : pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),S2,Y4))),Bd),bNF_Wellorder_ordLeq(B,C)))
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),aa(fun(D,A),fun(D,set(B)),aa(fun(A,set(B)),fun(fun(D,A),fun(D,set(B))),comp(A,set(B),D),S2),Rep),X2))),Bd),bNF_Wellorder_ordLeq(B,C))) ) ).
% type_copy_set_bd
tff(fact_5223_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ( A4 != bot_bot(set(A)) )
=> ( ? [G5: fun(B,A)] : aa(set(B),set(A),image2(B,A,G5),B5) = A4
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B))) ) ) ).
% card_of_ordLeq2
tff(fact_5224_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B5: set(A),F: fun(B,A),A4: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),aa(set(B),set(A),image2(B,A,F),A4)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordLeq(A,B))) ) ).
% surj_imp_ordLeq
tff(fact_5225_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set(A),B2: B] :
( ( A4 != bot_bot(set(A)) )
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),insert2(B,B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))) ) ).
% card_of_singl_ordLeq
tff(fact_5226_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B5: set(A),A4: set(B)] :
( ( B5 != bot_bot(set(A)) )
=> ( ~ ? [F6: fun(B,A)] : aa(set(B),set(A),image2(B,A,F6),A4) = B5
<=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_We4044943003108391690rdLess(B,A))) ) ) ).
% card_of_ordLess2
tff(fact_5227_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A4: set(A),C5: set(B)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,C5)),bNF_Wellorder_ordLeq(A,B)))
<=> ? [B9: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B9),C5))
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B9)),bNF_Wellorder_ordIso(A,B)))
& pp(member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B9)),bNF_Ca6860139660246222851ard_of(B,C5)),bNF_Wellorder_ordLeq(B,B))) ) ) ).
% internalize_card_of_ordLeq2
tff(fact_5228_card__of__Field__ordLess,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R),bNF_Wellorder_ordLeq(A,A))) ) ).
% card_of_Field_ordLess
tff(fact_5229_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),P3: set(product_prod(C,C)),P7: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P7),bNF_Wellorder_ordLeq(C,D)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),field2(C,P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R3),field2(D,P7)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).
% ordLeq_Plus_mono
tff(fact_5230_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R3),C5))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).
% ordLeq_Plus_mono1
tff(fact_5231_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),A4: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A4,field2(A,R)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A4,field2(B,R3)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).
% ordLeq_Plus_mono2
tff(fact_5232_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),P3: set(product_prod(C,C)),P7: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P3),P7),bNF_Wellorder_ordIso(C,D)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),field2(C,P3)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R3),field2(D,P7)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).
% ordIso_Plus_cong
tff(fact_5233_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,C),sum_Plus(A,C,field2(A,R),C5))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R3),C5))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).
% ordIso_Plus_cong1
tff(fact_5234_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),A4: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,A),sum_Plus(C,A,A4,field2(A,R)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A4,field2(B,R3)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).
% ordIso_Plus_cong2
tff(fact_5235_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),C5: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Ca6860139660246222851ard_of(product_prod(A,C),product_Sigma(A,C,field2(A,R),aTP_Lamp_to(set(C),fun(A,set(C)),C5)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,field2(B,R3),aTP_Lamp_mz(set(C),fun(B,set(C)),C5)))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).
% ordLeq_Times_mono1
tff(fact_5236_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),A4: set(C)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Ca6860139660246222851ard_of(product_prod(C,A),product_Sigma(C,A,A4,aTP_Lamp_uv(set(product_prod(A,A)),fun(C,set(A)),R)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,A4,aTP_Lamp_uw(set(product_prod(B,B)),fun(C,set(B)),R3)))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).
% ordLeq_Times_mono2
tff(fact_5237_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] :
( ? [F6: fun(A,B)] :
( inj_on(A,B,F6,A4)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F6),A4)),B5)) )
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B5)),bNF_Wellorder_ordLeq(A,B))) ) ).
% card_of_ordLeq
tff(fact_5238_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set(A),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
<=> ? [B9: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B9),field2(B,R)))
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B9)),bNF_Wellorder_ordIso(A,B)))
& pp(member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(B,B9)),R),bNF_Wellorder_ordLeq(B,B))) ) ) ).
% internalize_card_of_ordLeq
tff(fact_5239_regularCard__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca7133664381575040944arCard(A,R)
<=> ! [K10: set(A)] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K10),field2(A,R)))
& bNF_Ca7293521722713021262ofinal(A,K10,R) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,K10)),R),bNF_Wellorder_ordIso(A,A))) ) ) ).
% regularCard_def
tff(fact_5240_card__of__ordIso__subst,axiom,
! [A: $tType,A4: set(A),B5: set(A)] :
( ( A4 = B5 )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(A,B5)),bNF_Wellorder_ordIso(A,A))) ) ).
% card_of_ordIso_subst
tff(fact_5241_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X2: C,X7: set(fun(C,set(set(A)))),Hbd: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(fun(C,set(set(A)))),set(set(set(A))),image2(fun(C,set(set(A))),set(set(A)),aTP_Lamp_ux(C,fun(fun(C,set(set(A))),set(set(A))),X2)),X7))))),Hbd),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),aa(fun(set(set(A)),set(A)),fun(fun(C,set(set(A))),fun(C,set(A))),comp(set(set(A)),set(A),C),complete_Sup_Sup(set(A))),bNF_collect(C,set(A),X7)),X2))),Hbd),bNF_Wellorder_ordLeq(A,B))) ) ).
% comp_set_bd_Union_o_collect
tff(fact_5242_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A4: set(B),B5: set(C)] :
( ~ finite_finite2(A,field2(A,R))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R),bNF_We4044943003108391690rdLess(B,A)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),R),bNF_We4044943003108391690rdLess(C,A)))
=> pp(member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A4,B5))),R),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
tff(fact_5243_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A4: set(B),B5: set(C)] :
( ~ finite_finite2(A,field2(A,R))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),R),bNF_Wellorder_ordLeq(C,A)))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,A4,aTP_Lamp_mz(set(C),fun(B,set(C)),B5)))),R),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
tff(fact_5244_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(C))] :
( ~ finite_finite2(A,field2(A,R))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),R),bNF_Wellorder_ordLeq(B,A)))
=> ( ! [X3: B] :
( pp(member(B,X3,I4))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),R),bNF_Wellorder_ordLeq(C,A))) )
=> pp(member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I4,A4))),R),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
tff(fact_5245_Card__order__trans,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y: A,Z2: A] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( X2 != Y )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( ( Y != Z2 )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R))
=> ( ( X2 != Z2 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),R)) ) ) ) ) ) ) ).
% Card_order_trans
tff(fact_5246_card__order__on__ordIso,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,A4,R)
=> ( bNF_Ca8970107618336181345der_on(A,A4,R3)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R3),bNF_Wellorder_ordIso(A,A))) ) ) ).
% card_order_on_ordIso
tff(fact_5247_infinite__Card__order__limit,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ~ finite_finite2(A,field2(A,R))
=> ( pp(member(A,A3,field2(A,R)))
=> ? [X3: A] :
( pp(member(A,X3,field2(A,R)))
& ( A3 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X3),R)) ) ) ) ) ).
% infinite_Card_order_limit
tff(fact_5248_ordLeq__refl,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R),bNF_Wellorder_ordLeq(A,A))) ) ).
% ordLeq_refl
tff(fact_5249_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordIso(B,A)))
=> bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3) ) ) ).
% Card_order_ordIso
tff(fact_5250_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B)))
=> bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3) ) ) ).
% Card_order_ordIso2
tff(fact_5251_ordIso__refl,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R),bNF_Wellorder_ordIso(A,A))) ) ).
% ordIso_refl
tff(fact_5252_card__of__unique,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,A4,R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(A,A))) ) ).
% card_of_unique
tff(fact_5253_card__order__on__def,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,A4,R)
<=> ( order_well_order_on(A,A4,R)
& ! [R11: set(product_prod(A,A))] :
( order_well_order_on(A,A4,R11)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R11),bNF_Wellorder_ordLeq(A,A))) ) ) ) ).
% card_order_on_def
tff(fact_5254_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_Wellorder_ordLeq(A,A))) ) ).
% Card_order_iff_ordLeq_card_of
tff(fact_5255_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),A4: set(B)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordIso(A,B)))
=> bNF_Ca8970107618336181345der_on(A,field2(A,R),R) ) ).
% ordIso_card_of_imp_Card_order
tff(fact_5256_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_Wellorder_ordIso(A,A))) ) ).
% Card_order_iff_ordIso_card_of
tff(fact_5257_card__of__Field__ordIso,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R),bNF_Wellorder_ordIso(A,A))) ) ).
% card_of_Field_ordIso
tff(fact_5258_dir__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R: set(product_prod(A,A))] :
( ! [X3: A,Y4: A] :
( ( aa(A,B,F,X3) = aa(A,B,F,Y4) )
<=> ( X3 = Y4 ) )
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_We2720479622203943262_image(A,B,R,F)),bNF_Wellorder_ordIso(A,B))) ) ) ).
% dir_image
tff(fact_5259_exists__minim__Card__order,axiom,
! [A: $tType,R4: set(set(product_prod(A,A)))] :
( ( R4 != bot_bot(set(set(product_prod(A,A)))) )
=> ( ! [X3: set(product_prod(A,A))] :
( pp(member(set(product_prod(A,A)),X3,R4))
=> bNF_Ca8970107618336181345der_on(A,field2(A,X3),X3) )
=> ? [X3: set(product_prod(A,A))] :
( pp(member(set(product_prod(A,A)),X3,R4))
& ! [Xa3: set(product_prod(A,A))] :
( pp(member(set(product_prod(A,A)),Xa3,R4))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),X3),Xa3),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).
% exists_minim_Card_order
tff(fact_5260_Card__order__empty,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R),bNF_Wellorder_ordLeq(B,A))) ) ).
% Card_order_empty
tff(fact_5261_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),A4: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordIso(A,B)))
=> ( finite_finite2(A,field2(A,R))
<=> finite_finite2(B,A4) ) ) ) ).
% card_of_ordIso_finite_Field
tff(fact_5262_card__of__underS,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(A,A3,field2(A,R)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,order_underS(A,R,A3))),R),bNF_We4044943003108391690rdLess(A,A))) ) ) ).
% card_of_underS
tff(fact_5263_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),B5: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),R),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R),B5))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).
% Card_order_Plus1
tff(fact_5264_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),A4: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),R),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A4,field2(A,R)))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).
% Card_order_Plus2
tff(fact_5265_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),B2: B] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( field2(A,R) != bot_bot(set(A)) )
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),insert2(B,B2),bot_bot(set(B))))),R),bNF_Wellorder_ordLeq(B,A))) ) ) ).
% Card_order_singl_ordLeq
tff(fact_5266_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),A4: set(B),B5: set(B)] :
( ~ finite_finite2(A,field2(A,R))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R),bNF_Wellorder_ordLeq(B,A)))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B5))),R),bNF_Wellorder_ordLeq(B,A))) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
tff(fact_5267_card__of__empty1,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A))] :
( ( order_well_order_on(A,field2(A,R),R)
| bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R),bNF_Wellorder_ordLeq(B,A))) ) ).
% card_of_empty1
tff(fact_5268_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),B5: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( B5 != bot_bot(set(B)) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),R),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_jm(set(B),fun(A,set(B)),B5)))),bNF_Wellorder_ordLeq(A,product_prod(A,B)))) ) ) ).
% Card_order_Times1
tff(fact_5269_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),A4: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( A4 != bot_bot(set(B)) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),R),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A4,aTP_Lamp_uy(set(product_prod(A,A)),fun(B,set(A)),R)))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ) ) ).
% Card_order_Times2
tff(fact_5270_Card__order__Times__same__infinite,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ~ finite_finite2(A,field2(A,R))
=> pp(member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,field2(A,R),aTP_Lamp_kw(set(product_prod(A,A)),fun(A,set(A)),R)))),R),bNF_Wellorder_ordLeq(product_prod(A,A),A))) ) ) ).
% Card_order_Times_same_infinite
tff(fact_5271_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(C))] :
( ~ finite_finite2(A,field2(A,R))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),R),bNF_Wellorder_ordLeq(B,A)))
=> ( ! [X3: B] :
( pp(member(B,X3,I4))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),R),bNF_Wellorder_ordLeq(C,A))) )
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I4)))),R),bNF_Wellorder_ordLeq(C,A))) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
tff(fact_5272_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
( ~ finite_finite2(A,field2(A,R))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R),field2(B,P3)))),R),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
& pp(member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,field2(B,P3),field2(A,R)))),R),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ) ).
% Card_order_Plus_infinite
tff(fact_5273_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
<=> ! [X4: A] :
( pp(member(A,X4,field2(A,R)))
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),inf_inf(set(product_prod(A,A)),R),product_Sigma(A,A,order_underS(A,R,X4),aa(A,fun(A,set(A)),aTP_Lamp_ss(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X4)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% Card_order_iff_Restr_underS
tff(fact_5274_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),As3: fun(A,set(B)),B5: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca7133664381575040944arCard(A,R)
=> ( bNF_Ca3754400796208372196lChain(A,set(B),R,As3)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As3),field2(A,R)))))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R),bNF_We4044943003108391690rdLess(B,A)))
=> ? [X3: A] :
( pp(member(A,X3,field2(A,R)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(A,set(B),As3,X3))) ) ) ) ) ) ) ).
% regularCard_UNION
tff(fact_5275_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A4: set(B),B5: set(C)] :
( ~ finite_finite2(A,field2(A,R))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A4)),R),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,B5)),R),bNF_Wellorder_ordLeq(C,A)))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A4,B5))),R),bNF_Wellorder_ordLeq(sum_sum(B,C),A))) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
tff(fact_5276_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
( ~ finite_finite2(A,field2(A,R))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( field2(B,P3) != bot_bot(set(B)) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R),bNF_Wellorder_ordLeq(B,A)))
=> ( pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_uz(set(product_prod(B,B)),fun(A,set(B)),P3)))),R),bNF_Wellorder_ordIso(product_prod(A,B),A)))
& pp(member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,field2(B,P3),aTP_Lamp_uy(set(product_prod(A,A)),fun(B,set(A)),R)))),R),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ) ).
% Card_order_Times_infinite
tff(fact_5277_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R),R)
=> ? [X_13: fun(A,B)] : bNF_Gr1419584066657907630d_pred(A,B,A4,R,X_13) ) ) ).
% ex_toCard_pred
tff(fact_5278_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),As3: fun(set(A),set(B)),B5: set(B)] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ~ finite_finite2(A,field2(A,R))
=> ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As3)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R),bNF_Wellorder_ordLeq(B,A)))
=> ? [X3: set(A)] :
( pp(member(set(A),X3,field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),As3,X3))) ) ) ) ) ) ) ).
% cardSuc_UNION
tff(fact_5279_toCard__inj,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(B,B)),X2: A,Y: A] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R),R)
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( ( aa(A,B,bNF_Greatest_toCard(A,B,A4,R),X2) = aa(A,B,bNF_Greatest_toCard(A,B,A4,R),Y) )
<=> ( X2 = Y ) ) ) ) ) ) ).
% toCard_inj
tff(fact_5280_cardSuc__greater,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).
% cardSuc_greater
tff(fact_5281_cardSuc__ordLeq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_Wellorder_ordLeq(A,set(A)))) ) ).
% cardSuc_ordLeq
tff(fact_5282_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A4: set(A),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R),R)
=> bNF_Gr1419584066657907630d_pred(A,B,A4,R,bNF_Greatest_toCard(A,B,A4,R)) ) ) ).
% toCard_pred_toCard
tff(fact_5283_cardSuc__least__aux,axiom,
! [A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(set(A),set(A)))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R3),R3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),R3),bNF_We4044943003108391690rdLess(A,set(A))))
=> pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),bNF_Ca8387033319878233205ardSuc(A,R)),R3),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).
% cardSuc_least_aux
tff(fact_5284_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),aa(set(product_prod(B,B)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(B,B)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(B,B)),set(product_prod(set(A),set(A)))),R3),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_We4044943003108391690rdLess(B,set(A))))
<=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R3),R),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).
% cardSuc_ordLeq_ordLess
tff(fact_5285_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
<=> pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R)),R3),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).
% cardSuc_ordLess_ordLeq
tff(fact_5286_cardSuc__least,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_We4044943003108391690rdLess(A,B)))
=> pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(B,B)),product_prod(set(product_prod(set(A),set(A))),set(product_prod(B,B)))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(B,B))),bNF_Ca8387033319878233205ardSuc(A,R)),R3),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).
% cardSuc_least
tff(fact_5287_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_Ca8387033319878233205ardSuc(B,R3)),bNF_Wellorder_ordLeq(set(A),set(B))))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).
% cardSuc_mono_ordLeq
tff(fact_5288_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R3),R3)
=> ( pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(B),set(B))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(B),set(B)))),bNF_Ca8387033319878233205ardSuc(A,R)),bNF_Ca8387033319878233205ardSuc(B,R3)),bNF_Wellorder_ordIso(set(A),set(B))))
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3),bNF_Wellorder_ordIso(A,B))) ) ) ) ).
% cardSuc_invar_ordIso
tff(fact_5289_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(B,B)),B2: A] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R),R)
=> ( pp(member(A,B2,A4))
=> ( bNF_Gr5436034075474128252omCard(A,B,A4,R,aa(A,B,bNF_Greatest_toCard(A,B,A4,R),B2)) = B2 ) ) ) ) ).
% fromCard_toCard
tff(fact_5290_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),As3: fun(set(A),set(B)),B5: set(B)] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As3)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As3),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))))
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B5)),R),bNF_Wellorder_ordLeq(B,A)))
=> ? [X3: set(A)] :
( pp(member(set(A),X3,field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),aa(set(A),set(B),As3,X3))) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
tff(fact_5291_isCardSuc__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(set(A),set(A)))] :
( bNF_Ca6246979054910435723ardSuc(A,R,R3)
<=> ( bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R3),R3)
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),R3),bNF_We4044943003108391690rdLess(A,set(A))))
& ! [R13: set(product_prod(set(A),set(A)))] :
( ( bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R13),R13)
& pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),R13),bNF_We4044943003108391690rdLess(A,set(A)))) )
=> pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),R3),R13),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).
% isCardSuc_def
tff(fact_5292_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca4139267488887388095finite(A,R1)
=> bNF_Ca4139267488887388095finite(B,R22) ) ) ).
% cinfinite_mono
tff(fact_5293_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R: set(product_prod(A,A)),X22: A] :
( pp(member(A,X1,field2(A,R)))
=> ( pp(member(A,X22,field2(A,R)))
=> ( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ? [X3: A] :
( pp(member(A,X3,field2(A,R)))
& ( X1 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X3),R))
& ( X22 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X22),X3),R)) ) ) ) ) ).
% Cinfinite_limit2
tff(fact_5294_Cinfinite__limit,axiom,
! [A: $tType,X2: A,R: set(product_prod(A,A))] :
( pp(member(A,X2,field2(A,R)))
=> ( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ? [X3: A] :
( pp(member(A,X3,field2(A,R)))
& ( X2 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X3),R)) ) ) ) ).
% Cinfinite_limit
tff(fact_5295_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R1),R22),bNF_Wellorder_ordIso(A,B)))
=> ( ( bNF_Ca4139267488887388095finite(A,R1)
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) ) ) ) ).
% Cinfinite_cong
tff(fact_5296_Cinfinite__limit__finite,axiom,
! [A: $tType,X7: set(A),R: set(product_prod(A,A))] :
( finite_finite2(A,X7)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),field2(A,R)))
=> ( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ? [X3: A] :
( pp(member(A,X3,field2(A,R)))
& ! [Xa3: A] :
( pp(member(A,Xa3,X7))
=> ( ( Xa3 != X3 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X3),R)) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
tff(fact_5297_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(B,B)),B5: set(A)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B5)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,field2(B,R),R) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))),R),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).
% Un_Cinfinite_bound
tff(fact_5298_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set(A),R: set(product_prod(B,B)),A4: fun(A,set(C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,I4)),R),bNF_Wellorder_ordLeq(A,B)))
=> ( ! [X3: A] :
( pp(member(A,X3,I4))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),A4,X3))),R),bNF_Wellorder_ordLeq(C,B))) )
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,field2(B,R),R) )
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A4),I4)))),R),bNF_Wellorder_ordLeq(C,B))) ) ) ) ).
% UNION_Cinfinite_bound
tff(fact_5299_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
( ( bNF_Cardinal_cfinite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( ( bNF_Ca4139267488887388095finite(B,S)
& bNF_Ca8970107618336181345der_on(B,field2(B,S),S) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),S),bNF_We4044943003108391690rdLess(A,B))) ) ) ).
% Cfinite_ordLess_Cinfinite
tff(fact_5300_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(C)),Gset: fun(D,set(B)),Gbd: set(product_prod(E,E)),X2: D] :
( bNF_Ca8970107618336181345der_on(A,field2(A,Fbd),Fbd)
=> ( ! [X3: B] : pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Fset,X3))),Fbd),bNF_Wellorder_ordLeq(C,A)))
=> ( ! [X3: D] : pp(member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),Gset,X3))),Gbd),bNF_Wellorder_ordLeq(B,E)))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(C,C)),fun(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),product_Pair(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),Fset),aa(D,set(B),Gset,X2))))),bNF_Cardinal_cprod(E,A,Gbd,Fbd)),bNF_Wellorder_ordLeq(C,product_prod(E,A)))) ) ) ) ).
% comp_single_set_bd
tff(fact_5301_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),P3: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
( bNF_Ca4139267488887388095finite(A,R)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(A,A),product_prod(A,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(A,A),product_prod(A,A)))),bNF_Cardinal_cprod(B,C,P3,P7)),bNF_Cardinal_cprod(A,A,R,R)),bNF_Wellorder_ordIso(product_prod(B,C),product_prod(A,A))))
=> pp(member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),bNF_Cardinal_cprod(B,C,P3,P7)),R),bNF_Wellorder_ordIso(product_prod(B,C),A))) ) ) ) ).
% cprod_dup
tff(fact_5302_cprod__com,axiom,
! [B: $tType,A: $tType,P12: set(product_prod(A,A)),P23: set(product_prod(B,B))] : pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_cprod(A,B,P12,P23)),bNF_Cardinal_cprod(B,A,P23,P12)),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).
% cprod_com
tff(fact_5303_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
( ( bNF_Cardinal_cfinite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( ( bNF_Ca4139267488887388095finite(B,S)
& bNF_Ca8970107618336181345der_on(B,field2(B,S),S) )
=> pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,B,R,S)),S),bNF_Wellorder_ordLeq(product_prod(A,B),B))) ) ) ).
% Cfinite_cprod_Cinfinite
tff(fact_5304_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordLeq(C,D)))
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P12,P23)),bNF_Cardinal_cprod(B,D,R1,R22)),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,D)))) ) ) ).
% cprod_mono
tff(fact_5305_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Cardinal_cprod(A,C,P12,Q3)),bNF_Cardinal_cprod(B,C,R1,Q3)),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).
% cprod_mono1
tff(fact_5306_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Cardinal_cprod(C,A,Q3,P23)),bNF_Cardinal_cprod(C,B,Q3,R22)),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).
% cprod_mono2
tff(fact_5307_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordIso(C,D)))
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,D),product_prod(B,D))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,D),product_prod(B,D)))),bNF_Cardinal_cprod(A,C,P12,P23)),bNF_Cardinal_cprod(B,D,R1,R22)),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,D)))) ) ) ).
% cprod_cong
tff(fact_5308_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(product_prod(B,C),product_prod(B,C))),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C))))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(product_prod(B,C),product_prod(B,C)))),bNF_Cardinal_cprod(A,C,P12,P23)),bNF_Cardinal_cprod(B,C,R1,P23)),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,C)))) ) ).
% cprod_cong1
tff(fact_5309_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),aa(set(product_prod(product_prod(C,A),product_prod(C,A))),fun(set(product_prod(product_prod(C,B),product_prod(C,B))),product_prod(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B))))),product_Pair(set(product_prod(product_prod(C,A),product_prod(C,A))),set(product_prod(product_prod(C,B),product_prod(C,B)))),bNF_Cardinal_cprod(C,A,Q3,P23)),bNF_Cardinal_cprod(C,B,Q3,R22)),bNF_Wellorder_ordIso(product_prod(C,A),product_prod(C,B)))) ) ).
% cprod_cong2
tff(fact_5310_cprod__infinite,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> pp(member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Cardinal_cprod(A,A,R,R)),R),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).
% cprod_infinite
tff(fact_5311_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P3: set(product_prod(A,A)),R: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P3),R),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R),bNF_Wellorder_ordLeq(C,B)))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P3),P3)
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,field2(B,R),R) )
=> pp(member(product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(product_prod(A,C),product_prod(A,C))),set(product_prod(B,B))),bNF_Cardinal_cprod(A,C,P3,Q3)),R),bNF_Wellorder_ordLeq(product_prod(A,C),B))) ) ) ) ) ) ).
% cprod_cinfinite_bound
tff(fact_5312_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),I4: set(B),A4: fun(B,set(C))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ! [X3: B] :
( pp(member(B,X3,I4))
=> pp(member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),R),bNF_Wellorder_ordLeq(C,A))) )
=> pp(member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,C),product_prod(B,C))),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Cardinal_Csum(B,C,bNF_Ca6860139660246222851ard_of(B,I4),aTP_Lamp_va(fun(B,set(C)),fun(B,set(product_prod(C,C))),A4))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,I4),R)),bNF_Wellorder_ordLeq(product_prod(B,C),product_prod(B,A)))) ) ) ).
% card_of_Csum_Times'
tff(fact_5313_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),A4: fun(A,set(B)),B5: set(C)] :
( ! [X3: A] :
( pp(member(A,X3,I4))
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),A4,X3))),bNF_Ca6860139660246222851ard_of(C,B5)),bNF_Wellorder_ordLeq(B,C))) )
=> pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(product_prod(A,C),product_prod(A,C))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(product_prod(A,C),product_prod(A,C)))),bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I4),aTP_Lamp_vb(fun(A,set(B)),fun(A,set(product_prod(B,B))),A4))),bNF_Cardinal_cprod(A,C,bNF_Ca6860139660246222851ard_of(A,I4),bNF_Ca6860139660246222851ard_of(C,B5))),bNF_Wellorder_ordLeq(product_prod(A,B),product_prod(A,C)))) ) ).
% card_of_Csum_Times
tff(fact_5314_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( ( ~ pp(member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),P3),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(B,B)))
& bNF_Ca8970107618336181345der_on(B,field2(B,P3),P3) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P3),R),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Cardinal_cprod(A,B,R,P3)),R),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).
% cprod_infinite1'
tff(fact_5315_czero__ordIso,axiom,
! [B: $tType,A: $tType] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Cardinal_czero(A)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))) ).
% czero_ordIso
tff(fact_5316_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,B))] :
( bNF_Ca4139267488887388095finite(B,R)
=> ~ pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(B,A))) ) ).
% cinfinite_not_czero
tff(fact_5317_czeroE,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)))
=> ( field2(A,R) = bot_bot(set(A)) ) ) ).
% czeroE
tff(fact_5318_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A4: set(A)] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)))
<=> ( A4 = bot_bot(set(A)) ) ) ).
% card_of_ordIso_czero_iff_empty
tff(fact_5319_czeroI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( field2(A,R) = bot_bot(set(A)) )
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))) ) ) ).
% czeroI
tff(fact_5320_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( field2(A,R) != bot_bot(set(A)) ) ) ).
% Cnotzero_imp_not_empty
tff(fact_5321_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,Q3),Q3)
=> ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),Q3),bNF_Wellorder_ordLeq(A,B)))
=> ( ~ pp(member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),Q3),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(B,B)))
& bNF_Ca8970107618336181345der_on(B,field2(B,Q3),Q3) ) ) ) ) ).
% Cnotzero_mono
tff(fact_5322_Cinfinite__Cnotzero,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) ) ) ).
% Cinfinite_Cnotzero
tff(fact_5323_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A)))) ) ).
% Cnotzero_UNIV
tff(fact_5324_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) )
=> bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22)) ) ) ).
% cinfinite_cprod2
tff(fact_5325_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) )
=> ( bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22))
& bNF_Ca8970107618336181345der_on(product_prod(A,B),field2(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22)),bNF_Cardinal_cprod(A,B,R1,R22)) ) ) ) ).
% Cinfinite_cprod2
tff(fact_5326_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P12: set(product_prod(A,A)),P23: set(product_prod(B,B))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P12),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,P12),P12) )
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,P23),P23)
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),P23),bNF_Cardinal_cprod(A,B,P12,P23)),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ) ).
% ordLeq_cprod2
tff(fact_5327_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> pp(member(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(product_unit,product_unit)),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),product_Pair(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),bNF_Cardinal_cone),R),bNF_Wellorder_ordLeq(product_unit,A))) ) ).
% cone_ordLeq_Cnotzero
tff(fact_5328_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),P3: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
( bNF_Ca4139267488887388095finite(A,R)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( pp(member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Cardinal_csum(B,C,P3,P7)),bNF_Cardinal_csum(A,A,R,R)),bNF_Wellorder_ordIso(sum_sum(B,C),sum_sum(A,A))))
=> pp(member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),bNF_Cardinal_csum(B,C,P3,P7)),R),bNF_Wellorder_ordIso(sum_sum(B,C),A))) ) ) ) ).
% csum_dup
tff(fact_5329_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R1),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1) )
=> ( ~ pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Cardinal_csum(A,B,R1,R22)),bNF_Cardinal_czero(sum_sum(A,B))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B))))
& bNF_Ca8970107618336181345der_on(sum_sum(A,B),field2(sum_sum(A,B),bNF_Cardinal_csum(A,B,R1,R22)),bNF_Cardinal_csum(A,B,R1,R22)) ) ) ).
% csum_Cnotzero1
tff(fact_5330_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C)),R42: set(product_prod(D,D))] : pp(member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),fun(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),bNF_Cardinal_csum(sum_sum(A,B),sum_sum(C,D),bNF_Cardinal_csum(A,B,R1,R22),bNF_Cardinal_csum(C,D,R33,R42))),bNF_Cardinal_csum(sum_sum(A,C),sum_sum(B,D),bNF_Cardinal_csum(A,C,R1,R33),bNF_Cardinal_csum(B,D,R22,R42))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))) ).
% csum_csum
tff(fact_5331_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P12: set(product_prod(A,A)),P23: set(product_prod(B,B)),P32: set(product_prod(C,C))] : pp(member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),fun(set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),C),sum_sum(sum_sum(A,B),C))),set(product_prod(sum_sum(A,sum_sum(B,C)),sum_sum(A,sum_sum(B,C))))),bNF_Cardinal_csum(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,P12,P23),P32)),bNF_Cardinal_csum(A,sum_sum(B,C),P12,bNF_Cardinal_csum(B,C,P23,P32))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).
% csum_assoc
tff(fact_5332_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C))] : pp(member(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),fun(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),bNF_Cardinal_csum(product_prod(A,B),product_prod(A,C),bNF_Cardinal_cprod(A,B,R1,R22),bNF_Cardinal_cprod(A,C,R1,R33))),bNF_Cardinal_cprod(A,sum_sum(B,C),R1,bNF_Cardinal_csum(B,C,R22,R33))),bNF_Wellorder_ordIso(sum_sum(product_prod(A,B),product_prod(A,C)),product_prod(A,sum_sum(B,C))))) ).
% cprod_csum_distrib1
tff(fact_5333_cone__not__czero,axiom,
! [A: $tType] : ~ pp(member(product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),aa(set(product_prod(product_unit,product_unit)),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_unit,product_unit)),set(product_prod(A,A)))),product_Pair(set(product_prod(product_unit,product_unit)),set(product_prod(A,A))),bNF_Cardinal_cone),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(product_unit,A))) ).
% cone_not_czero
tff(fact_5334_csum__com,axiom,
! [B: $tType,A: $tType,P12: set(product_prod(A,A)),P23: set(product_prod(B,B))] : pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),bNF_Cardinal_csum(A,B,P12,P23)),bNF_Cardinal_csum(B,A,P23,P12)),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A)))) ).
% csum_com
tff(fact_5335_Un__csum,axiom,
! [A: $tType,A4: set(A),B5: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B5))),bNF_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,A4),bNF_Ca6860139660246222851ard_of(A,B5))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).
% Un_csum
tff(fact_5336_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P12: set(product_prod(A,A)),P23: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,P12),P12)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),P12),bNF_Cardinal_csum(A,B,P12,P23)),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).
% ordLeq_csum1
tff(fact_5337_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P23: set(product_prod(A,A)),P12: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,P23),P23)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),P23),bNF_Cardinal_csum(B,A,P12,P23)),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).
% ordLeq_csum2
tff(fact_5338_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordIso(C,D)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Cardinal_csum(A,C,P12,P23)),bNF_Cardinal_csum(B,D,R1,R22)),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).
% csum_cong
tff(fact_5339_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Cardinal_csum(A,C,P12,Q3)),bNF_Cardinal_csum(B,C,R1,Q3)),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).
% csum_cong1
tff(fact_5340_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordIso(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Cardinal_csum(C,A,Q3,P23)),bNF_Cardinal_csum(C,B,Q3,R22)),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).
% csum_cong2
tff(fact_5341_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordLeq(C,D)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,D),sum_sum(B,D))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D)))),bNF_Cardinal_csum(A,C,P12,P23)),bNF_Cardinal_csum(B,D,R1,R22)),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).
% csum_mono
tff(fact_5342_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(sum_sum(B,C),sum_sum(B,C))),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C))))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,C),sum_sum(B,C)))),bNF_Cardinal_csum(A,C,P12,Q3)),bNF_Cardinal_csum(B,C,R1,Q3)),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).
% csum_mono1
tff(fact_5343_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordLeq(A,B)))
=> pp(member(product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),aa(set(product_prod(sum_sum(C,A),sum_sum(C,A))),fun(set(product_prod(sum_sum(C,B),sum_sum(C,B))),product_prod(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B))))),product_Pair(set(product_prod(sum_sum(C,A),sum_sum(C,A))),set(product_prod(sum_sum(C,B),sum_sum(C,B)))),bNF_Cardinal_csum(C,A,Q3,P23)),bNF_Cardinal_csum(C,B,Q3,R22)),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).
% csum_mono2
tff(fact_5344_Plus__csum,axiom,
! [B: $tType,A: $tType,A4: set(A),B5: set(B)] : pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B5))),bNF_Cardinal_csum(A,B,bNF_Ca6860139660246222851ard_of(A,A4),bNF_Ca6860139660246222851ard_of(B,B5))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B)))) ).
% Plus_csum
tff(fact_5345_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R22),R22)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R1),R22),bNF_Wellorder_ordLeq(B,A)))
=> ( ( bNF_Ca4139267488887388095finite(B,R1)
| bNF_Ca4139267488887388095finite(A,R22) )
=> pp(member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Cardinal_csum(B,A,R1,R22)),R22),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ).
% csum_absorb2'
tff(fact_5346_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R22),R22)
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R1),R22),bNF_Wellorder_ordLeq(B,A)))
=> ( ( bNF_Ca4139267488887388095finite(B,R1)
| bNF_Ca4139267488887388095finite(A,R22) )
=> pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Cardinal_csum(A,B,R22,R1)),R22),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ) ).
% csum_absorb1'
tff(fact_5347_csum__absorb1,axiom,
! [B: $tType,A: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
( ( bNF_Ca4139267488887388095finite(A,R22)
& bNF_Ca8970107618336181345der_on(A,field2(A,R22),R22) )
=> ( pp(member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),R1),R22),bNF_Wellorder_ordLeq(B,A)))
=> pp(member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Cardinal_csum(A,B,R22,R1)),R22),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ).
% csum_absorb1
tff(fact_5348_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P3: set(product_prod(A,A)),R: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P3),R),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),Q3),R),bNF_Wellorder_ordLeq(C,B)))
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P3),P3)
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,field2(B,R),R) )
=> pp(member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),bNF_Cardinal_csum(A,C,P3,Q3)),R),bNF_Wellorder_ordLeq(sum_sum(A,C),B))) ) ) ) ) ) ).
% csum_cinfinite_bound
tff(fact_5349_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),R),bNF_Wellorder_ordLeq(nat,A))) ) ).
% natLeq_ordLeq_cinfinite
tff(fact_5350_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),P23),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,P23),P23) )
=> pp(member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P23)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).
% cexp_mono2_Cnotzero
tff(fact_5351_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D))),aa(set(product_prod(A,A)),fun(set(product_prod(D,D)),product_prod(set(product_prod(A,A)),set(product_prod(D,D)))),product_Pair(set(product_prod(A,A)),set(product_prod(D,D))),P23),bNF_Cardinal_czero(D)),bNF_Wellorder_ordIso(A,D)))
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),R22),bNF_Cardinal_czero(E)),bNF_Wellorder_ordIso(B,E))) )
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P23),P23)
=> pp(member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P23)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ) ).
% cexp_mono2
tff(fact_5352_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(B,B)),S: set(product_prod(C,C)),T2: set(product_prod(A,A))] : pp(member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R,S),T2)),bNF_Cardinal_cprod(fun(A,B),fun(A,C),bNF_Cardinal_cexp(B,A,R,T2),bNF_Cardinal_cexp(C,A,S,T2))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).
% cprod_cexp
tff(fact_5353_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(C,C)),R33: set(product_prod(B,B))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1)
=> pp(member(product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),aa(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),fun(set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))),product_prod(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A))))),product_Pair(set(product_prod(fun(B,fun(C,A)),fun(B,fun(C,A)))),set(product_prod(fun(product_prod(C,B),A),fun(product_prod(C,B),A)))),bNF_Cardinal_cexp(fun(C,A),B,bNF_Cardinal_cexp(A,C,R1,R22),R33)),bNF_Cardinal_cexp(A,product_prod(C,B),R1,bNF_Cardinal_cprod(C,B,R22,R33))),bNF_Wellorder_ordIso(fun(B,fun(C,A)),fun(product_prod(C,B),A)))) ) ).
% cexp_cprod
tff(fact_5354_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set(product_prod(A,A)),R: set(product_prod(B,B)),S: set(product_prod(C,C))] :
( ( bNF_Ca4139267488887388095finite(A,T2)
& bNF_Ca8970107618336181345der_on(A,field2(A,T2),T2) )
=> pp(member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R,S),T2)),bNF_Cardinal_cexp(sum_sum(B,C),A,bNF_Cardinal_csum(B,C,R,S),T2)),bNF_Wellorder_ordLeq(fun(A,product_prod(B,C)),fun(A,sum_sum(B,C))))) ) ).
% cprod_cexp_csum_cexp_Cinfinite
tff(fact_5355_card__of__Field__natLeq,axiom,
pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,field2(nat,bNF_Ca8665028551170535155natLeq))),bNF_Ca8665028551170535155natLeq),bNF_Wellorder_ordIso(nat,nat))) ).
% card_of_Field_natLeq
tff(fact_5356_cexp__cone,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),aa(set(product_prod(fun(product_unit,A),fun(product_unit,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(fun(product_unit,A),fun(product_unit,A))),set(product_prod(A,A))),bNF_Cardinal_cexp(A,product_unit,R,bNF_Cardinal_cone)),R),bNF_Wellorder_ordIso(fun(product_unit,A),A))) ) ).
% cexp_cone
tff(fact_5357_finite__iff__ordLess__natLeq,axiom,
! [A: $tType,A4: set(A)] :
( finite_finite2(A,A4)
<=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(A,A)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(A,A)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca8665028551170535155natLeq),bNF_We4044943003108391690rdLess(A,nat))) ) ).
% finite_iff_ordLess_natLeq
tff(fact_5358_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R1),R1)
=> ( ( bNF_Ca4139267488887388095finite(B,R22)
& bNF_Ca8970107618336181345der_on(B,field2(B,R22),R22) )
=> ( ( ~ pp(member(product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C))),aa(set(product_prod(C,C)),fun(set(product_prod(C,C)),product_prod(set(product_prod(C,C)),set(product_prod(C,C)))),product_Pair(set(product_prod(C,C)),set(product_prod(C,C))),R33),bNF_Cardinal_czero(C)),bNF_Wellorder_ordIso(C,C)))
& bNF_Ca8970107618336181345der_on(C,field2(C,R33),R33) )
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),R33),R22),bNF_Wellorder_ordLeq(C,B)))
=> pp(member(product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(fun(C,fun(B,A)),fun(C,fun(B,A)))),set(product_prod(fun(B,A),fun(B,A)))),bNF_Cardinal_cexp(fun(B,A),C,bNF_Cardinal_cexp(A,B,R1,R22),R33)),bNF_Cardinal_cexp(A,B,R1,R22)),bNF_Wellorder_ordIso(fun(C,fun(B,A)),fun(B,A)))) ) ) ) ) ).
% cexp_cprod_ordLeq
tff(fact_5359_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordLeq(C,D)))
=> ( ( ( field2(C,P23) = bot_bot(set(C)) )
=> ( field2(D,R22) = bot_bot(set(D)) ) )
=> pp(member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P12,P23)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ).
% cexp_mono'
tff(fact_5360_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> pp(member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P12,Q3)),bNF_Cardinal_cexp(B,C,R1,Q3)),bNF_Wellorder_ordLeq(fun(C,A),fun(C,B)))) ) ) ).
% cexp_mono1
tff(fact_5361_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordIso(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( bNF_Ca8970107618336181345der_on(A,field2(A,P23),P23)
=> pp(member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P23)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordIso(fun(A,C),fun(B,C)))) ) ) ) ).
% cexp_cong2
tff(fact_5362_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordIso(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> pp(member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(C,B),fun(C,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(C,B),fun(C,B)))),bNF_Cardinal_cexp(A,C,P12,Q3)),bNF_Cardinal_cexp(B,C,R1,Q3)),bNF_Wellorder_ordIso(fun(C,A),fun(C,B)))) ) ) ).
% cexp_cong1
tff(fact_5363_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordIso(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordIso(C,D)))
=> ( bNF_Ca8970107618336181345der_on(D,field2(D,R22),R22)
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,P23),P23)
=> pp(member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P12,P23)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordIso(fun(C,A),fun(D,B)))) ) ) ) ) ).
% cexp_cong
tff(fact_5364_card__of__nat,axiom,
pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca8665028551170535155natLeq),bNF_Wellorder_ordIso(nat,nat))) ).
% card_of_nat
tff(fact_5365_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P23: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q3: set(product_prod(C,C))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P23),R22),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,Q3),Q3)
=> ( ( ( field2(A,P23) = bot_bot(set(A)) )
=> ( field2(B,R22) = bot_bot(set(B)) ) )
=> pp(member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(A,C),fun(A,C))),fun(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),bNF_Cardinal_cexp(C,A,Q3,P23)),bNF_Cardinal_cexp(C,B,Q3,R22)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).
% cexp_mono2'
tff(fact_5366_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),Q3: set(product_prod(B,B))] :
( ( ~ pp(member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A)))
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,Q3),Q3)
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(A,B),fun(A,B)))),Q3),bNF_Cardinal_cexp(B,A,Q3,R)),bNF_Wellorder_ordLeq(B,fun(A,B)))) ) ) ).
% ordLeq_cexp1
tff(fact_5367_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A4: set(A)] :
~ ( finite_finite2(A,A4)
<=> pp(member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(nat,nat)),fun(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A))),bNF_Ca8665028551170535155natLeq),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(nat,A))) ) ).
% infinite_iff_natLeq_ordLeq
tff(fact_5368_cexp__mono,axiom,
! [E: $tType,F4: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P12: set(product_prod(A,A)),R1: set(product_prod(B,B)),P23: set(product_prod(C,C)),R22: set(product_prod(D,D))] :
( pp(member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),P12),R1),bNF_Wellorder_ordLeq(A,B)))
=> ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(C,C)),fun(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D)))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D))),P23),R22),bNF_Wellorder_ordLeq(C,D)))
=> ( ( pp(member(product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E))),aa(set(product_prod(C,C)),fun(set(product_prod(E,E)),product_prod(set(product_prod(C,C)),set(product_prod(E,E)))),product_Pair(set(product_prod(C,C)),set(product_prod(E,E))),P23),bNF_Cardinal_czero(E)),bNF_Wellorder_ordIso(C,E)))
=> pp(member(product_prod(set(product_prod(D,D)),set(product_prod(F4,F4))),aa(set(product_prod(F4,F4)),product_prod(set(product_prod(D,D)),set(product_prod(F4,F4))),aa(set(product_prod(D,D)),fun(set(product_prod(F4,F4)),product_prod(set(product_prod(D,D)),set(product_prod(F4,F4)))),product_Pair(set(product_prod(D,D)),set(product_prod(F4,F4))),R22),bNF_Cardinal_czero(F4)),bNF_Wellorder_ordIso(D,F4))) )
=> ( bNF_Ca8970107618336181345der_on(C,field2(C,P23),P23)
=> pp(member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(C,A),fun(C,A))),fun(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),bNF_Cardinal_cexp(A,C,P12,P23)),bNF_Cardinal_cexp(B,D,R1,R22)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ) ).
% cexp_mono
tff(fact_5369_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3),bNF_Wellorder_ordLeq(bool,A)))
=> ( bNF_Ca8970107618336181345der_on(B,field2(B,R),R)
=> pp(member(product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),aa(set(product_prod(B,B)),fun(set(product_prod(fun(B,A),fun(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A))))),product_Pair(set(product_prod(B,B)),set(product_prod(fun(B,A),fun(B,A)))),R),bNF_Cardinal_cexp(A,B,Q3,R)),bNF_Wellorder_ordLeq(B,fun(B,A)))) ) ) ).
% ordLeq_cexp2
tff(fact_5370_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S: set(product_prod(A,A)),T2: set(product_prod(B,B))] :
( ( bNF_Cardinal_cfinite(A,S)
& bNF_Ca8970107618336181345der_on(A,field2(A,S),S) )
=> ( ( bNF_Ca4139267488887388095finite(B,T2)
& bNF_Ca8970107618336181345der_on(B,field2(B,T2),T2) )
=> pp(member(product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool)))),aa(set(product_prod(fun(B,bool),fun(B,bool))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool)))),aa(set(product_prod(fun(B,A),fun(B,A))),fun(set(product_prod(fun(B,bool),fun(B,bool))),product_prod(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool))))),product_Pair(set(product_prod(fun(B,A),fun(B,A))),set(product_prod(fun(B,bool),fun(B,bool)))),bNF_Cardinal_cexp(A,B,S,T2)),bNF_Cardinal_cexp(bool,B,bNF_Cardinal_ctwo,T2)),bNF_Wellorder_ordLeq(fun(B,A),fun(B,bool)))) ) ) ).
% Cfinite_cexp_Cinfinite
tff(fact_5371_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3),bNF_Wellorder_ordLeq(bool,A)))
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,field2(B,R),R) )
=> bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R)) ) ) ).
% cinfinite_cexp
tff(fact_5372_ctwo__Cnotzero,axiom,
( ~ pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool))),aa(set(product_prod(bool,bool)),product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool))),aa(set(product_prod(bool,bool)),fun(set(product_prod(bool,bool)),product_prod(set(product_prod(bool,bool)),set(product_prod(bool,bool)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(bool,bool))),bNF_Cardinal_ctwo),bNF_Cardinal_czero(bool)),bNF_Wellorder_ordIso(bool,bool)))
& bNF_Ca8970107618336181345der_on(bool,field2(bool,bNF_Cardinal_ctwo),bNF_Cardinal_ctwo) ) ).
% ctwo_Cnotzero
tff(fact_5373_ctwo__ordLess__natLeq,axiom,
pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat))),aa(set(product_prod(bool,bool)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(bool,bool)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(nat,nat))),bNF_Cardinal_ctwo),bNF_Ca8665028551170535155natLeq),bNF_We4044943003108391690rdLess(bool,nat))) ).
% ctwo_ordLess_natLeq
tff(fact_5374_ordLess__ctwo__cexp,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(A,A)),fun(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool))))),product_Pair(set(product_prod(A,A)),set(product_prod(fun(A,bool),fun(A,bool)))),R),bNF_Cardinal_cexp(bool,A,bNF_Cardinal_ctwo,R)),bNF_We4044943003108391690rdLess(A,fun(A,bool)))) ) ).
% ordLess_ctwo_cexp
tff(fact_5375_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] : pp(member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B))))),aa(set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B))))),bNF_Cardinal_cprod(A,B,R1,R22)),bNF_Cardinal_cexp(sum_sum(A,B),bool,bNF_Cardinal_csum(A,B,R1,R22),bNF_Cardinal_ctwo)),bNF_Wellorder_ordLeq(product_prod(A,B),fun(bool,sum_sum(A,B))))) ).
% cprod_csum_cexp
tff(fact_5376_ctwo__not__czero,axiom,
! [A: $tType] : ~ pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(bool,A))) ).
% ctwo_not_czero
tff(fact_5377_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R),bNF_Wellorder_ordLeq(bool,A))) ) ).
% ctwo_ordLeq_Cinfinite
tff(fact_5378_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,field2(A,R),R) )
=> pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),R),bNF_We4044943003108391690rdLess(bool,A))) ) ).
% ctwo_ordLess_Cinfinite
tff(fact_5379_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B)),T2: set(product_prod(C,C))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> ( ( bNF_Cardinal_cfinite(B,S)
& bNF_Ca8970107618336181345der_on(B,field2(B,S),S) )
=> ( ( bNF_Ca4139267488887388095finite(C,T2)
& bNF_Ca8970107618336181345der_on(C,field2(C,T2),T2) )
=> pp(member(product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool))))),aa(set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool))))),aa(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),fun(set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))))),product_Pair(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool))))),bNF_Cardinal_cexp(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,R,S),T2)),bNF_Cardinal_cexp(sum_sum(A,bool),C,bNF_Cardinal_csum(A,bool,R,bNF_Cardinal_ctwo),T2)),bNF_Wellorder_ordLeq(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,bool))))) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
tff(fact_5380_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set(product_prod(A,A)),R: set(product_prod(B,B))] :
( pp(member(product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A))),aa(set(product_prod(bool,bool)),fun(set(product_prod(A,A)),product_prod(set(product_prod(bool,bool)),set(product_prod(A,A)))),product_Pair(set(product_prod(bool,bool)),set(product_prod(A,A))),bNF_Cardinal_ctwo),Q3),bNF_Wellorder_ordLeq(bool,A)))
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,field2(B,R),R) )
=> ( bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R))
& bNF_Ca8970107618336181345der_on(fun(B,A),field2(fun(B,A),bNF_Cardinal_cexp(A,B,Q3,R)),bNF_Cardinal_cexp(A,B,Q3,R)) ) ) ) ).
% Cinfinite_cexp
tff(fact_5381_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B5: set(B)] : pp(member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),B5))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_vc(set(B),fun(fun(A,B),bool),B5)))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B)))) ).
% card_of_Func_UNIV
tff(fact_5382_Card__order__Pow,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,field2(A,R),R)
=> pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca6860139660246222851ard_of(set(A),pow(A,field2(A,R)))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).
% Card_order_Pow
tff(fact_5383_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] : pp(member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(product_prod(fun(A,B),fun(A,C)),product_prod(fun(A,B),fun(A,C))))),bNF_Ca6860139660246222851ard_of(fun(A,product_prod(B,C)),bNF_Wellorder_Func(A,product_prod(B,C),A4,product_Sigma(B,C,B5,aTP_Lamp_mz(set(C),fun(B,set(C)),C5))))),bNF_Ca6860139660246222851ard_of(product_prod(fun(A,B),fun(A,C)),product_Sigma(fun(A,B),fun(A,C),bNF_Wellorder_Func(A,B,A4,B5),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_vd(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),A4),C5)))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).
% Func_Times_Range
tff(fact_5384_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set(A),B5: set(B),C5: set(C)] : pp(member(product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),aa(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),fun(set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))),product_prod(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C)))))),product_Pair(set(product_prod(fun(product_prod(A,B),C),fun(product_prod(A,B),C))),set(product_prod(fun(A,fun(B,C)),fun(A,fun(B,C))))),bNF_Ca6860139660246222851ard_of(fun(product_prod(A,B),C),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A4,aTP_Lamp_jm(set(B),fun(A,set(B)),B5)),C5))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),A4,bNF_Wellorder_Func(B,C,B5,C5)))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C))))) ).
% card_of_Func_Times
tff(fact_5385_card__of__Pow__Func,axiom,
! [A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(fun(A,bool),fun(A,bool))),product_prod(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool))))),product_Pair(set(product_prod(set(A),set(A))),set(product_prod(fun(A,bool),fun(A,bool)))),bNF_Ca6860139660246222851ard_of(set(A),pow(A,A4))),bNF_Ca6860139660246222851ard_of(fun(A,bool),bNF_Wellorder_Func(A,bool,A4,top_top(set(bool))))),bNF_Wellorder_ordIso(set(A),fun(A,bool)))) ).
% card_of_Pow_Func
tff(fact_5386_Cantors__paradox,axiom,
! [A: $tType,A4: set(A)] :
~ ? [F7: fun(A,set(A))] : aa(set(A),set(set(A)),image2(A,set(A),F7),A4) = pow(A,A4) ).
% Cantors_paradox
tff(fact_5387_inj__on__image__Pow,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,A4)
=> inj_on(set(A),set(B),image2(A,B,F),pow(A,A4)) ) ).
% inj_on_image_Pow
tff(fact_5388_card__of__Pow,axiom,
! [A: $tType,A4: set(A)] : pp(member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(set(A),pow(A,A4))),bNF_We4044943003108391690rdLess(A,set(A)))) ).
% card_of_Pow
tff(fact_5389_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,bool))] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2),complete_lattice_lfp(set(product_prod(A,B)),F)))
=> ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F)
=> ( ! [A5: A,B3: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3),aa(set(product_prod(A,B)),set(product_prod(A,B)),F,aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),complete_lattice_lfp(set(product_prod(A,B)),F)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P))))))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3)) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) ) ) ) ).
% lfp_induct2
tff(fact_5390_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_sort_key(A,A,aTP_Lamp_ve(A,A)) = quicksort_by_rel(A,ord_less_eq(A),nil(A)) ) ) ).
% sort_quicksort_by_rel
tff(fact_5391_Gr__def,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : bNF_Gr(A,B,A4,F) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_vf(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A4),F)) ).
% Gr_def
tff(fact_5392_GrD2,axiom,
! [A: $tType,B: $tType,X2: A,Fx: B,A4: set(A),F: fun(A,B)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Fx),bNF_Gr(A,B,A4,F)))
=> ( aa(A,B,F,X2) = Fx ) ) ).
% GrD2
tff(fact_5393_GrD1,axiom,
! [B: $tType,A: $tType,X2: A,Fx: B,A4: set(A),F: fun(A,B)] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Fx),bNF_Gr(A,B,A4,F)))
=> pp(member(A,X2,A4)) ) ).
% GrD1
tff(fact_5394_sort__mergesort,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_sort_key(A,A,aTP_Lamp_ve(A,A)) = mergesort(A) ) ) ).
% sort_mergesort
tff(fact_5395_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( linorder(A)
=> ( linorder_sort_key(A,A,aTP_Lamp_ve(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).
% sort_mergesort_by_rel
tff(fact_5396_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( linorder(A)
=> ( aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(fun(list(A),list(A)),fun(list(A),list(A))),comp(list(A),list(A),list(A)),remdups(A)),linorder_sort_key(A,A,aTP_Lamp_ve(A,A))) = mergesort_remdups(A) ) ) ).
% remdup_sort_mergesort_remdups
tff(fact_5397_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ( groups1962203154675924110t_prod(C,A) = groups_comm_monoid_G(A,C,times_times(A),one_one(A)) ) ) ).
% prod'_def
tff(fact_5398_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> groups4802862169904069756st_set(A,times_times(A),one_one(A)) ) ).
% prod.comm_monoid_list_set_axioms
tff(fact_5399_length__remdups__card,axiom,
! [A: $tType,L: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),remdups(A),L)) = finite_card(A,aa(list(A),set(A),set2(A),L)) ).
% length_remdups_card
tff(fact_5400_sum_H__def,axiom,
! [C: $tType,A: $tType] :
( comm_monoid_add(A)
=> ( groups1027152243600224163dd_sum(C,A) = groups_comm_monoid_G(A,C,plus_plus(A),zero_zero(A)) ) ) ).
% sum'_def
tff(fact_5401_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> groups4802862169904069756st_set(A,plus_plus(A),zero_zero(A)) ) ).
% sum.comm_monoid_list_set_axioms
tff(fact_5402_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> comm_monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).
% and.comm_monoid_axioms
tff(fact_5403_bijective__alt,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B))] :
( bijective(A,B,R4)
<=> ( single_valued(A,B,R4)
& single_valued(B,A,converse(A,B,R4)) ) ) ).
% bijective_alt
tff(fact_5404_Heap_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),nat),aa(fun(heap_Time_Heap(B),nat),fun(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),nat)),comp(heap_Time_Heap(B),nat,heap_Time_Heap(A)),heap_Time_size_Heap(B,F)),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G)) = heap_Time_size_Heap(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% Heap.size_gen_o_map
tff(fact_5405_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( single_valued(A,B,R4)
=> single_valued(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),R4),S2)) ) ).
% single_valued_inter1
tff(fact_5406_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
( single_valued(A,B,R4)
=> single_valued(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),inf_inf(set(product_prod(A,B)),S2),R4)) ) ).
% single_valued_inter2
tff(fact_5407_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: A] :
( comm_monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),Z2) = A3 ) ) ).
% comm_monoid.comm_neutral
tff(fact_5408_Heap_Omap__ident,axiom,
! [A: $tType,T2: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(fun(A,A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),heap_Time_map_Heap(A,A),aTP_Lamp_ae(A,A)),T2) = T2 ).
% Heap.map_ident
tff(fact_5409_Heap_Omap__id0,axiom,
! [A: $tType] : aa(fun(A,A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),heap_Time_map_Heap(A,A),id(A)) = id(heap_Time_Heap(A)) ).
% Heap.map_id0
tff(fact_5410_Heap_Omap__id,axiom,
! [A: $tType,T2: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(fun(A,A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),heap_Time_map_Heap(A,A),id(A)),T2) = T2 ).
% Heap.map_id
tff(fact_5411_Heap_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: heap_Time_Heap(A)] : aa(heap_Time_Heap(B),heap_Time_Heap(C),aa(fun(B,C),fun(heap_Time_Heap(B),heap_Time_Heap(C)),heap_Time_map_Heap(B,C),G),aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),V)) = aa(heap_Time_Heap(A),heap_Time_Heap(C),aa(fun(A,C),fun(heap_Time_Heap(A),heap_Time_Heap(C)),heap_Time_map_Heap(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% Heap.map_comp
tff(fact_5412_Heap_Omap,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),map_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_map_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),F,id(product_prod(heap_ext(product_unit),nat))))),X2)) ).
% Heap.map
tff(fact_5413_single__valuedD,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,B)),X2: A,Y: B,Z2: B] :
( single_valued(A,B,R)
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y),R))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z2),R))
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
tff(fact_5414_single__valuedI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( ! [X3: A,Y4: B,Z3: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y4),R))
=> ( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Z3),R))
=> ( Y4 = Z3 ) ) )
=> single_valued(A,B,R) ) ).
% single_valuedI
tff(fact_5415_single__valued__def,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,B))] :
( single_valued(A,B,R)
<=> ! [X4: A,Y3: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),R))
=> ! [Z4: B] :
( pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4),R))
=> ( Y3 = Z4 ) ) ) ) ).
% single_valued_def
tff(fact_5416_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> comm_monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.comm_monoid_axioms
tff(fact_5417_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> comm_monoid(A,times_times(A),one_one(A)) ) ).
% mult.comm_monoid_axioms
tff(fact_5418_Heap_Oinj__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),top_top(set(heap_Time_Heap(A)))) ) ).
% Heap.inj_map
tff(fact_5419_single__valued__confluent,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y: A,Z2: A] :
( single_valued(A,A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2),transitive_rtrancl(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),transitive_rtrancl(A,R)))
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y),transitive_rtrancl(A,R))) ) ) ) ) ).
% single_valued_confluent
tff(fact_5420_single__valued__below__Id,axiom,
! [A: $tType,R4: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R4),id2(A)))
=> single_valued(A,A,R4) ) ).
% single_valued_below_Id
tff(fact_5421_Heap_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),F: fun(A,B)] : aa(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),C),aa(fun(heap_Time_Heap(B),C),fun(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),C)),comp(heap_Time_Heap(B),C,heap_Time_Heap(A)),aa(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(B),C),heap_Time_rec_Heap(B,C),G)),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F)) = aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C),heap_Time_rec_Heap(A,C),aa(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_vg(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),G),F)) ).
% Heap.rec_o_map
tff(fact_5422_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R))
<=> single_valued(A,B,R) ) ).
% single_valuedp_single_valued_eq
tff(fact_5423_disjnt__equiv__class,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,A4,R)
=> ( disjnt(A,image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))),image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))
<=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)) ) ) ).
% disjnt_equiv_class
tff(fact_5424_equiv__class__eq__iff,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
<=> ( ( image(A,A,R,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) )
& pp(member(A,X2,A4))
& pp(member(A,Y,A4)) ) ) ) ).
% equiv_class_eq_iff
tff(fact_5425_eq__equiv__class__iff,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( ( image(A,A,R,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert2(A,Y),bot_bot(set(A)))) )
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R)) ) ) ) ) ).
% eq_equiv_class_iff
tff(fact_5426_equiv__class__eq,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> ( image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ) ) ).
% equiv_class_eq
tff(fact_5427_eq__equiv__class,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,A4: set(A)] :
( ( image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) )
=> ( equiv_equiv(A,A4,R)
=> ( pp(member(A,B2,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)) ) ) ) ).
% eq_equiv_class
tff(fact_5428_Heap_Orec,axiom,
! [C: $tType,A: $tType,F: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C),heap_Time_rec_Heap(A,C),F),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,F,X2) ).
% Heap.rec
tff(fact_5429_subset__equiv__class,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),B2: A,A3: A] :
( equiv_equiv(A,A4,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))))
=> ( pp(member(A,B2,A4))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)) ) ) ) ).
% subset_equiv_class
tff(fact_5430_equiv__class__subset,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))) ) ) ).
% equiv_class_subset
tff(fact_5431_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,A3: A,B2: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(A,X2,aa(set(A),set(A),inf_inf(set(A),image(A,A,R,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),image(A,A,R,aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R)) ) ) ).
% equiv_class_nondisjoint
tff(fact_5432_proj__iff,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X2),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))),A4))
=> ( ( equiv_proj(A,A,R,X2) = equiv_proj(A,A,R,Y) )
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R)) ) ) ) ).
% proj_iff
tff(fact_5433_in__quotient__imp__in__rel,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X7: set(A),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(set(A),X7,equiv_quotient(A,A4,R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X2),aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))))),X7))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R)) ) ) ) ).
% in_quotient_imp_in_rel
tff(fact_5434_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(A,A)),F: fun(A,fun(A,B))] :
( equiv_equiv(A,A4,R)
=> ( ! [Y4: A,Z3: A] :
( pp(member(A,Y4,A4))
=> ( pp(member(A,Z3,A4))
=> ( aa(A,B,aa(A,fun(A,B),F,Y4),Z3) = aa(A,B,aa(A,fun(A,B),F,Z3),Y4) ) ) )
=> ( ! [Y4: A,Z3: A,W3: A] :
( pp(member(A,W3,A4))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> ( aa(A,B,aa(A,fun(A,B),F,W3),Y4) = aa(A,B,aa(A,fun(A,B),F,W3),Z3) ) ) )
=> equiv_congruent2(A,A,B,R,R,F) ) ) ) ).
% congruent2_commuteI
tff(fact_5435_quotient__eqI,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X7: set(A),Y7: set(A),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(set(A),X7,equiv_quotient(A,A4,R)))
=> ( pp(member(set(A),Y7,equiv_quotient(A,A4,R)))
=> ( pp(member(A,X2,X7))
=> ( pp(member(A,Y,Y7))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> ( X7 = Y7 ) ) ) ) ) ) ) ).
% quotient_eqI
tff(fact_5436_quotient__eq__iff,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X7: set(A),Y7: set(A),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(set(A),X7,equiv_quotient(A,A4,R)))
=> ( pp(member(set(A),Y7,equiv_quotient(A,A4,R)))
=> ( pp(member(A,X2,X7))
=> ( pp(member(A,Y,Y7))
=> ( ( X7 = Y7 )
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R)) ) ) ) ) ) ) ).
% quotient_eq_iff
tff(fact_5437_in__quotient__imp__closed,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X7: set(A),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(set(A),X7,equiv_quotient(A,A4,R)))
=> ( pp(member(A,X2,X7))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R))
=> pp(member(A,Y,X7)) ) ) ) ) ).
% in_quotient_imp_closed
tff(fact_5438_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,fun(B,C)),Y1: A,Z1: A,Y2: B,Z22: B] :
( equiv_congruent2(A,B,C,R1,R22,F)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1),R1))
=> ( pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z22),R22))
=> ( aa(B,C,aa(A,fun(B,C),F,Y1),Y2) = aa(B,C,aa(A,fun(B,C),F,Z1),Z22) ) ) ) ) ).
% congruent2D
tff(fact_5439_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( ! [Y12: A,Z12: A,Y22: B,Z23: B] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z12),R1))
=> ( pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z23),R22))
=> ( aa(B,C,aa(A,fun(B,C),F,Y12),Y22) = aa(B,C,aa(A,fun(B,C),F,Z12),Z23) ) ) )
=> equiv_congruent2(A,B,C,R1,R22,F) ) ).
% congruent2I'
tff(fact_5440_eq__equiv__class__iff2,axiom,
! [A: $tType,A4: set(A),R: set(product_prod(A,A)),X2: A,Y: A] :
( equiv_equiv(A,A4,R)
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( ( equiv_quotient(A,aa(set(A),set(A),insert2(A,X2),bot_bot(set(A))),R) = equiv_quotient(A,aa(set(A),set(A),insert2(A,Y),bot_bot(set(A))),R) )
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),R)) ) ) ) ) ).
% eq_equiv_class_iff2
tff(fact_5441_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A17: set(A),R1: set(product_prod(A,A)),A25: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( equiv_equiv(A,A17,R1)
=> ( equiv_equiv(B,A25,R22)
=> ( ! [Y4: A,Z3: A,W3: B] :
( pp(member(B,W3,A25))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R1))
=> ( aa(B,C,aa(A,fun(B,C),F,Y4),W3) = aa(B,C,aa(A,fun(B,C),F,Z3),W3) ) ) )
=> ( ! [Y4: B,Z3: B,W3: A] :
( pp(member(A,W3,A17))
=> ( pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y4),Z3),R22))
=> ( aa(B,C,aa(A,fun(B,C),F,W3),Y4) = aa(B,C,aa(A,fun(B,C),F,W3),Z3) ) ) )
=> equiv_congruent2(A,B,C,R1,R22,F) ) ) ) ) ).
% congruent2I
tff(fact_5442_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A4: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),X7: set(A),Y7: set(A)] :
( equiv_equiv(A,A4,R)
=> ( equiv_congruent(A,set(B),R,F)
=> ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X7)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),Y7)) )
=> ( pp(member(set(A),X7,equiv_quotient(A,A4,R)))
=> ( pp(member(set(A),Y7,equiv_quotient(A,A4,R)))
=> ( ! [X3: A,Y4: A] :
( pp(member(A,X3,A4))
=> ( pp(member(A,Y4,A4))
=> ( ( aa(A,set(B),F,X3) = aa(A,set(B),F,Y4) )
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y4),R)) ) ) )
=> ( X7 = Y7 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
tff(fact_5443_equivp__equiv,axiom,
! [A: $tType,A4: set(product_prod(A,A))] :
( equiv_equiv(A,top_top(set(A)),A4)
<=> equiv_equivp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),A4)) ) ).
% equivp_equiv
tff(fact_5444_prod__mset__def,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ( comm_m9189036328036947845d_mset(A) = comm_monoid_F(A,times_times(A),one_one(A)) ) ) ).
% prod_mset_def
tff(fact_5445_congruentI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
( ! [Y4: A,Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z3),R))
=> ( aa(A,B,F,Y4) = aa(A,B,F,Z3) ) )
=> equiv_congruent(A,B,R,F) ) ).
% congruentI
tff(fact_5446_congruentD,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B),Y: A,Z2: A] :
( equiv_congruent(A,B,R,F)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2),R))
=> ( aa(A,B,F,Y) = aa(A,B,F,Z2) ) ) ) ).
% congruentD
tff(fact_5447_sum__mset__def,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ( comm_m7189776963980413722m_mset(A) = comm_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).
% sum_mset_def
tff(fact_5448_or__num_Opelims,axiom,
! [X2: num,Xa: num,Y: num] :
( ( bit_un6697907153464112080or_num(X2,Xa) = Y )
=> ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),X2),Xa)))
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(num,num,bit1,N3) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit0,N3)))) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(num,num,bit1,N3) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,num,bit1,M4) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(num,num,bit0,bit_un6697907153464112080or_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit0,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit0,M4)),aa(num,num,bit1,N3)))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ( ( Xa = one2 )
=> ( ( Y = aa(num,num,bit1,M4) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2))) ) ) )
=> ( ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit0,N3) )
=> ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit0,N3)))) ) ) )
=> ~ ! [M4: num] :
( ( X2 = aa(num,num,bit1,M4) )
=> ! [N3: num] :
( ( Xa = aa(num,num,bit1,N3) )
=> ( ( Y = aa(num,num,bit1,bit_un6697907153464112080or_num(M4,N3)) )
=> ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4773296044027857193um_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N3)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
tff(fact_5449_Heap_Oset,axiom,
! [A: $tType,X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_set_Heap(A,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),set(set(A)),image2(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A),aTP_Lamp_vh(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A))),aa(set(heap_ext(product_unit)),set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),image2(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X2),top_top(set(heap_ext(product_unit)))))) ).
% Heap.set
tff(fact_5450_under__incr,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( trans(A,R)
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_under(A,R,A3)),order_under(A,R,B2))) ) ) ).
% under_incr
tff(fact_5451_Heap_Oset__intros,axiom,
! [A: $tType,Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Ya: product_prod(A,product_prod(heap_ext(product_unit),nat)),Yb2: A] :
( pp(member(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y,aa(set(heap_ext(product_unit)),set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),image2(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X2),top_top(set(heap_ext(product_unit))))))
=> ( pp(member(product_prod(A,product_prod(heap_ext(product_unit),nat)),Ya,aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y)))
=> ( pp(member(A,Yb2,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),set(A),basic_fsts(A,product_prod(heap_ext(product_unit),nat)),Ya)))
=> pp(member(A,Yb2,heap_Time_set_Heap(A,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)))) ) ) ) ).
% Heap.set_intros
tff(fact_5452_Heap_Oset__cases,axiom,
! [A: $tType,E2: A,A3: heap_Time_Heap(A)] :
( pp(member(A,E2,heap_Time_set_Heap(A,A3)))
=> ~ ! [Z3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( ( A3 = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Z3) )
=> ! [Xa4: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
( pp(member(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa4,aa(set(heap_ext(product_unit)),set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),image2(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Z3),top_top(set(heap_ext(product_unit))))))
=> ! [Xb3: product_prod(A,product_prod(heap_ext(product_unit),nat))] :
( pp(member(product_prod(A,product_prod(heap_ext(product_unit),nat)),Xb3,aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa4)))
=> ~ pp(member(A,E2,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),set(A),basic_fsts(A,product_prod(heap_ext(product_unit),nat)),Xb3))) ) ) ) ) ).
% Heap.set_cases
tff(fact_5453_under__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_under(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_sk(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% under_def
tff(fact_5454_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: heap_Time_Heap(A),Xa: heap_Time_Heap(A),F: fun(A,B),Fa: fun(A,B)] :
( ! [Z3: A,Za: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X2)))
=> ( pp(member(A,Za,heap_Time_set_Heap(A,Xa)))
=> ( ( aa(A,B,F,Z3) = aa(A,B,Fa,Za) )
=> ( Z3 = Za ) ) ) )
=> ( ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X2) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),Fa),Xa) )
=> ( X2 = Xa ) ) ) ).
% Heap.inj_map_strong
tff(fact_5455_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X2)))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X2) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G),X2) ) ) ).
% Heap.map_cong0
tff(fact_5456_Heap_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: heap_Time_Heap(A),Ya: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,Ya)))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X2) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G),Ya) ) ) ) ).
% Heap.map_cong
tff(fact_5457_Heap_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: heap_Time_Heap(A)] : heap_Time_set_Heap(B,aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),V)) = aa(set(A),set(B),image2(A,B,F),heap_Time_set_Heap(A,V)) ).
% Heap.set_map
tff(fact_5458_Heap_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set(A),F: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),bNF_Grp(A,B,A4,F)) = bNF_Grp(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(heap_Time_Heap(A),bool),set(heap_Time_Heap(A)),collect(heap_Time_Heap(A)),aTP_Lamp_vi(set(A),fun(heap_Time_Heap(A),bool),A4)),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F)) ).
% Heap.rel_Grp
tff(fact_5459_Heap_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4) = relcompp(heap_Time_Heap(A),heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B),conversep(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A),bNF_Grp(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A),aa(fun(heap_Time_Heap(product_prod(A,B)),bool),set(heap_Time_Heap(product_prod(A,B))),collect(heap_Time_Heap(product_prod(A,B))),aTP_Lamp_vj(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),A),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A)),heap_Time_map_Heap(product_prod(A,B),A),product_fst(A,B)))),bNF_Grp(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B),aa(fun(heap_Time_Heap(product_prod(A,B)),bool),set(heap_Time_Heap(product_prod(A,B))),collect(heap_Time_Heap(product_prod(A,B))),aTP_Lamp_vj(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),B),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B)),heap_Time_map_Heap(product_prod(A,B),B),product_snd(A,B)))) ).
% Heap.rel_compp_Grp
tff(fact_5460_Heap_Oin__rel,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: heap_Time_Heap(A),B2: heap_Time_Heap(B)] :
( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),A3),B2))
<=> ? [Z4: heap_Time_Heap(product_prod(A,B))] :
( pp(member(heap_Time_Heap(product_prod(A,B)),Z4,aa(fun(heap_Time_Heap(product_prod(A,B)),bool),set(heap_Time_Heap(product_prod(A,B))),collect(heap_Time_Heap(product_prod(A,B))),aTP_Lamp_vj(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R4))))
& ( aa(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A),aa(fun(product_prod(A,B),A),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A)),heap_Time_map_Heap(product_prod(A,B),A),product_fst(A,B)),Z4) = A3 )
& ( aa(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B),aa(fun(product_prod(A,B),B),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B)),heap_Time_map_Heap(product_prod(A,B),B),product_snd(A,B)),Z4) = B2 ) ) ) ).
% Heap.in_rel
tff(fact_5461_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X2: heap_Time_Heap(A),Ya: heap_Time_Heap(A),Y: heap_Time_Heap(B),Xa: heap_Time_Heap(B),R4: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
( ( X2 = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( pp(member(A,Z3,heap_Time_set_Heap(A,Ya)))
=> ( pp(member(B,Yb,heap_Time_set_Heap(B,Xa)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z3),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z3),Yb)) ) ) )
=> ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),X2),Y))
<=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra2),Ya),Xa)) ) ) ) ) ).
% Heap.rel_cong
tff(fact_5462_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: heap_Time_Heap(A),Y: heap_Time_Heap(B),Ra2: fun(A,fun(B,bool))] :
( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),X2),Y))
=> ( ! [Z3: A,Yb: B] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X2)))
=> ( pp(member(B,Yb,heap_Time_set_Heap(B,Y)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z3),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z3),Yb)) ) ) )
=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra2),X2),Y)) ) ) ).
% Heap.rel_mono_strong
tff(fact_5463_Heap_Orel__refl__strong,axiom,
! [A: $tType,X2: heap_Time_Heap(A),Ra2: fun(A,fun(A,bool))] :
( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra2,Z3),Z3)) )
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),Ra2),X2),X2)) ) ).
% Heap.rel_refl_strong
tff(fact_5464_Heap_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X2: heap_Time_Heap(A),G: fun(B,C),Y: heap_Time_Heap(B)] :
( pp(aa(heap_Time_Heap(C),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool),aa(fun(A,fun(C,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(A,C),Sa),X2),aa(heap_Time_Heap(B),heap_Time_Heap(C),aa(fun(B,C),fun(heap_Time_Heap(B),heap_Time_Heap(C)),heap_Time_map_Heap(B,C),G),Y)))
<=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_jt(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X2),Y)) ) ).
% Heap.rel_map(2)
tff(fact_5465_Heap_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X2: heap_Time_Heap(A),Y: heap_Time_Heap(B)] :
( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(C),fun(heap_Time_Heap(B),bool),aa(fun(C,fun(B,bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(C,B),Sb),aa(heap_Time_Heap(A),heap_Time_Heap(C),aa(fun(A,C),fun(heap_Time_Heap(A),heap_Time_Heap(C)),heap_Time_map_Heap(A,C),I),X2)),Y))
<=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_js(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X2),Y)) ) ).
% Heap.rel_map(1)
tff(fact_5466_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R4),Ra2))
=> pp(aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),bool,aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),fun(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),bool),ord_less_eq(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra2))) ) ).
% Heap.rel_mono
tff(fact_5467_Heap_Orel__transp,axiom,
! [A: $tType,R4: fun(A,fun(A,bool))] :
( transp(A,R4)
=> transp(heap_Time_Heap(A),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),R4)) ) ).
% Heap.rel_transp
tff(fact_5468_Heap_Orel__refl,axiom,
! [B: $tType,Ra2: fun(B,fun(B,bool)),X2: heap_Time_Heap(B)] :
( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra2,X3),X3))
=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(B),fun(heap_Time_Heap(B),bool),aa(fun(B,fun(B,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(B,B),Ra2),X2),X2)) ) ).
% Heap.rel_refl
tff(fact_5469_Heap_Orel__eq,axiom,
! [A: $tType] : aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),fequal(A)) = fequal(heap_Time_Heap(A)) ).
% Heap.rel_eq
tff(fact_5470_Heap_Orel__conversep,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(B,A),conversep(A,B,R4)) = conversep(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) ).
% Heap.rel_conversep
tff(fact_5471_Heap_Orel__flip,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: heap_Time_Heap(B),B2: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(B),fun(heap_Time_Heap(A),bool),aa(fun(B,fun(A,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(B,A),conversep(A,B,R4)),A3),B2))
<=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),B2),A3)) ) ).
% Heap.rel_flip
tff(fact_5472_Heap_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(A,C),relcompp(A,B,C,R4,S2)) = relcompp(heap_Time_Heap(A),heap_Time_Heap(B),heap_Time_Heap(C),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),aa(fun(B,fun(C,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(B,C),S2)) ).
% Heap.rel_compp
tff(fact_5473_Heap_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(C),fun(heap_Time_Heap(B),bool),fun(heap_Time_Heap(D),bool),aa(fun(A,fun(C,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(A,C),Sa),bNF_rel_fun(heap_Time_Heap(B),heap_Time_Heap(D),bool,bool,aa(fun(B,fun(D,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(D),bool)),heap_Time_rel_Heap(B,D),Sc),fequal(bool)))),heap_Time_rel_Heap(A,B)),heap_Time_rel_Heap(C,D))) ).
% Heap.rel_transfer
tff(fact_5474_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F4: $tType,E: $tType,Rb2: fun(A,fun(E,bool)),Sd: fun(B,fun(F4,bool))] : pp(aa(fun(fun(E,F4),fun(heap_Time_Heap(E),heap_Time_Heap(F4))),bool,aa(fun(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B))),fun(fun(fun(E,F4),fun(heap_Time_Heap(E),heap_Time_Heap(F4))),bool),bNF_rel_fun(fun(A,B),fun(E,F4),fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(E),heap_Time_Heap(F4)),bNF_rel_fun(A,E,B,F4,Rb2,Sd),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(E),heap_Time_Heap(B),heap_Time_Heap(F4),aa(fun(A,fun(E,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(E),bool)),heap_Time_rel_Heap(A,E),Rb2),aa(fun(B,fun(F4,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(F4),bool)),heap_Time_rel_Heap(B,F4),Sd))),heap_Time_map_Heap(A,B)),heap_Time_map_Heap(E,F4))) ).
% Heap.map_transfer
tff(fact_5475_those_Osimps_I2_J,axiom,
! [A: $tType,X2: option(A),Xs: list(option(A))] : those(A,aa(list(option(A)),list(option(A)),aa(option(A),fun(list(option(A)),list(option(A))),cons(option(A)),X2),Xs)) = aa(option(A),option(list(A)),aa(fun(A,option(list(A))),fun(option(A),option(list(A))),aa(option(list(A)),fun(fun(A,option(list(A))),fun(option(A),option(list(A)))),case_option(option(list(A)),A),none(list(A))),aTP_Lamp_vk(list(option(A)),fun(A,option(list(A))),Xs)),X2) ).
% those.simps(2)
tff(fact_5476_Quotient__natural,axiom,
quotient(nat,code_natural,fequal(nat),code_natural_of_nat,code_nat_of_natural,code_cr_natural) ).
% Quotient_natural
tff(fact_5477_Quotient__integer,axiom,
quotient(int,code_integer,fequal(int),code_integer_of_int,code_int_of_integer,code_cr_integer) ).
% Quotient_integer
tff(fact_5478_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T5: fun(A,fun(B,bool))] :
( quotient(A,B,R4,Abs,Rep,T5)
=> quotient(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),R4),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),Abs),aa(fun(B,A),fun(heap_Time_Heap(B),heap_Time_Heap(A)),heap_Time_map_Heap(B,A),Rep),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),T5)) ) ).
% Heap.Quotient
tff(fact_5479_option_OQuotient,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T5: fun(A,fun(B,bool))] :
( quotient(A,B,R4,Abs,Rep,T5)
=> quotient(option(A),option(B),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R4),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Abs),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),Rep),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),T5)) ) ).
% option.Quotient
tff(fact_5480_Quotient__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),T1: fun(A,fun(B,bool)),R23: fun(B,fun(B,bool)),Abs22: fun(B,C),Rep22: fun(C,B),T22: fun(B,fun(C,bool))] :
( quotient(A,B,R12,Abs1,Rep1,T1)
=> ( quotient(B,C,R23,Abs22,Rep22,T22)
=> quotient(A,C,relcompp(A,B,A,T1,relcompp(B,B,A,R23,conversep(A,B,T1))),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22),relcompp(A,B,C,T1,T22)) ) ) ).
% Quotient_compose
tff(fact_5481_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(heap_Time_Heap(B),bool)),bool,aa(fun(fun(A,bool),fun(heap_Time_Heap(A),bool)),fun(fun(fun(B,bool),fun(heap_Time_Heap(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(heap_Time_Heap(A),bool),fun(heap_Time_Heap(B),bool),bNF_rel_fun(A,B,bool,bool,R4,fequal(bool)),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),bool,bool,aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),fequal(bool))),heap_Time_pred_Heap(A)),heap_Time_pred_Heap(B))) ).
% Heap.pred_transfer
tff(fact_5482_Heap_Orec__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(C,fun(D,bool))] : pp(aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool,aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C)),fun(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool),bNF_rel_fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(A),C),fun(heap_Time_Heap(B),D),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C,D,bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),S2),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),C,D,aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),S2)),heap_Time_rec_Heap(A,C)),heap_Time_rec_Heap(B,D))) ).
% Heap.rec_transfer
tff(fact_5483_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A3: A,B2: C,C2: B,D3: D] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C2),D3)))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A3),C2))
& pp(aa(D,bool,aa(C,fun(D,bool),R23,B2),D3)) ) ) ).
% rel_prod_inject
tff(fact_5484_Heap_Opred__mono,axiom,
! [A: $tType,P: fun(A,bool),Pa: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Pa))
=> pp(aa(fun(heap_Time_Heap(A),bool),bool,aa(fun(heap_Time_Heap(A),bool),fun(fun(heap_Time_Heap(A),bool),bool),ord_less_eq(fun(heap_Time_Heap(A),bool)),aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P)),aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),Pa))) ) ).
% Heap.pred_mono
tff(fact_5485_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: fun(A,fun(B,bool)),A3: A,B2: B,R23: fun(C,fun(D,bool)),C2: C,D3: D] :
( pp(aa(B,bool,aa(A,fun(B,bool),R12,A3),B2))
=> ( pp(aa(D,bool,aa(C,fun(D,bool),R23,C2),D3))
=> pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D3))) ) ) ).
% rel_prod.intros
tff(fact_5486_rel__prod_Osimps,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),A1),A22))
<=> ? [A7: A,B7: B,C4: C,D5: D] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),C4) )
& ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B7),D5) )
& pp(aa(B,bool,aa(A,fun(B,bool),R12,A7),B7))
& pp(aa(D,bool,aa(C,fun(D,bool),R23,C4),D5)) ) ) ).
% rel_prod.simps
tff(fact_5487_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),A1),A22))
=> ~ ! [A5: A,B3: B,C3: C] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),C3) )
=> ! [D2: D] :
( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B3),D2) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A5),B3))
=> ~ pp(aa(D,bool,aa(C,fun(D,bool),R23,C3),D2)) ) ) ) ) ).
% rel_prod.cases
tff(fact_5488_Heap_Opred__True,axiom,
! [A: $tType,X: heap_Time_Heap(A)] : pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aTP_Lamp_af(A,bool)),X)) ).
% Heap.pred_True
tff(fact_5489_Heap_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,bool),X2: heap_Time_Heap(A),Pa: fun(A,bool)] :
( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X2))
=> ( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X2)))
=> ( pp(aa(A,bool,P,Z3))
=> pp(aa(A,bool,Pa,Z3)) ) )
=> pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),Pa),X2)) ) ) ).
% Heap.pred_mono_strong
tff(fact_5490_Heap_Opred__cong,axiom,
! [A: $tType,X2: heap_Time_Heap(A),Ya: heap_Time_Heap(A),P: fun(A,bool),Pa: fun(A,bool)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,Ya)))
=> ( pp(aa(A,bool,P,Z3))
<=> pp(aa(A,bool,Pa,Z3)) ) )
=> ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X2))
<=> pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),Pa),Ya)) ) ) ) ).
% Heap.pred_cong
tff(fact_5491_Heap_Octr__transfer,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B)),bool,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A)),fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B)),bool),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap(B),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)),heap_Time_Heap2(A)),heap_Time_Heap2(B))) ).
% Heap.ctr_transfer
tff(fact_5492_Heap_Orel__intros,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Y: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
( pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),X2),Y))
=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)),aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),Y))) ) ).
% Heap.rel_intros
tff(fact_5493_Heap_Orel__inject,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Y: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)),aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),Y)))
<=> pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),X2),Y)) ) ).
% Heap.rel_inject
tff(fact_5494_Heap_Orel__induct,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: heap_Time_Heap(A),Y: heap_Time_Heap(B),Q: fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))] :
( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),X2),Y))
=> ( ! [A5: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B3: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
( pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),A5),B3))
=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),Q,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),A5)),aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),B3))) )
=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),Q,X2),Y)) ) ) ).
% Heap.rel_induct
tff(fact_5495_Heap_Orel__cases,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: heap_Time_Heap(A),B2: heap_Time_Heap(B)] :
( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),A3),B2))
=> ~ ! [Xa4: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( ( A3 = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Xa4) )
=> ! [Ya2: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
( ( B2 = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),Ya2) )
=> ~ pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),Xa4),Ya2)) ) ) ) ).
% Heap.rel_cases
tff(fact_5496_Heap_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X2: heap_Time_Heap(A),Ya: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_ud(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
=> ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X2) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G),Ya) ) ) ) ).
% Heap.map_cong_pred
tff(fact_5497_Heap_Orel__eq__onp,axiom,
! [A: $tType,P: fun(A,bool)] : aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),bNF_eq_onp(A,P)) = bNF_eq_onp(heap_Time_Heap(A),aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P)) ).
% Heap.rel_eq_onp
tff(fact_5498_Heap_Opred__rel,axiom,
! [A: $tType,P: fun(A,bool),X2: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X2))
<=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),bNF_eq_onp(A,P)),X2),X2)) ) ).
% Heap.pred_rel
tff(fact_5499_Heap_Opred__set,axiom,
! [A: $tType,P: fun(A,bool),X: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X))
<=> ! [Xa2: A] :
( pp(member(A,Xa2,heap_Time_set_Heap(A,X)))
=> pp(aa(A,bool,P,Xa2)) ) ) ).
% Heap.pred_set
tff(fact_5500_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] : pp(aa(fun(B,fun(D,product_prod(B,D))),bool,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),bool),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),A4,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),B5,basic_rel_prod(A,B,C,D,A4,B5))),product_Pair(A,C)),product_Pair(B,D))) ).
% Pair_transfer
tff(fact_5501_Heap_Opred__map,axiom,
! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X2: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(B),bool,aa(fun(B,bool),fun(heap_Time_Heap(B),bool),heap_Time_pred_Heap(B),Q),aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X2)))
<=> pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X2)) ) ).
% Heap.pred_map
tff(fact_5502_Heap_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(C,fun(D,bool))] : pp(aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool,aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C)),fun(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool),bNF_rel_fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(A),C),fun(heap_Time_Heap(B),D),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C,D,bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),S2),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),C,D,aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),S2)),heap_Time_case_Heap(A,C)),heap_Time_case_Heap(B,D))) ).
% Heap.case_transfer
tff(fact_5503_sndOp__def,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(C,fun(A,bool)),Q: fun(A,fun(B,bool)),Ac: product_prod(C,B)] : bNF_sndOp(C,A,B,P,Q,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(C,A,B,P,Q,aa(product_prod(C,B),C,product_fst(C,B),Ac),aa(product_prod(C,B),B,product_snd(C,B),Ac))),aa(product_prod(C,B),B,product_snd(C,B),Ac)) ).
% sndOp_def
tff(fact_5504_fstOp__def,axiom,
! [B: $tType,C: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(B,fun(C,bool)),Ac: product_prod(A,C)] : bNF_fstOp(A,B,C,P,Q,Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Ac)),bNF_pick_middlep(A,B,C,P,Q,aa(product_prod(A,C),A,product_fst(A,C),Ac),aa(product_prod(A,C),C,product_snd(A,C),Ac))) ).
% fstOp_def
tff(fact_5505_Heap_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),Heap: heap_Time_Heap(A)] : aa(B,C,H,aa(heap_Time_Heap(A),B,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(heap_Time_Heap(A),B),heap_Time_case_Heap(A,B),F),Heap)) = aa(heap_Time_Heap(A),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C),heap_Time_case_Heap(A,C),aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_vl(fun(B,C),fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),H),F)),Heap) ).
% Heap.case_distrib
tff(fact_5506_Heap_Ocase,axiom,
! [B: $tType,A: $tType,F: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),B,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(heap_Time_Heap(A),B),heap_Time_case_Heap(A,B),F),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X2)) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B,F,X2) ).
% Heap.case
tff(fact_5507_trancl__def,axiom,
! [A: $tType,X: set(product_prod(A,A))] : transitive_trancl(A,X) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_tranclp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),X)))) ).
% trancl_def
tff(fact_5508_Nitpick_Otranclp__unfold,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,R),A3),B2))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),transitive_trancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% Nitpick.tranclp_unfold
tff(fact_5509_reflp__refl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( reflp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> refl_on(A,top_top(set(A)),R) ) ).
% reflp_refl_eq
tff(fact_5510_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( ! [A5: A,B3: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3)) )
=> ( ! [A5: A,B3: B,Aa2: A,Ba: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)))
=> ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).
% tranclp_induct2
tff(fact_5511_Heap_Orel__reflp,axiom,
! [A: $tType,R4: fun(A,fun(A,bool))] :
( reflp(A,R4)
=> reflp(heap_Time_Heap(A),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),R4)) ) ).
% Heap.rel_reflp
tff(fact_5512_option_Orel__reflp,axiom,
! [A: $tType,R4: fun(A,fun(A,bool))] :
( reflp(A,R4)
=> reflp(option(A),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R4)) ) ).
% option.rel_reflp
tff(fact_5513_tranclp__trancl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,Xa3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X),Xa3))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa3),transitive_trancl(A,R))) ) ).
% tranclp_trancl_eq
tff(fact_5514_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) )
<=> ( ! [X4: A] :
( pp(member(A,X4,A4))
=> pp(dvd_dvd(A,X4,one_one(A))) )
& finite_finite2(A,A4) ) ) ) ).
% Lcm_fin_1_iff
tff(fact_5515_heap__interpretation,axiom,
! [A: $tType] : partia7178651479351089652itions(heap_Time_Heap(A),heap_Time_Heap_ord(A),heap_Time_Heap_lub(A)) ).
% heap_interpretation
tff(fact_5516_old_Orec__bool__def,axiom,
! [T: $tType,X: T,Xa3: T,Xb2: bool] : product_rec_bool(T,X,Xa3,Xb2) = the(T,product_rec_set_bool(T,X,Xa3,Xb2)) ).
% old.rec_bool_def
tff(fact_5517_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fFalse) = F22 ).
% old.bool.simps(6)
tff(fact_5518_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fTrue) = F1 ).
% old.bool.simps(5)
tff(fact_5519_Lcm__fin_Oempty,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_fin.empty
tff(fact_5520_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( pp(dvd_dvd(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A4),one_one(A)))
<=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = one_one(A) ) ) ) ).
% is_unit_Lcm_fin_iff
tff(fact_5521_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A4: set(A)] :
( ( finite_finite2(A,A4)
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = finite_fold(A,A,gcd_lcm(A),one_one(A),A4) ) )
& ( ~ finite_finite2(A,A4)
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A4) = zero_zero(A) ) ) ) ) ).
% Lcm_fin.eq_fold
tff(fact_5522_Lcm__fin__def,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( semiring_gcd_Lcm_fin(A) = bounde2362111253966948842tice_F(A,gcd_lcm(A),one_one(A),zero_zero(A)) ) ) ).
% Lcm_fin_def
tff(fact_5523_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: fun(B,bool),F: fun(A,B),X2: fun(D,A)] :
( basic_pred_fun(D,B,aTP_Lamp_vm(D,bool),Q,aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2))
<=> basic_pred_fun(D,A,aTP_Lamp_vm(D,bool),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F),X2) ) ).
% fun.pred_map
tff(fact_5524_lcm__eq__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = one_one(A) )
<=> ( pp(dvd_dvd(A,A3,one_one(A)))
& pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).
% lcm_eq_1_iff
tff(fact_5525_unit__factor__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = zero_zero(A) ) )
& ( ~ ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = one_one(A) ) ) ) ) ).
% unit_factor_lcm
tff(fact_5526_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),Ya: fun(D,A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( basic_pred_fun(D,A,aTP_Lamp_vm(D,bool),aa(fun(A,B),fun(A,bool),aTP_Lamp_ud(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G),Ya)
=> ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),Ya) ) ) ) ).
% fun.map_cong_pred
tff(fact_5527_lcm__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_mult_unit2
tff(fact_5528_lcm__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_mult_unit1
tff(fact_5529_lcm__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_div_unit1
tff(fact_5530_lcm__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).
% lcm_div_unit2
tff(fact_5531_Lcm__nat__def,axiom,
! [M2: set(nat)] :
( ( finite_finite2(nat,M2)
=> ( gcd_Lcm(nat,M2) = lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M2) ) )
& ( ~ finite_finite2(nat,M2)
=> ( gcd_Lcm(nat,M2) = zero_zero(nat) ) ) ) ).
% Lcm_nat_def
tff(fact_5532_Heap_Opred__inject,axiom,
! [A: $tType,P: fun(A,bool),A3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),A3)))
<=> basic_pred_fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_vn(heap_ext(product_unit),bool),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),bool),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool),pred_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),basic_pred_prod(A,product_prod(heap_ext(product_unit),nat),P,top_top(fun(product_prod(heap_ext(product_unit),nat),bool)))),A3) ) ).
% Heap.pred_inject
tff(fact_5533_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).
% Lcm_fin.set_eq_fold
tff(fact_5534_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P13: fun(A,bool),P24: fun(B,bool),A3: A,B2: B] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P13,P24),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
<=> ( pp(aa(A,bool,P13,A3))
& pp(aa(B,bool,P24,B2)) ) ) ).
% pred_prod_inject
tff(fact_5535_fold__append,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : fold(B,A,F,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),fold(B,A,F,Ys)),fold(B,A,F,Xs)) ).
% fold_append
tff(fact_5536_pred__prod__split,axiom,
! [A: $tType,B: $tType,P: fun(bool,bool),Q: fun(A,bool),R4: fun(B,bool),Xy: product_prod(A,B)] :
( pp(aa(bool,bool,P,aa(product_prod(A,B),bool,basic_pred_prod(A,B,Q,R4),Xy)))
<=> ! [X4: A,Y3: B] :
( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3) )
=> pp(aa(bool,bool,P,fconj(aa(A,bool,Q,X4),aa(B,bool,R4,Y3)))) ) ) ).
% pred_prod_split
tff(fact_5537_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P13: fun(A,bool),P24: fun(B,bool),A3: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P13,P24),A3))
=> ~ ! [A5: A,B3: B] :
( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) )
=> ( pp(aa(A,bool,P13,A5))
=> ~ pp(aa(B,bool,P24,B3)) ) ) ) ).
% pred_prod.cases
tff(fact_5538_pred__prod_Osimps,axiom,
! [A: $tType,B: $tType,P13: fun(A,bool),P24: fun(B,bool),A3: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P13,P24),A3))
<=> ? [A7: A,B7: B] :
( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
& pp(aa(A,bool,P13,A7))
& pp(aa(B,bool,P24,B7)) ) ) ).
% pred_prod.simps
tff(fact_5539_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P13: fun(A,bool),A3: A,P24: fun(B,bool),B2: B] :
( pp(aa(A,bool,P13,A3))
=> ( pp(aa(B,bool,P24,B2))
=> pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P13,P24),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ) ).
% pred_prod.intros
tff(fact_5540_fold__Cons,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A,Xs: list(A)] : fold(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),fold(A,B,F,Xs)),aa(A,fun(B,B),F,X2)) ).
% fold_Cons
tff(fact_5541_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C)),S: B] :
( ! [X3: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),aa(A,fun(B,B),G,X3)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(A,fun(C,C),F,X3)),H) ) )
=> ( aa(B,C,H,aa(B,B,fold(A,B,G,Xs),S)) = aa(C,C,fold(A,C,F,Xs),aa(B,C,H,S)) ) ) ).
% fold_commute_apply
tff(fact_5542_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C))] :
( ! [X3: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),aa(A,fun(B,B),G,X3)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(A,fun(C,C),F,X3)),H) ) )
=> ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),fold(A,B,G,Xs)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),fold(A,C,F,Xs)),H) ) ) ).
% fold_commute
tff(fact_5543_Lcm__nat__set__eq__fold,axiom,
! [Xs: list(nat)] : gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_lcm(nat),Xs),one_one(nat)) ).
% Lcm_nat_set_eq_fold
tff(fact_5544_fold__multiset__equiv,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B)),Ys: list(A)] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member(A,Y4,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X3)) ) ) )
=> ( ( mset(A,Xs) = mset(A,Ys) )
=> ( fold(A,B,F,Xs) = fold(A,B,F,Ys) ) ) ) ).
% fold_multiset_equiv
tff(fact_5545_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member(A,Y4,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y4)) ) ) )
=> ( fold(A,B,F,rev(A,Xs)) = fold(A,B,F,Xs) ) ) ).
% fold_rev
tff(fact_5546_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member(A,Y4,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y4)) ) ) )
=> ( foldr(A,B,F,Xs) = fold(A,B,F,Xs) ) ) ).
% foldr_fold
tff(fact_5547_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : fold(A,A,plus_plus(A),Xs) = aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),rev(A,Xs))) ) ).
% fold_plus_sum_list_rev
tff(fact_5548_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B)),X2: A] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,aa(list(A),set(A),set2(A),Xs)))
=> ( pp(member(A,Y4,aa(list(A),set(A),set2(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X3)) ) ) )
=> ( pp(member(A,X2,aa(list(A),set(A),set2(A),Xs)))
=> ( fold(A,B,F,Xs) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),fold(A,B,F,remove1(A,X2,Xs))),aa(A,fun(B,B),F,X2)) ) ) ) ).
% fold_remove1_split
tff(fact_5549_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Xs: list(A)] : gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).
% Lcm_set_eq_fold
tff(fact_5550_subset__mset_Omax__add__distrib__left,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A),Z2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),max(multiset(A),subseteq_mset(A),X2,Y)),Z2) = max(multiset(A),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Z2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y),Z2)) ).
% subset_mset.max_add_distrib_left
tff(fact_5551_subset__mset_Omax__add__distrib__right,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A),Z2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),max(multiset(A),subseteq_mset(A),Y,Z2)) = max(multiset(A),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Y),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Z2)) ).
% subset_mset.max_add_distrib_right
tff(fact_5552_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> groups1828464146339083142d_list(A,times_times(A),one_one(A)) ) ).
% prod_list.comm_monoid_list_axioms
tff(fact_5553_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> groups1828464146339083142d_list(A,plus_plus(A),zero_zero(A)) ) ).
% sum_list.comm_monoid_list_axioms
tff(fact_5554_lazI,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> ( ! [I2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),B2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A3),I2)),aa(nat,B,nth(B,B2),I2))) )
=> list_all_zip(A,B,P,A3,B2) ) ) ).
% lazI
tff(fact_5555_list__all__zip__alt,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),As2: list(A),Bs2: list(B)] :
( list_all_zip(A,B,P,As2,Bs2)
<=> ( ( aa(list(A),nat,size_size(list(A)),As2) = aa(list(B),nat,size_size(list(B)),Bs2) )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),As2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,As2),I3)),aa(nat,B,nth(B,Bs2),I3))) ) ) ) ).
% list_all_zip_alt
tff(fact_5556_is__none__bind,axiom,
! [A: $tType,B: $tType,F: option(B),G: fun(B,option(A))] :
( is_none(A,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),F),G))
<=> ( is_none(B,F)
| is_none(A,aa(B,option(A),G,aa(option(B),B,the2(B),F))) ) ) ).
% is_none_bind
tff(fact_5557_laz__conj,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( list_all_zip(A,B,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_vo(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q),A3,B2)
<=> ( list_all_zip(A,B,P,A3,B2)
& list_all_zip(A,B,Q,A3,B2) ) ) ).
% laz_conj
tff(fact_5558_is__none__code_I2_J,axiom,
! [B: $tType,X2: B] : ~ is_none(B,aa(B,option(B),some(B),X2)) ).
% is_none_code(2)
tff(fact_5559_is__none__code_I1_J,axiom,
! [A: $tType] : is_none(A,none(A)) ).
% is_none_code(1)
tff(fact_5560_is__none__map__option,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: option(B)] :
( is_none(A,aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),X2))
<=> is_none(B,X2) ) ).
% is_none_map_option
tff(fact_5561_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: fun(B,bool),A4: list(A),B5: list(B)] :
( list_all_zip(A,B,aTP_Lamp_vp(fun(B,bool),fun(A,fun(B,bool)),P),A4,B5)
<=> ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B5) )
& ! [X4: B] :
( pp(member(B,X4,aa(list(B),set(B),set2(B),B5)))
=> pp(aa(B,bool,P,X4)) ) ) ) ).
% laz_weak_Pb
tff(fact_5562_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),A4: list(A),B5: list(B)] :
( list_all_zip(A,B,aTP_Lamp_vq(fun(A,bool),fun(A,fun(B,bool)),P),A4,B5)
<=> ( ( aa(list(A),nat,size_size(list(A)),A4) = aa(list(B),nat,size_size(list(B)),B5) )
& ! [X4: A] :
( pp(member(A,X4,aa(list(A),set(A),set2(A),A4)))
=> pp(aa(A,bool,P,X4)) ) ) ) ).
% laz_weak_Pa
tff(fact_5563_list__all__zip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : list_all_zip(A,B,P,nil(A),nil(B)) ).
% list_all_zip.simps(1)
tff(fact_5564_laz__eq,axiom,
! [A: $tType,A3: list(A),B2: list(A)] :
( list_all_zip(A,A,fequal(A),A3,B2)
<=> ( A3 = B2 ) ) ).
% laz_eq
tff(fact_5565_laz__len,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( list_all_zip(A,B,P,A3,B2)
=> ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) ) ) ).
% laz_len
tff(fact_5566_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: fun(A,fun(B,fun(C,bool))),A4: list(A),B5: list(B)] :
( list_all_zip(A,B,aTP_Lamp_vr(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,bool)),P),A4,B5)
=> ~ ! [C8: list(C)] :
( list_all_zip(A,C,aTP_Lamp_vs(fun(A,fun(B,fun(C,bool))),fun(A,fun(C,bool)),P),A4,C8)
=> ~ list_all_zip(B,C,aTP_Lamp_vt(fun(A,fun(B,fun(C,bool))),fun(B,fun(C,bool)),P),B5,C8) ) ) ).
% laz_swap_ex
tff(fact_5567_is__none__simps_I2_J,axiom,
! [B: $tType,X2: B] : ~ is_none(B,aa(B,option(B),some(B),X2)) ).
% is_none_simps(2)
tff(fact_5568_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As2: list(A),F: fun(C,B),Bs2: list(C)] :
( list_all_zip(A,B,P,As2,aa(list(C),list(B),map(C,B,F),Bs2))
<=> list_all_zip(A,C,aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_vu(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F),As2,Bs2) ) ).
% list_all_zip_map2
tff(fact_5569_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F: fun(C,A),As2: list(C),Bs2: list(B)] :
( list_all_zip(A,B,P,aa(list(C),list(A),map(C,A,F),As2),Bs2)
<=> list_all_zip(C,B,aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_vv(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F),As2,Bs2) ) ).
% list_all_zip_map1
tff(fact_5570_is__none__simps_I1_J,axiom,
! [A: $tType] : is_none(A,none(A)) ).
% is_none_simps(1)
tff(fact_5571_Option_Ois__none__def,axiom,
! [A: $tType,X2: option(A)] :
( is_none(A,X2)
<=> ( X2 = none(A) ) ) ).
% Option.is_none_def
tff(fact_5572_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A3: A,As2: list(A),B2: B,Bs2: list(B)] :
( list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),As2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B2),Bs2))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2))
& list_all_zip(A,B,P,As2,Bs2) ) ) ).
% list_all_zip.simps(2)
tff(fact_5573_list__all__zip_Oelims_I3_J,axiom,
! [A: $tType,B: $tType,X2: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X2,Xa,Xb)
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),X2,A5),B3))
& list_all_zip(A,B,X2,As,Bs) ) ) )
=> ( ( ? [V4: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)
=> ( Xb != nil(B) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V4: B,Va: list(B)] : Xb != aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) ) ) ) ) ).
% list_all_zip.elims(3)
tff(fact_5574_list__all__zip_Oelims_I2_J,axiom,
! [A: $tType,B: $tType,X2: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X2,Xa,Xb)
=> ( ( ( Xa = nil(A) )
=> ( Xb != nil(B) ) )
=> ~ ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X2,A5),B3))
& list_all_zip(A,B,X2,As,Bs) ) ) ) ) ) ).
% list_all_zip.elims(2)
tff(fact_5575_list__all__zip_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X2: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B),Y: bool] :
( ( list_all_zip(A,B,X2,Xa,Xb)
<=> pp(Y) )
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ~ pp(Y) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( pp(Y)
<=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X2,A5),B3))
& list_all_zip(A,B,X2,As,Bs) ) ) ) )
=> ( ( ? [V4: A,Va: list(A)] : Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)
=> ( ( Xb = nil(B) )
=> pp(Y) ) )
=> ~ ( ( Xa = nil(A) )
=> ( ? [V4: B,Va: list(B)] : Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va)
=> pp(Y) ) ) ) ) ) ) ).
% list_all_zip.elims(1)
tff(fact_5576_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),V: B,Va2: list(B)] : ~ list_all_zip(A,B,P,nil(A),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va2)) ).
% list_all_zip.simps(4)
tff(fact_5577_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),V: A,Va2: list(A)] : ~ list_all_zip(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va2),nil(B)) ).
% list_all_zip.simps(3)
tff(fact_5578_the__map__option,axiom,
! [B: $tType,A: $tType,X2: option(A),F: fun(A,B)] :
( ~ is_none(A,X2)
=> ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X2)) = aa(A,B,F,aa(option(A),A,the2(A),X2)) ) ) ).
% the_map_option
tff(fact_5579_rel__optionI,axiom,
! [A: $tType,B: $tType,X2: option(A),Y: option(B),P: fun(A,fun(B,bool))] :
( ( is_none(A,X2)
<=> is_none(B,Y) )
=> ( ( ~ is_none(A,X2)
=> ( ~ is_none(B,Y)
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(option(A),A,the2(A),X2)),aa(option(B),B,the2(B),Y))) ) )
=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),X2),Y)) ) ) ).
% rel_optionI
tff(fact_5580_rel__option__unfold,axiom,
! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: option(A),Y: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X2),Y))
<=> ( ( is_none(A,X2)
<=> is_none(B,Y) )
& ( ~ is_none(A,X2)
=> ( ~ is_none(B,Y)
=> pp(aa(B,bool,aa(A,fun(B,bool),R4,aa(option(A),A,the2(A),X2)),aa(option(B),B,the2(B),Y))) ) ) ) ) ).
% rel_option_unfold
tff(fact_5581_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X2: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B),Y: bool] :
( ( list_all_zip(A,B,X2,Xa,Xb)
<=> pp(Y) )
=> ( pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb))))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ( pp(Y)
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))))) ) ) )
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( ( pp(Y)
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),X2,A5),B3))
& list_all_zip(A,B,X2,As,Bs) ) )
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs))))) ) ) )
=> ( ! [V4: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( ( Xb = nil(B) )
=> ( ~ pp(Y)
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(B))))) ) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V4: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
=> ( ~ pp(Y)
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va))))) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
tff(fact_5582_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X2: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( list_all_zip(A,B,X2,Xa,Xb)
=> ( pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb))))
=> ( ( ( Xa = nil(A) )
=> ( ( Xb = nil(B) )
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B))))) ) )
=> ~ ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))))
=> ~ ( pp(aa(B,bool,aa(A,fun(B,bool),X2,A5),B3))
& list_all_zip(A,B,X2,As,Bs) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
tff(fact_5583_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X2: fun(A,fun(B,bool)),Xa: list(A),Xb: list(B)] :
( ~ list_all_zip(A,B,X2,Xa,Xb)
=> ( pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xa),Xb))))
=> ( ! [A5: A,As: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As) )
=> ! [B3: B,Bs: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs) )
=> ( pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),X2,A5),B3))
& list_all_zip(A,B,X2,As,Bs) ) ) ) )
=> ( ! [V4: A,Va: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va) )
=> ( ( Xb = nil(B) )
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V4),Va)),nil(B))))) ) )
=> ~ ( ( Xa = nil(A) )
=> ! [V4: B,Va: list(B)] :
( ( Xb = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va) )
=> ~ pp(aa(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),bool,accp(product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),list_all_zip_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B))),aa(fun(A,fun(B,bool)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,bool)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,bool)),product_prod(list(A),list(B))),X2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V4),Va))))) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
tff(fact_5584_plus__multiset_Otransfer,axiom,
! [A: $tType] : pp(aa(fun(multiset(A),fun(multiset(A),multiset(A))),bool,aa(fun(fun(A,nat),fun(fun(A,nat),fun(A,nat))),fun(fun(multiset(A),fun(multiset(A),multiset(A))),bool),bNF_rel_fun(fun(A,nat),multiset(A),fun(fun(A,nat),fun(A,nat)),fun(multiset(A),multiset(A)),pcr_multiset(A,A,fequal(A)),bNF_rel_fun(fun(A,nat),multiset(A),fun(A,nat),multiset(A),pcr_multiset(A,A,fequal(A)),pcr_multiset(A,A,fequal(A)))),aTP_Lamp_tq(fun(A,nat),fun(fun(A,nat),fun(A,nat)))),plus_plus(multiset(A)))) ).
% plus_multiset.transfer
tff(fact_5585_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(D,C),G: fun(A,option(D)),H: fun(B,option(D))] : aa(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C)),aa(fun(option(D),option(C)),fun(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C))),comp(option(D),option(C),sum_sum(A,B)),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F)),sum_case_sum(A,option(D),B,G,H)) = sum_case_sum(A,option(C),B,aa(fun(A,option(D)),fun(A,option(C)),aa(fun(option(D),option(C)),fun(fun(A,option(D)),fun(A,option(C))),comp(option(D),option(C),A),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F)),G),aa(fun(B,option(D)),fun(B,option(C)),aa(fun(option(D),option(C)),fun(fun(B,option(D)),fun(B,option(C))),comp(option(D),option(C),B),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F)),H)) ).
% map_option_o_case_sum
tff(fact_5586_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( lattic1715443433743089157tice_F(A,F,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_vw(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F),none(A),A4)) ) ) ).
% semilattice_set.eq_fold'
tff(fact_5587_o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,H: fun(D,C),F: fun(A,D),G: fun(B,D)] : aa(fun(sum_sum(A,B),D),fun(sum_sum(A,B),C),aa(fun(D,C),fun(fun(sum_sum(A,B),D),fun(sum_sum(A,B),C)),comp(D,C,sum_sum(A,B)),H),sum_case_sum(A,D,B,F,G)) = sum_case_sum(A,C,B,aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),H),F),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),H),G)) ).
% o_case_sum
tff(fact_5588_semilattice__set_Oinfinite,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A4: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( ~ finite_finite2(A,A4)
=> ( lattic1715443433743089157tice_F(A,F,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).
% semilattice_set.infinite
tff(fact_5589_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> group_axioms(A,F,Z2,Inverse) ) ).
% group.axioms(2)
tff(fact_5590_normalize__div,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,normal6383669964737779283malize(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) ) ).
% normalize_div
tff(fact_5591_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( linorder(B)
& linorder(A) )
=> ! [F: fun(A,B),G: fun(B,A)] :
( order_strict_mono(A,B,F)
=> ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
=> ( ! [X3: A] : aa(B,A,G,aa(A,B,F,X3)) = X3
=> order_strict_mono(B,A,G) ) ) ) ) ).
% strict_mono_inv
tff(fact_5592_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% gcd.normalize_bottom
tff(fact_5593_normalize__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% normalize_1
tff(fact_5594_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% lcm.top_left_normalize
tff(fact_5595_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% lcm.top_right_normalize
tff(fact_5596_normalize__mult__unit__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [B2: A,A3: A] :
( pp(dvd_dvd(A,B2,one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).
% normalize_mult_unit_right
tff(fact_5597_normalize__mult__unit__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).
% normalize_mult_unit_left
tff(fact_5598_unit__factor__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A3)) = one_one(A) ) ) ) ).
% unit_factor_normalize
tff(fact_5599_normalize__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A3)) = one_one(A) ) ) ) ).
% normalize_unit_factor
tff(fact_5600_associated__unit,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> ( pp(dvd_dvd(A,A3,one_one(A)))
=> pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).
% associated_unit
tff(fact_5601_normalize__1__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
<=> pp(dvd_dvd(A,A3,one_one(A))) ) ) ).
% normalize_1_iff
tff(fact_5602_is__unit__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( pp(dvd_dvd(A,A3,one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).
% is_unit_normalize
tff(fact_5603_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
=> ( pp(dvd_dvd(A,A3,one_one(A)))
<=> ( A3 = one_one(A) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
tff(fact_5604_group__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group_axioms(A,F,Z2,Inverse)
<=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A7) = A7
& ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A7)),A7) = Z2 ) ) ).
% group_axioms_def
tff(fact_5605_group__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A5) = A5
=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A5)),A5) = Z2
=> group_axioms(A,F,Z2,Inverse) ) ) ).
% group_axioms.intro
tff(fact_5606_strict__mono__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( order_strict_mono(A,B,F)
=> inj_on(A,B,F,A4) ) ) ).
% strict_mono_imp_inj_on
tff(fact_5607_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_5608_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde6485984586167503788ce_set(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
tff(fact_5609_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde8507323023520639062attice(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).
% lcm.bounded_quasi_semilattice_axioms
tff(fact_5610_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> bounde8507323023520639062attice(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).
% gcd.bounded_quasi_semilattice_axioms
tff(fact_5611_group__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
<=> ( semigroup(A,F)
& group_axioms(A,F,Z2,Inverse) ) ) ).
% group_def
tff(fact_5612_group_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( semigroup(A,F)
=> ( group_axioms(A,F,Z2,Inverse)
=> group(A,F,Z2,Inverse) ) ) ).
% group.intro
tff(fact_5613_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] :
( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),F),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N),M))))
=> ( iterate_upto(A,F,N,M) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_vx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M)) ) ) ).
% Predicate.iterate_upto.psimps
tff(fact_5614_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group(A,F,Z2,Inverse)
=> semigroup(A,F) ) ).
% group.axioms(1)
tff(fact_5615_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> semigroup(A,F) ) ).
% monoid.axioms(1)
tff(fact_5616_semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
<=> ! [A7: A,B7: A,C4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A7),B7)),C4) = aa(A,A,aa(A,fun(A,A),F,A7),aa(A,A,aa(A,fun(A,A),F,B7),C4)) ) ).
% semigroup_def
tff(fact_5617_semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A5: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A5),B3)),C3) = aa(A,A,aa(A,fun(A,A),F,A5),aa(A,A,aa(A,fun(A,A),F,B3),C3))
=> semigroup(A,F) ) ).
% semigroup.intro
tff(fact_5618_semigroup_Oassoc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
( semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).
% semigroup.assoc
tff(fact_5619_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> semigroup(A,times_times(A)) ) ).
% mult.semigroup_axioms
tff(fact_5620_Predicate_Oiterate__upto_Osimps,axiom,
! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] : iterate_upto(A,F,N,M) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_vx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M)) ).
% Predicate.iterate_upto.simps
tff(fact_5621_Predicate_Oiterate__upto_Oelims,axiom,
! [A: $tType,X2: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
( ( iterate_upto(A,X2,Xa,Xb) = Y )
=> ( Y = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_vx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X2),Xa),Xb)) ) ) ).
% Predicate.iterate_upto.elims
tff(fact_5622_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_add(A)
=> semigroup(A,plus_plus(A)) ) ).
% add.semigroup_axioms
tff(fact_5623_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X2: fun(code_natural,A),Xa: code_natural,Xb: code_natural,Y: pred(A)] :
( ( iterate_upto(A,X2,Xa,Xb) = Y )
=> ( pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),X2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Xa),Xb))))
=> ~ ( ( Y = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_vx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X2),Xa),Xb)) )
=> ~ pp(aa(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool,accp(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),iterate_upto_rel(A)),aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),aa(fun(code_natural,A),fun(product_prod(code_natural,code_natural),product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))),product_Pair(fun(code_natural,A),product_prod(code_natural,code_natural)),X2),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Xa),Xb)))) ) ) ) ).
% Predicate.iterate_upto.pelims
tff(fact_5624_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] : random_iterate_upto(A,F,N,M) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),iterate_upto(A,F,N,M)) ).
% Random_Pred.iterate_upto_def
tff(fact_5625_monoid_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semigroup(A,F)
=> ( monoid_axioms(A,F,Z2)
=> monoid(A,F,Z2) ) ) ).
% monoid.intro
tff(fact_5626_monoid__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
<=> ( semigroup(A,F)
& monoid_axioms(A,F,Z2) ) ) ).
% monoid_def
tff(fact_5627_monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A5) = A5
=> ( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),Z2) = A5
=> monoid_axioms(A,F,Z2) ) ) ).
% monoid_axioms.intro
tff(fact_5628_monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid_axioms(A,F,Z2)
<=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A7) = A7
& ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,A7),Z2) = A7 ) ) ).
% monoid_axioms_def
tff(fact_5629_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid(A,F,Z2)
=> monoid_axioms(A,F,Z2) ) ).
% monoid.axioms(2)
tff(fact_5630_mod__h__bot__normalize,axiom,
! [A: $tType,H: heap_ext(product_unit),P: assn] :
( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H)
=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_normalize
tff(fact_5631_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),A4: set(C),G: fun(C,set(B))] : aa(set(B),set(A),image2(B,A,F),bind3(C,B,A4,G)) = bind3(C,A,A4,aa(fun(C,set(B)),fun(C,set(A)),aa(fun(set(B),set(A)),fun(fun(C,set(B)),fun(C,set(A))),comp(set(B),set(A),C),image2(B,A,F)),G)) ).
% image_bind
tff(fact_5632_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod(bool,product_prod(A,set(B)))] :
~ ! [B3: bool,H2: A,As: set(B)] : X2 != aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),B3),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As)) ).
% pure_assn_raw.cases
tff(fact_5633_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H: heap_ext(product_unit)] :
( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P),Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
| rep_assn(Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_iff(7)
tff(fact_5634_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H: heap_ext(product_unit)] :
( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
& rep_assn(Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_iff(5)
tff(fact_5635_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H: heap_ext(product_unit)] :
( rep_assn(aa(assn,assn,inf_inf(assn,P),Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
& rep_assn(Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ) ).
% mod_h_bot_iff(6)
tff(fact_5636_mod__h__bot__iff_I2_J,axiom,
! [H: heap_ext(product_unit)] : rep_assn(top_top(assn),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ).
% mod_h_bot_iff(2)
tff(fact_5637_times__assn__raw_Ocases,axiom,
! [X2: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [P4: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q7: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H2: heap_ext(product_unit),As: set(nat)] : X2 != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),P4),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Q7),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))) ).
% times_assn_raw.cases
tff(fact_5638_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [R2: ref(A),X3: A,H2: heap_ext(product_unit),As: set(nat)] : X2 != aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),R2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))) ) ).
% sngr_assn_raw.cases
tff(fact_5639_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
~ ! [R2: array(A),X3: list(A),H2: heap_ext(product_unit),As: set(nat)] : X2 != aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),R2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),X3),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))) ) ).
% snga_assn_raw.cases
tff(fact_5640_one__assn__raw_Ocases,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
~ ! [H2: heap_ext(product_unit),As: set(nat)] : X2 != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) ).
% one_assn_raw.cases
tff(fact_5641_mod__star__conv,axiom,
! [A4: assn,B5: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A4),B5),H)
<=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
( ( H = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22)) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As1),As22) = bot_bot(set(nat)) )
& rep_assn(A4,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1))
& rep_assn(B5,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As22)) ) ) ).
% mod_star_conv
tff(fact_5642_star__assnI,axiom,
! [P: assn,H: heap_ext(product_unit),As2: set(nat),Q: assn,As4: set(nat)] :
( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ( rep_assn(Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
=> ( ( aa(set(nat),set(nat),inf_inf(set(nat),As2),As4) = bot_bot(set(nat)) )
=> rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4))) ) ) ) ).
% star_assnI
tff(fact_5643_mod__emp__simp,axiom,
! [H: heap_ext(product_unit)] : rep_assn(one_one(assn),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ).
% mod_emp_simp
tff(fact_5644_mod__h__bot__indep,axiom,
! [P: assn,H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),bot_bot(set(nat)))) ) ).
% mod_h_bot_indep
tff(fact_5645_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),A4: set(C),G: fun(B,set(A))] : bind3(B,A,aa(set(C),set(B),image2(C,B,F),A4),G) = bind3(C,A,A4,aa(fun(C,B),fun(C,set(A)),aa(fun(B,set(A)),fun(fun(C,B),fun(C,set(A))),comp(B,set(A),C),G),F)) ).
% bind_image
tff(fact_5646_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [P3: ref(A),X2: A,H: heap_ext(product_unit)] : ~ rep_assn(sngr_assn(A,P3,X2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ).
% mod_h_bot_iff(3)
tff(fact_5647_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( heap(B)
=> ! [Q3: array(B),Y: list(B),H: heap_ext(product_unit)] : ~ rep_assn(snga_assn(B,Q3,Y),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ).
% mod_h_bot_iff(4)
tff(fact_5648_times__assn__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat)] :
( times_assn_raw(P,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As2 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22))) ) ) ).
% times_assn_raw.simps
tff(fact_5649_times__assn__raw_Oelims_I3_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ times_assn_raw(X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ? [As12: set(nat),As23: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As12),As23) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As12)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As23))) ) ) ) ).
% times_assn_raw.elims(3)
tff(fact_5650_times__assn__raw_Oelims_I2_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( times_assn_raw(X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ~ ? [As13: set(nat),As24: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As13),As24) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As13)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As24))) ) ) ) ).
% times_assn_raw.elims(2)
tff(fact_5651_times__assn__raw_Oelims_I1_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( times_assn_raw(X2,Xa,Xb)
<=> pp(Y) )
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(Y)
<=> ~ ? [As1: set(nat),As22: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As22))) ) ) ) ) ).
% times_assn_raw.elims(1)
tff(fact_5652_times__assn__raw_Opelims_I1_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( times_assn_raw(X2,Xa,Xb)
<=> pp(Y) )
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( pp(Y)
<=> ? [As1: set(nat),As22: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As1),As22) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As22))) ) )
=> ~ pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))))) ) ) ) ) ).
% times_assn_raw.pelims(1)
tff(fact_5653_times__assn__raw_Opelims_I2_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( times_assn_raw(X2,Xa,Xb)
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ~ ? [As13: set(nat),As24: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As13),As24) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As13)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As24))) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
tff(fact_5654_times__assn__raw_Opelims_I3_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ times_assn_raw(X2,Xa,Xb)
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ? [As12: set(nat),As23: set(nat)] :
( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
& ( aa(set(nat),set(nat),inf_inf(set(nat),As12),As23) = bot_bot(set(nat)) )
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As12)))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As23))) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
tff(fact_5655_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R4: fun(C,assn),H: heap_ext(product_unit)] :
( rep_assn(ex_assn(C,R4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> ? [X4: C] : rep_assn(aa(C,assn,R4,X4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ).
% mod_h_bot_iff(8)
tff(fact_5656_make__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,N,F),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_vy(nat,fun(fun(nat,A),fun(array(A),assn)),N),F)) ) ).
% make_rule
tff(fact_5657_mod__h__bot__iff_I1_J,axiom,
! [B2: bool,H: heap_ext(product_unit)] :
( rep_assn(pure_assn(B2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
<=> pp(B2) ) ).
% mod_h_bot_iff(1)
tff(fact_5658_bind__rule,axiom,
! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap(A),R4: fun(A,assn),G: fun(A,heap_Time_Heap(B)),Q: fun(B,assn)] :
( hoare_hoare_triple(A,P,F,R4)
=> ( ! [X3: A] : hoare_hoare_triple(B,aa(A,assn,R4,X3),aa(A,heap_Time_Heap(B),G,X3),Q)
=> hoare_hoare_triple(B,P,heap_Time_bind(A,B,F,G),Q) ) ) ).
% bind_rule
tff(fact_5659_raise__iff,axiom,
! [A: $tType,P: assn,S: list(char),Q: fun(A,assn)] :
( hoare_hoare_triple(A,P,heap_Time_raise(A,S),Q)
<=> ( P = bot_bot(assn) ) ) ).
% raise_iff
tff(fact_5660_raise__rule,axiom,
! [A: $tType,S: list(char),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),heap_Time_raise(A,S),Q) ).
% raise_rule
tff(fact_5661_bind__rule_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap(B),R4: fun(B,assn),G: fun(B,heap_Time_Heap(C)),Q: fun(C,assn)] :
( ! [R2: A] : hoare_hoare_triple(B,P,F,aa(fun(B,assn),fun(B,assn),aTP_Lamp_vz(assn,fun(fun(B,assn),fun(B,assn)),P),R4))
=> ( ! [X3: B] : hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,R4,X3)),P),aa(B,heap_Time_Heap(C),G,X3),Q)
=> hoare_hoare_triple(C,P,heap_Time_bind(B,C,F,G),Q) ) ) ).
% bind_rule'
tff(fact_5662_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X2: product_prod(A,B),P: assn,F: fun(A,fun(B,heap_Time_Heap(C))),Q: fun(C,assn)] :
( ! [A5: A,B3: B] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3) )
=> hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A5),B3),Q) )
=> hoare_hoare_triple(C,P,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F),X2),Q) ) ).
% case_prod_rule
tff(fact_5663_hoare__triple__success,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As2: set(nat)] :
( hoare_hoare_triple(A,P,C2,Q)
=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> heap_Time_success(A,C2,H) ) ) ).
% hoare_triple_success
tff(fact_5664_case__option__rule,axiom,
! [A: $tType,B: $tType,V: option(A),P: assn,Fn: heap_Time_Heap(B),Q: fun(B,assn),Fs: fun(A,heap_Time_Heap(B))] :
( ( ( V = none(A) )
=> hoare_hoare_triple(B,P,Fn,Q) )
=> ( ! [X3: A] :
( ( V = aa(A,option(A),some(A),X3) )
=> hoare_hoare_triple(B,P,aa(A,heap_Time_Heap(B),Fs,X3),Q) )
=> hoare_hoare_triple(B,P,aa(option(A),heap_Time_Heap(B),aa(fun(A,heap_Time_Heap(B)),fun(option(A),heap_Time_Heap(B)),aa(heap_Time_Heap(B),fun(fun(A,heap_Time_Heap(B)),fun(option(A),heap_Time_Heap(B))),case_option(heap_Time_Heap(B),A),Fn),Fs),V),Q) ) ) ).
% case_option_rule
tff(fact_5665_of__list__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : hoare_hoare_triple(array(A),one_one(assn),array_of_list(A,Xs),aTP_Lamp_wa(list(A),fun(array(A),assn),Xs)) ) ).
% of_list_rule
tff(fact_5666_new__rule,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X2: A] : hoare_hoare_triple(array(A),one_one(assn),aa(A,heap_Time_Heap(array(A)),aa(nat,fun(A,heap_Time_Heap(array(A))),array_new(A),N),X2),aa(A,fun(array(A),assn),aTP_Lamp_wb(nat,fun(A,fun(array(A),assn)),N),X2)) ) ).
% new_rule
tff(fact_5667_hoare__triple__effect,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As2: set(nat)] :
( hoare_hoare_triple(A,P,C2,Q)
=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ? [H5: heap_ext(product_unit),R2: A,T6: nat] :
( heap_Time_effect(A,C2,H,H5,R2,T6)
& rep_assn(aa(A,assn,Q,R2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),hoare_new_addrs(H,As2,H5))) ) ) ) ).
% hoare_triple_effect
tff(fact_5668_hoare__triple__def,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( hoare_hoare_triple(A,P,C2,Q)
<=> ! [H7: heap_ext(product_unit),As5: set(nat)] :
( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As5))
=> ? [R5: A,H8: heap_ext(product_unit)] :
( ? [T7: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H7) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R5),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H8),T7)))
& rep_assn(aa(A,assn,Q,R5),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),hoare_new_addrs(H7,As5,H8)))
& relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_wc(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H7),As5)),H7,H8)
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H7)),lim(product_unit,H8))) ) ) ) ).
% hoare_triple_def
tff(fact_5669_hoare__tripleI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
( ! [H2: heap_ext(product_unit),As: set(nat)] :
( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
=> ? [R14: A,H9: heap_ext(product_unit),T8: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H2) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R14),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H9),T8))) )
& rep_assn(aa(A,assn,Q,R14),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),hoare_new_addrs(H2,As,H9)))
& relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_wc(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H2),As)),H2,H9)
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H2)),lim(product_unit,H9))) ) )
=> hoare_hoare_triple(A,P,C2,Q) ) ).
% hoare_tripleI
tff(fact_5670_mod__relH,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),P: assn] :
( relH(As2,H,H3)
=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2)) ) ) ).
% mod_relH
tff(fact_5671_hoare__tripleE,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As2: set(nat)] :
( hoare_hoare_triple(A,P,C2,Q)
=> ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ~ ! [R2: A,H5: heap_ext(product_unit)] :
( ? [T6: nat] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H5),T6)))
=> ( rep_assn(aa(A,assn,Q,R2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),hoare_new_addrs(H,As2,H5)))
=> ( relH(aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_wc(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),H),As2)),H,H5)
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),lim(product_unit,H)),lim(product_unit,H5))) ) ) ) ) ) ).
% hoare_tripleE
tff(fact_5672_Array__Time_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : array_alloc(A,Xs,H) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),array2(A,lim(product_unit,H))),aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,array2(A,lim(product_unit,H)),Xs),lim_update(product_unit,aTP_Lamp_wd(heap_ext(product_unit),fun(nat,nat),H),H))) ) ).
% Array_Time.alloc_def
tff(fact_5673_snga__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( snga_assn_raw(A,X2,Xa,Xb)
<=> pp(Y) )
=> ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( pp(Y)
<=> ( ( array_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X2)),lim(product_unit,H2))) ) )
=> ~ pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))))) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
tff(fact_5674_snga__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( snga_assn_raw(A,X2,Xa,Xb)
=> ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ~ ( ( array_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X2)),lim(product_unit,H2))) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
tff(fact_5675_Array__Time_Opresent__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A)] :
( pp(aa(array(A),bool,array_present(A,H),A3))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,A3)),lim(product_unit,H))) ) ) ).
% Array_Time.present_def
tff(fact_5676_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X2: list(A),H: heap_ext(product_unit),As2: set(nat)] :
( snga_assn_raw(A,R,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> ( ( array_get(A,H,R) = X2 )
& ( As2 = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,R)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,R)),lim(product_unit,H))) ) ) ) ).
% snga_assn_raw.simps
tff(fact_5677_snga__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( snga_assn_raw(A,X2,Xa,Xb)
<=> pp(Y) )
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(Y)
<=> ~ ( ( array_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X2)),lim(product_unit,H2))) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
tff(fact_5678_snga__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( snga_assn_raw(A,X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ~ ( ( array_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X2)),lim(product_unit,H2))) ) ) ) ) ).
% snga_assn_raw.elims(2)
tff(fact_5679_snga__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ snga_assn_raw(A,X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( array_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X2)),lim(product_unit,H2))) ) ) ) ) ).
% snga_assn_raw.elims(3)
tff(fact_5680_snga__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: array(A),Xa: list(A),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ snga_assn_raw(A,X2,Xa,Xb)
=> ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),snga_assn_raw_rel(A)),aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ( ( array_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_array(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,X2)),lim(product_unit,H2))) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
tff(fact_5681_relH__set__array,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),As2: set(nat),H: heap_ext(product_unit),X2: list(A)] :
( ~ pp(member(nat,addr_of_array(A,R),As2))
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> relH(As2,H,aa(heap_ext(product_unit),heap_ext(product_unit),array_set(A,R,X2),H)) ) ) ) ).
% relH_set_array
tff(fact_5682_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( sngr_assn_raw(A,X2,Xa,Xb)
<=> pp(Y) )
=> ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( pp(Y)
<=> ( ( ref_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X2)),lim(product_unit,H2))) ) )
=> ~ pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))))) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
tff(fact_5683_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( sngr_assn_raw(A,X2,Xa,Xb)
=> ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ~ ( ( ref_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X2)),lim(product_unit,H2))) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
tff(fact_5684_in__range__empty,axiom,
! [H: heap_ext(product_unit)] : in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ).
% in_range_empty
tff(fact_5685_in__range__dist__union,axiom,
! [H: heap_ext(product_unit),As2: set(nat),As4: set(nat)] :
( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As4)))
<=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4)) ) ) ).
% in_range_dist_union
tff(fact_5686_relH__set__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),As2: set(nat),H: heap_ext(product_unit),X2: A] :
( ~ pp(member(nat,addr_of_ref(A,R),As2))
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> relH(As2,H,ref_set(A,R,X2,H)) ) ) ) ).
% relH_set_ref
tff(fact_5687_relH__refl,axiom,
! [H: heap_ext(product_unit),As2: set(nat)] :
( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> relH(As2,H,H) ) ).
% relH_refl
tff(fact_5688_relH__in__rangeI_I1_J,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
( relH(As2,H,H3)
=> in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ).
% relH_in_rangeI(1)
tff(fact_5689_relH__in__rangeI_I2_J,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
( relH(As2,H,H3)
=> in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2)) ) ).
% relH_in_rangeI(2)
tff(fact_5690_in__range__subset,axiom,
! [As2: set(nat),As4: set(nat),H: heap_ext(product_unit)] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),As2),As4))
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As4))
=> in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ) ).
% in_range_subset
tff(fact_5691_in__range_Oelims_I3_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( ~ in_range(X2)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ! [X3: nat] :
( pp(member(nat,X3,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H2))) ) ) ) ).
% in_range.elims(3)
tff(fact_5692_in__range_Oelims_I2_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( in_range(X2)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ~ ! [X: nat] :
( pp(member(nat,X,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),lim(product_unit,H2))) ) ) ) ).
% in_range.elims(2)
tff(fact_5693_in__range_Oelims_I1_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( in_range(X2)
<=> pp(Y) )
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(Y)
<=> ~ ! [X4: nat] :
( pp(member(nat,X4,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H2))) ) ) ) ) ).
% in_range.elims(1)
tff(fact_5694_in__range_Osimps,axiom,
! [H: heap_ext(product_unit),As2: set(nat)] :
( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> ! [X4: nat] :
( pp(member(nat,X4,As2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H))) ) ) ).
% in_range.simps
tff(fact_5695_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ sngr_assn_raw(A,X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( ref_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X2)),lim(product_unit,H2))) ) ) ) ) ).
% sngr_assn_raw.elims(3)
tff(fact_5696_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( sngr_assn_raw(A,X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ~ ( ( ref_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X2)),lim(product_unit,H2))) ) ) ) ) ).
% sngr_assn_raw.elims(2)
tff(fact_5697_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( sngr_assn_raw(A,X2,Xa,Xb)
<=> pp(Y) )
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(Y)
<=> ~ ( ( ref_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X2)),lim(product_unit,H2))) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
tff(fact_5698_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),X2: A,H: heap_ext(product_unit),As2: set(nat)] :
( sngr_assn_raw(A,R,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> ( ( ref_get(A,H,R) = X2 )
& ( As2 = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,R)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,R)),lim(product_unit,H))) ) ) ) ).
% sngr_assn_raw.simps
tff(fact_5699_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: ref(A),Xa: A,Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ sngr_assn_raw(A,X2,Xa,Xb)
=> ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),sngr_assn_raw_rel(A)),aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ( ( ref_get(A,H2,X2) = Xa )
& ( As = aa(set(nat),set(nat),insert2(nat,addr_of_ref(A,X2)),bot_bot(set(nat))) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_ref(A,X2)),lim(product_unit,H2))) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
tff(fact_5700_wand__assnI,axiom,
! [H: heap_ext(product_unit),As2: set(nat),Q: assn,R4: assn] :
( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
=> ( ! [H5: heap_ext(product_unit),As6: set(nat)] :
( ( aa(set(nat),set(nat),inf_inf(set(nat),As2),As6) = bot_bot(set(nat)) )
=> ( relH(As2,H,H5)
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))
=> ( rep_assn(Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6))
=> rep_assn(R4,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As6))) ) ) ) )
=> rep_assn(wand_assn(Q,R4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ) ).
% wand_assnI
tff(fact_5701_wand__raw_Osimps,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Q: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat)] :
( wand_raw(P,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& ! [H8: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As2),As7) = bot_bot(set(nat)) )
& relH(As2,H,H8)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As2))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Q,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As2),As7)))) ) ) ) ).
% wand_raw.simps
tff(fact_5702_wand__raw_Oelims_I1_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( wand_raw(X2,Xa,Xb)
<=> pp(Y) )
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(Y)
<=> ~ ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H8: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As),As7) = bot_bot(set(nat)) )
& relH(As,H2,H8)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7)))) ) ) ) ) ) ).
% wand_raw.elims(1)
tff(fact_5703_wand__raw_Oelims_I3_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ wand_raw(X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H5: heap_ext(product_unit),As6: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As),As6) = bot_bot(set(nat)) )
& relH(As,H2,H5)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6)))) ) ) ) ) ).
% wand_raw.elims(3)
tff(fact_5704_wand__raw_Oelims_I2_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( wand_raw(X2,Xa,Xb)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ~ ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H9: heap_ext(product_unit),As8: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As),As8) = bot_bot(set(nat)) )
& relH(As,H2,H9)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As8))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As8)))) ) ) ) ) ).
% wand_raw.elims(2)
tff(fact_5705_wand__raw_Opelims_I3_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( ~ wand_raw(X2,Xa,Xb)
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H5: heap_ext(product_unit),As6: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As),As6) = bot_bot(set(nat)) )
& relH(As,H2,H5)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As6))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As6)))) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
tff(fact_5706_wand__raw_Opelims_I2_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat))] :
( wand_raw(X2,Xa,Xb)
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))))
=> ~ ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H9: heap_ext(product_unit),As8: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As),As8) = bot_bot(set(nat)) )
& relH(As,H2,H9)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),As8))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H9),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As8)))) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
tff(fact_5707_wand__raw_Opelims_I1_J,axiom,
! [X2: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xa: fun(product_prod(heap_ext(product_unit),set(nat)),bool),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( wand_raw(X2,Xa,Xb)
<=> pp(Y) )
=> ( pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),Xb))))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( pp(Y)
<=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
& ! [H8: heap_ext(product_unit),As7: set(nat)] :
( ( ( aa(set(nat),set(nat),inf_inf(set(nat),As),As7) = bot_bot(set(nat)) )
& relH(As,H2,H8)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As))
& pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,X2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As7))) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As),As7)))) ) ) )
=> ~ pp(aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),bool,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),wand_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),X2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),bool),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),bool),product_prod(heap_ext(product_unit),set(nat))),Xa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))))) ) ) ) ) ).
% wand_raw.pelims(1)
tff(fact_5708_in__range_Opelims_I3_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( ~ in_range(X2)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X2))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))
=> ! [X3: nat] :
( pp(member(nat,X3,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),lim(product_unit,H2))) ) ) ) ) ) ).
% in_range.pelims(3)
tff(fact_5709_in__range_Opelims_I2_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( in_range(X2)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X2))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))
=> ~ ! [X: nat] :
( pp(member(nat,X,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),lim(product_unit,H2))) ) ) ) ) ) ).
% in_range.pelims(2)
tff(fact_5710_in__range_Opelims_I1_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( in_range(X2)
<=> pp(Y) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),X2))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( pp(Y)
<=> ! [X4: nat] :
( pp(member(nat,X4,As))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),lim(product_unit,H2))) ) )
=> ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),in_range_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))) ) ) ) ) ).
% in_range.pelims(1)
tff(fact_5711_Ref__Time_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: A,H: heap_ext(product_unit)] : ref_alloc(A,X2,H) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),ref2(A,lim(product_unit,H))),ref_set(A,ref2(A,lim(product_unit,H)),X2,lim_update(product_unit,aTP_Lamp_wd(heap_ext(product_unit),fun(nat,nat),H),H))) ) ).
% Ref_Time.alloc_def
tff(fact_5712_pure__assn__raw_Osimps,axiom,
! [B: $tType,A: $tType,B2: bool,H: A,As2: set(B)] :
( pure_assn_raw(A,B,B2,aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As2))
<=> ( ( As2 = bot_bot(set(B)) )
& pp(B2) ) ) ).
% pure_assn_raw.simps
tff(fact_5713_pure__assn__raw_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X2: bool,Xa: product_prod(A,set(B)),Y: bool] :
( ( pure_assn_raw(A,B,X2,Xa)
<=> pp(Y) )
=> ~ ! [H2: A,As: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As) )
=> ( pp(Y)
<=> ~ ( ( As = bot_bot(set(B)) )
& pp(X2) ) ) ) ) ).
% pure_assn_raw.elims(1)
tff(fact_5714_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X2: bool,Xa: product_prod(A,set(B))] :
( ~ pure_assn_raw(A,B,X2,Xa)
=> ~ ! [H2: A,As: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As) )
=> ( ( As = bot_bot(set(B)) )
& pp(X2) ) ) ) ).
% pure_assn_raw.elims(3)
tff(fact_5715_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X2: bool,Xa: product_prod(A,set(B))] :
( pure_assn_raw(A,B,X2,Xa)
=> ~ ! [H2: A,As: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As) )
=> ~ ( ( As = bot_bot(set(B)) )
& pp(X2) ) ) ) ).
% pure_assn_raw.elims(2)
tff(fact_5716_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X2: bool,Xa: product_prod(A,set(B))] :
( ~ pure_assn_raw(A,B,X2,Xa)
=> ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X2),Xa)))
=> ~ ! [H2: A,As: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As) )
=> ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X2),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As))))
=> ( ( As = bot_bot(set(B)) )
& pp(X2) ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
tff(fact_5717_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X2: bool,Xa: product_prod(A,set(B))] :
( pure_assn_raw(A,B,X2,Xa)
=> ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X2),Xa)))
=> ~ ! [H2: A,As: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As) )
=> ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X2),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As))))
=> ~ ( ( As = bot_bot(set(B)) )
& pp(X2) ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
tff(fact_5718_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X2: bool,Xa: product_prod(A,set(B)),Y: bool] :
( ( pure_assn_raw(A,B,X2,Xa)
<=> pp(Y) )
=> ( pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X2),Xa)))
=> ~ ! [H2: A,As: set(B)] :
( ( Xa = aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As) )
=> ( ( pp(Y)
<=> ( ( As = bot_bot(set(B)) )
& pp(X2) ) )
=> ~ pp(aa(product_prod(bool,product_prod(A,set(B))),bool,accp(product_prod(bool,product_prod(A,set(B))),pure_assn_raw_rel(A,B)),aa(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B))),aa(bool,fun(product_prod(A,set(B)),product_prod(bool,product_prod(A,set(B)))),product_Pair(bool,product_prod(A,set(B))),X2),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As)))) ) ) ) ) ).
% pure_assn_raw.pelims(1)
tff(fact_5719_effect__refI,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H3: heap_ext(product_unit),V: A,H: heap_ext(product_unit),N: nat] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R),H3) = ref_alloc(A,V,H) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(ref(A),ref_ref(A,V),H,H3,R,N) ) ) ) ).
% effect_refI
tff(fact_5720_ref__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A] : ref_ref(A,V) = heap_Time_heap(ref(A),aTP_Lamp_wf(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),V)) ) ).
% ref_def
tff(fact_5721_execute__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(ref(A)),fun(heap_ext(product_unit),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(ref(A)),ref_ref(A,V)),H) = aa(product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_we(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,V,H))) ) ).
% execute_ref
tff(fact_5722_effect__refE,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: ref(A),N: nat] :
( heap_Time_effect(ref(A),ref_ref(A,V),H,H3,R,N)
=> ~ ( ( ref_get(A,H3,R) = V )
=> ( ref_present(A,H3,R)
=> ( ~ ref_present(A,H,R)
=> ( N != one_one(nat) ) ) ) ) ) ) ).
% effect_refE
tff(fact_5723_next__fresh,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H3: heap_ext(product_unit),X2: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R),H3) = ref_alloc(A,X2,H) )
=> ~ ref_present(A,H,R) ) ) ).
% next_fresh
tff(fact_5724_next__present,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H3: heap_ext(product_unit),X2: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R),H3) = ref_alloc(A,X2,H) )
=> ref_present(A,H3,R) ) ) ).
% next_present
tff(fact_5725_one__assn__raw_Oelims_I3_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( ~ one_assn_raw(X2)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( As = bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(3)
tff(fact_5726_one__assn__raw_Oelims_I2_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( one_assn_raw(X2)
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( As != bot_bot(set(nat)) ) ) ) ).
% one_assn_raw.elims(2)
tff(fact_5727_one__assn__raw_Oelims_I1_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( one_assn_raw(X2)
<=> pp(Y) )
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(Y)
<=> ( As != bot_bot(set(nat)) ) ) ) ) ).
% one_assn_raw.elims(1)
tff(fact_5728_one__assn__raw_Osimps,axiom,
! [H: heap_ext(product_unit),As2: set(nat)] :
( one_assn_raw(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
<=> ( As2 = bot_bot(set(nat)) ) ) ).
% one_assn_raw.simps
tff(fact_5729_one__assn__raw_Opelims_I3_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( ~ one_assn_raw(X2)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X2))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))
=> ( As = bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
tff(fact_5730_one__assn__raw_Opelims_I2_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat))] :
( one_assn_raw(X2)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X2))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))
=> ( As != bot_bot(set(nat)) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
tff(fact_5731_one__assn__raw_Opelims_I1_J,axiom,
! [X2: product_prod(heap_ext(product_unit),set(nat)),Y: bool] :
( ( one_assn_raw(X2)
<=> pp(Y) )
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),X2))
=> ~ ! [H2: heap_ext(product_unit),As: set(nat)] :
( ( X2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As) )
=> ( ( pp(Y)
<=> ( As = bot_bot(set(nat)) ) )
=> ~ pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,accp(product_prod(heap_ext(product_unit),set(nat)),one_assn_raw_rel),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))) ) ) ) ) ).
% one_assn_raw.pelims(1)
tff(fact_5732_proper__iff,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),As2: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
( proper(P)
=> ( relH(As2,H,H3)
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2))
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
<=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2))) ) ) ) ) ).
% proper_iff
tff(fact_5733_proper__def,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( proper(P)
<=> ! [H7: heap_ext(product_unit),H8: heap_ext(product_unit),As5: set(nat)] :
( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As5)))
=> in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As5)) )
& ( ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H7),As5)))
& relH(As5,H7,H8)
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As5)) )
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),As5))) ) ) ) ).
% proper_def
tff(fact_5734_properD2,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat),H3: heap_ext(product_unit)] :
( proper(P)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
=> ( relH(As2,H,H3)
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2))) ) ) ) ) ).
% properD2
tff(fact_5735_properD1,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool),H: heap_ext(product_unit),As2: set(nat)] :
( proper(P)
=> ( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)))
=> in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2)) ) ) ).
% properD1
tff(fact_5736_properI,axiom,
! [P: fun(product_prod(heap_ext(product_unit),set(nat)),bool)] :
( ! [As: set(nat),H2: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))
=> in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)) )
=> ( ! [As: set(nat),H2: heap_ext(product_unit),H5: heap_ext(product_unit)] :
( pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As)))
=> ( relH(As,H2,H5)
=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))
=> pp(aa(product_prod(heap_ext(product_unit),set(nat)),bool,P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As))) ) ) )
=> proper(P) ) ) ).
% properI
tff(fact_5737_Array__Time_Onoteq__def,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [R: array(A),S: array(B)] :
( array_noteq(A,B,R,S)
<=> ( ( typerep_typerep(A,type2(A)) != typerep_typerep(B,type2(B)) )
| ( addr_of_array(A,R) != addr_of_array(B,S) ) ) ) ) ).
% Array_Time.noteq_def
tff(fact_5738_comm__monoid__mset_Ounion,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M2: multiset(A),N5: multiset(A)] :
( comm_monoid_mset(A,F,Z2)
=> ( aa(multiset(A),A,comm_monoid_F(A,F,Z2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(A,A,aa(A,fun(A,A),F,aa(multiset(A),A,comm_monoid_F(A,F,Z2),M2)),aa(multiset(A),A,comm_monoid_F(A,F,Z2),N5)) ) ) ).
% comm_monoid_mset.union
tff(fact_5739_prod__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> groups_monoid_list(A,times_times(A),one_one(A)) ) ).
% prod_list.monoid_list_axioms
tff(fact_5740_sum__mset_Ocomm__monoid__mset__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> comm_monoid_mset(A,plus_plus(A),zero_zero(A)) ) ).
% sum_mset.comm_monoid_mset_axioms
tff(fact_5741_prod__mset_Ocomm__monoid__mset__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> comm_monoid_mset(A,times_times(A),one_one(A)) ) ).
% prod_mset.comm_monoid_mset_axioms
tff(fact_5742_sum__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> groups_monoid_list(A,plus_plus(A),zero_zero(A)) ) ).
% sum_list.monoid_list_axioms
tff(fact_5743_Array__Time_Oset__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),X2: list(A)] : array_set(A,A3,X2) = arrays_update(product_unit,aa(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat)))),aTP_Lamp_wg(array(A),fun(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat))))),A3),X2)) ) ).
% Array_Time.set_def
tff(fact_5744_Array__Time_Oget__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A)] : array_get(A,H,A3) = aa(list(nat),list(A),map(nat,A,from_nat(A)),arrays(product_unit,H,typerep_typerep(A,type2(A)),addr_of_array(A,A3))) ) ).
% Array_Time.get_def
tff(fact_5745_Int_Osub__code_I8_J,axiom,
! [M: num,N: num] : sub(aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),dup(sub(M,N))),one_one(int)) ).
% Int.sub_code(8)
tff(fact_5746_Int_Odup__def,axiom,
! [K: int] : dup(K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),K) ).
% Int.dup_def
tff(fact_5747_relH__def,axiom,
! [As2: set(nat),H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
( relH(As2,H,H3)
<=> ( in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As2))
& in_range(aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),As2))
& ! [T7: typerep,X4: nat] :
( pp(member(nat,X4,As2))
=> ( ( refs(product_unit,H,T7,X4) = refs(product_unit,H3,T7,X4) )
& ( arrays(product_unit,H,T7,X4) = arrays(product_unit,H3,T7,X4) ) ) ) ) ) ).
% relH_def
tff(fact_5748_case__sum__o__map__sum__id,axiom,
! [A: $tType,B: $tType,C: $tType,G: fun(B,A),F: fun(C,A),X2: sum_sum(C,B)] : aa(sum_sum(C,B),A,aa(fun(sum_sum(C,B),sum_sum(A,B)),fun(sum_sum(C,B),A),aa(fun(sum_sum(A,B),A),fun(fun(sum_sum(C,B),sum_sum(A,B)),fun(sum_sum(C,B),A)),comp(sum_sum(A,B),A,sum_sum(C,B)),sum_case_sum(A,A,B,id(A),G)),sum_map_sum(C,A,B,B,F,id(B))),X2) = aa(sum_sum(C,B),A,sum_case_sum(C,A,B,aa(fun(C,C),fun(C,A),aa(fun(C,A),fun(fun(C,C),fun(C,A)),comp(C,A,C),F),id(C)),G),X2) ).
% case_sum_o_map_sum_id
tff(fact_5749_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid.axioms(2)
tff(fact_5750_sum_Omap__comp,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: fun(C,E),G22: fun(D,F4),F1: fun(A,C),F22: fun(B,D),V: sum_sum(A,B)] : aa(sum_sum(C,D),sum_sum(E,F4),sum_map_sum(C,E,D,F4,G1,G22),aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),V)) = aa(sum_sum(A,B),sum_sum(E,F4),sum_map_sum(A,E,B,F4,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),G1),F1),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G22),F22)),V) ).
% sum.map_comp
tff(fact_5751_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A5: A] : aa(A,A,aa(A,fun(A,A),F,A5),Z2) = A5
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid_axioms.intro
tff(fact_5752_comm__monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_axioms(A,F,Z2)
<=> ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,A7),Z2) = A7 ) ).
% comm_monoid_axioms_def
tff(fact_5753_case__sum__map__sum,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: fun(B,A),R: fun(C,A),F: fun(D,B),G: fun(E,C),X2: sum_sum(D,E)] : aa(sum_sum(B,C),A,sum_case_sum(B,A,C,L,R),aa(sum_sum(D,E),sum_sum(B,C),sum_map_sum(D,B,E,C,F,G),X2)) = aa(sum_sum(D,E),A,sum_case_sum(D,A,E,aa(fun(D,B),fun(D,A),aa(fun(B,A),fun(fun(D,B),fun(D,A)),comp(B,A,D),L),F),aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),R),G)),X2) ).
% case_sum_map_sum
tff(fact_5754_case__sum__o__map__sum,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(D,C),G: fun(E,C),H1: fun(A,D),H22: fun(B,E)] : aa(fun(sum_sum(A,B),sum_sum(D,E)),fun(sum_sum(A,B),C),aa(fun(sum_sum(D,E),C),fun(fun(sum_sum(A,B),sum_sum(D,E)),fun(sum_sum(A,B),C)),comp(sum_sum(D,E),C,sum_sum(A,B)),sum_case_sum(D,C,E,F,G)),sum_map_sum(A,D,B,E,H1,H22)) = sum_case_sum(A,C,B,aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),F),H1),aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),G),H22)) ).
% case_sum_o_map_sum
tff(fact_5755_map__sum_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F4: $tType,D: $tType,B: $tType,F: fun(C,E),G: fun(D,F4),H: fun(A,C),I: fun(B,D)] : aa(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),sum_sum(E,F4)),aa(fun(sum_sum(C,D),sum_sum(E,F4)),fun(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),sum_sum(E,F4))),comp(sum_sum(C,D),sum_sum(E,F4),sum_sum(A,B)),sum_map_sum(C,E,D,F4,F,G)),sum_map_sum(A,C,B,D,H,I)) = sum_map_sum(A,E,B,F4,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G),I)) ).
% map_sum.comp
tff(fact_5756_map__sum_Ocompositionality,axiom,
! [D: $tType,F4: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: fun(C,E),G: fun(D,F4),H: fun(A,C),I: fun(B,D),Sum: sum_sum(A,B)] : aa(sum_sum(C,D),sum_sum(E,F4),sum_map_sum(C,E,D,F4,F,G),aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,H,I),Sum)) = aa(sum_sum(A,B),sum_sum(E,F4),sum_map_sum(A,E,B,F4,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F4),aa(fun(D,F4),fun(fun(B,D),fun(B,F4)),comp(D,F4,B),G),I)),Sum) ).
% map_sum.compositionality
tff(fact_5757_comm__monoid__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
<=> ( abel_semigroup(A,F)
& comm_monoid_axioms(A,F,Z2) ) ) ).
% comm_monoid_def
tff(fact_5758_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> abel_semigroup(A,times_times(A)) ) ).
% mult.abel_semigroup_axioms
tff(fact_5759_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid(A,F,Z2)
=> abel_semigroup(A,F) ) ).
% comm_monoid.axioms(1)
tff(fact_5760_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),B2: A,A3: A,C2: A] :
( abel_semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C2)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).
% abel_semigroup.left_commute
tff(fact_5761_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( abel_semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).
% abel_semigroup.commute
tff(fact_5762_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> abel_semigroup(A,plus_plus(A)) ) ).
% add.abel_semigroup_axioms
tff(fact_5763_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> semigroup(A,F) ) ).
% abel_semigroup.axioms(1)
tff(fact_5764_comm__monoid_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( abel_semigroup(A,F)
=> ( comm_monoid_axioms(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ) ).
% comm_monoid.intro
tff(fact_5765_abel__semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
=> ( abel_s757365448890700780axioms(A,F)
=> abel_semigroup(A,F) ) ) ).
% abel_semigroup.intro
tff(fact_5766_abel__semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
<=> ( semigroup(A,F)
& abel_s757365448890700780axioms(A,F) ) ) ).
% abel_semigroup_def
tff(fact_5767_curr__def,axiom,
! [A: $tType,C: $tType,B: $tType,A4: set(A),F: fun(product_prod(A,B),C),X: A] :
( ( pp(member(A,X,A4))
=> ( bNF_Wellorder_curr(A,B,C,A4,F,X) = aa(A,fun(B,C),aTP_Lamp_bp(fun(product_prod(A,B),C),fun(A,fun(B,C)),F),X) ) )
& ( ~ pp(member(A,X,A4))
=> ( bNF_Wellorder_curr(A,B,C,A4,F,X) = undefined(fun(B,C)) ) ) ) ).
% curr_def
tff(fact_5768_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A5: A,B3: A] : aa(A,A,aa(A,fun(A,A),F,A5),B3) = aa(A,A,aa(A,fun(A,A),F,B3),A5)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup_axioms.intro
tff(fact_5769_abel__semigroup__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_s757365448890700780axioms(A,F)
<=> ! [A7: A,B7: A] : aa(A,A,aa(A,fun(A,A),F,A7),B7) = aa(A,A,aa(A,fun(A,A),F,B7),A7) ) ).
% abel_semigroup_axioms_def
tff(fact_5770_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup.axioms(2)
tff(fact_5771_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P3))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_wh(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z2),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_wi(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).
% comm_monoid_set.zero_middle
tff(fact_5772_iterate__add__diff__add__eq2,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [I: nat,J: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M)),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),N)) ) ) ) ).
% iterate_add_diff_add_eq2
tff(fact_5773_iterate__add__add__eq1,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [J: nat,I: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M)),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J),U)),M)),N) ) ) ) ).
% iterate_add_add_eq1
tff(fact_5774_iterate__add__simps_I2_J,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [N: nat,A3: A] : iterate_add(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),iterate_add(A,N,A3)) ) ).
% iterate_add_simps(2)
tff(fact_5775_iterate__add__distrib,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [M: nat,N: nat,A3: A] : iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,M,A3)),iterate_add(A,N,A3)) ) ).
% iterate_add_distrib
tff(fact_5776_iterate__add__1,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add(A)
& semiring_1(A) )
=> ! [N: nat] : iterate_add(A,N,one_one(A)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).
% iterate_add_1
tff(fact_5777_prod_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> groups778175481326437816id_set(A,times_times(A),one_one(A)) ) ).
% prod.comm_monoid_set_axioms
tff(fact_5778_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),K: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_wj(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),F),Z2),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ) ).
% comm_monoid_set.nat_group
tff(fact_5779_comm__monoid__set_Oub__add__nat,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A),P3: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P3)))) ) ) ) ).
% comm_monoid_set.ub_add_nat
tff(fact_5780_comm__monoid__set_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ord(B)
=> ! [F: fun(A,fun(A,A)),Z2: A,A3: B,C2: B,B2: B,D3: B,G: fun(B,A),H: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D3))
=> ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set_or7035219750837199246ssThan(B,A3,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H),set_or7035219750837199246ssThan(B,C2,D3)) ) ) ) ) ) ) ).
% comm_monoid_set.ivl_cong
tff(fact_5781_comm__monoid__set_Ohead,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).
% comm_monoid_set.head
tff(fact_5782_comm__monoid__set_Olast__plus,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).
% comm_monoid_set.last_plus
tff(fact_5783_comm__monoid__set_OatLeast__Suc__atMost,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost
tff(fact_5784_comm__monoid__set_Onat__ivl__Suc_H,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% comm_monoid_set.nat_ivl_Suc'
tff(fact_5785_comm__monoid__set_OatLeastLessThan__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: nat,B2: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).
% comm_monoid_set.atLeastLessThan_Suc
tff(fact_5786_comm__monoid__set_OatLeastLessThan__concat,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,P3: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P3))
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,N,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_concat
tff(fact_5787_comm__monoid__set_OSuc__reindex__ivl,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% comm_monoid_set.Suc_reindex_ivl
tff(fact_5788_comm__monoid__set_OatLeast1__atMost__eq,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).
% comm_monoid_set.atLeast1_atMost_eq
tff(fact_5789_comm__monoid__set_OatLeastLessThan__rev,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat,M: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_wl(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ) ).
% comm_monoid_set.atLeastLessThan_rev
tff(fact_5790_comm__monoid__set_Onested__swap,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: fun(nat,fun(nat,A)),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_wm(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_wo(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F),Z2),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).
% comm_monoid_set.nested_swap
tff(fact_5791_comm__monoid__set_Onat__diff__reindex,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aTP_Lamp_wp(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).
% comm_monoid_set.nat_diff_reindex
tff(fact_5792_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.shift_bounds_cl_Suc_ivl
tff(fact_5793_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.shift_bounds_Suc_ivl
tff(fact_5794_comm__monoid__set_OlessThan__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ) ).
% comm_monoid_set.lessThan_Suc
tff(fact_5795_comm__monoid__set_OatMost__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ).
% comm_monoid_set.atMost_Suc
tff(fact_5796_comm__monoid__set_Ohead__if,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,N: nat,M: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) = Z2 ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).
% comm_monoid_set.head_if
tff(fact_5797_comm__monoid__set_Ocl__ivl__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,N: nat,M: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = Z2 ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).
% comm_monoid_set.cl_ivl_Suc
tff(fact_5798_comm__monoid__set_Oop__ivl__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,N: nat,M: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = Z2 ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).
% comm_monoid_set.op_ivl_Suc
tff(fact_5799_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan
tff(fact_5800_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc
tff(fact_5801_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc
tff(fact_5802_comm__monoid__set_OatMost__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ) ).
% comm_monoid_set.atMost_Suc_shift
tff(fact_5803_comm__monoid__set_OatLeastAtMost__rev,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat,M: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_wq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ) ).
% comm_monoid_set.atLeastAtMost_rev
tff(fact_5804_comm__monoid__set_OlessThan__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).
% comm_monoid_set.lessThan_Suc_shift
tff(fact_5805_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.shift_bounds_cl_nat_ivl
tff(fact_5806_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.shift_bounds_nat_ivl
tff(fact_5807_sum_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> groups778175481326437816id_set(A,plus_plus(A),zero_zero(A)) ) ).
% sum.comm_monoid_set_axioms
tff(fact_5808_comm__monoid__set_OatMost__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aTP_Lamp_wk(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).
% comm_monoid_set.atMost_shift
tff(fact_5809_comm__monoid__set_Onested__swap_H,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A3: fun(nat,fun(nat,A)),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ws(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_wo(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F),Z2),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).
% comm_monoid_set.nested_swap'
tff(fact_5810_comm__monoid__set_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A4: set(B),H: fun(B,C),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,A4)
=> ( ! [X3: B,Y4: B] :
( pp(member(B,X3,A4))
=> ( pp(member(B,Y4,A4))
=> ( ( X3 != Y4 )
=> ( ( aa(B,C,H,X3) = aa(B,C,H,Y4) )
=> ( aa(C,A,G,aa(B,C,H,X3)) = Z2 ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A4) ) ) ) ) ).
% comm_monoid_set.reindex_nontrivial
tff(fact_5811_comm__monoid__set_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,H: fun(B,C),A4: set(B),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( inj_on(B,C,H,A4)
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(B),set(C),image2(B,C,H),A4)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A4) ) ) ) ).
% comm_monoid_set.reindex
tff(fact_5812_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
tff(fact_5813_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
tff(fact_5814_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_bounds
tff(fact_5815_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_bounds
tff(fact_5816_comm__monoid__set_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,fun(A,A)),Z2: A,H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_reindex
tff(fact_5817_comm__monoid__set_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,fun(A,A)),Z2: A,H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_reindex
tff(fact_5818_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
tff(fact_5819_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
tff(fact_5820_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_0
tff(fact_5821_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat,M: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_wq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ) ).
% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5822_iterate__add__eq__add__iff1,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [I: nat,J: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),M) = aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),N) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),M) = N ) ) ) ) ).
% iterate_add_eq_add_iff1
tff(fact_5823_iterate__add__eq__add__iff2,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [I: nat,J: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M) = aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N) )
<=> ( M = aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),N) ) ) ) ) ).
% iterate_add_eq_add_iff2
tff(fact_5824_iterate__add__def,axiom,
! [A: $tType] :
( cancel1802427076303600483id_add(A)
=> ! [N: nat,A3: A] : iterate_add(A,N,A3) = aa(A,A,compow(fun(A,A),N,aa(A,fun(A,A),plus_plus(A),A3)),zero_zero(A)) ) ).
% iterate_add_def
tff(fact_5825_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_bt(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
tff(fact_5826_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_bt(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
tff(fact_5827_comm__monoid__add__class_Osum__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ( groups7311177749621191930dd_sum(B,A) = groups_comm_monoid_F(A,B,plus_plus(A),zero_zero(A)) ) ) ).
% comm_monoid_add_class.sum_def
tff(fact_5828_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ( groups7121269368397514597t_prod(B,A) = groups_comm_monoid_F(A,B,times_times(A),one_one(A)) ) ) ).
% comm_monoid_mult_class.prod_def
tff(fact_5829_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M: nat,N: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_0
tff(fact_5830_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,A),fun(nat,A),aTP_Lamp_wt(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.in_pairs
tff(fact_5831_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,A),fun(nat,A),aTP_Lamp_wt(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).
% comm_monoid_set.in_pairs_0
tff(fact_5832_iterate__add__less__eq__iff2,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add(A)
& ordere2412721322843649153imp_le(A) )
=> ! [I: nat,J: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M)),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),N))) ) ) ) ).
% iterate_add_less_eq_iff2
tff(fact_5833_iterate__add__less__eq__iff1,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add(A)
& ordere2412721322843649153imp_le(A) )
=> ! [J: nat,I: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M)),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J),U)),M)),N)) ) ) ) ).
% iterate_add_less_eq_iff1
tff(fact_5834_iterate__add__less__iff1,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add(A)
& ordere2412721322843649153imp_le(A) )
=> ! [J: nat,I: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M)),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J),U)),M)),N)) ) ) ) ).
% iterate_add_less_iff1
tff(fact_5835_iterate__add__less__iff2,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add(A)
& ordere2412721322843649153imp_le(A) )
=> ! [I: nat,J: nat,U: A,M: A,N: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,I,U)),M)),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,J,U)),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),iterate_add(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I),U)),N))) ) ) ) ).
% iterate_add_less_iff2
tff(fact_5836_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,fun(nat,A)),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ej(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_wv(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).
% comm_monoid_set.triangle_reindex
tff(fact_5837_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,fun(nat,A)),N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_ds(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_wv(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).
% comm_monoid_set.triangle_reindex_eq
tff(fact_5838_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(int,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F,Z2),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_int_lessThan_int_shift
tff(fact_5839_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(int,A),M: nat,N: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F,Z2),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_int_atMost_int_shift
tff(fact_5840_old_Ounit_Ocase,axiom,
! [A: $tType,F: A] : product_case_unit(A,F,product_Unity) = F ).
% old.unit.case
tff(fact_5841_wfP__wf__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wfP(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> wf(A,R) ) ).
% wfP_wf_eq
tff(fact_5842_subset__mset_Omin__add__distrib__left,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A),Z2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),min(multiset(A),subseteq_mset(A),X2,Y)),Z2) = min(multiset(A),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Z2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Y),Z2)) ).
% subset_mset.min_add_distrib_left
tff(fact_5843_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N: multiset(A),M: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),N),min(multiset(A),subseteq_mset(A),M,N)))
<=> ( min(multiset(A),subseteq_mset(A),M,N) = N ) ) ).
% subset_mset.min_arg_le(1)
tff(fact_5844_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M: multiset(A),N: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),subseteq_mset(A),M),min(multiset(A),subseteq_mset(A),M,N)))
<=> ( min(multiset(A),subseteq_mset(A),M,N) = M ) ) ).
% subset_mset.min_arg_le(2)
tff(fact_5845_wfP__accp__iff,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( wfP(A,R)
<=> ! [X_1: A] : pp(aa(A,bool,accp(A,R),X_1)) ) ).
% wfP_accp_iff
tff(fact_5846_accp__wfPI,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( ! [X_13: A] : pp(aa(A,bool,accp(A,R),X_13))
=> wfP(A,R) ) ).
% accp_wfPI
tff(fact_5847_accp__wfPD,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X2: A] :
( wfP(A,R)
=> pp(aa(A,bool,accp(A,R),X2)) ) ).
% accp_wfPD
tff(fact_5848_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: fun(A,B),F: A,Unit: product_unit] : aa(A,B,H,product_case_unit(A,F,Unit)) = product_case_unit(B,aa(A,B,H,F),Unit) ).
% unit.case_distrib
tff(fact_5849_subset__mset_Omin__add__distrib__right,axiom,
! [A: $tType,X2: multiset(A),Y: multiset(A),Z2: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),min(multiset(A),subseteq_mset(A),Y,Z2)) = min(multiset(A),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Y),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),X2),Z2)) ).
% subset_mset.min_add_distrib_right
tff(fact_5850_accp__eq__acc,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X: A] :
( pp(aa(A,bool,accp(A,R),X))
<=> pp(member(A,X,acc(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% accp_eq_acc
tff(fact_5851_rtrancl__def,axiom,
! [A: $tType,X: set(product_prod(A,A))] : transitive_rtrancl(A,X) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_rtranclp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),X)))) ).
% rtrancl_def
tff(fact_5852_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X2: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,R),X2),Y))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y),transitive_rtrancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% Enum.rtranclp_rtrancl_eq
tff(fact_5853_acc_Ocases,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] :
( pp(member(A,A3,acc(A,R)))
=> ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),A3),R))
=> pp(member(A,Y5,acc(A,R))) ) ) ).
% acc.cases
tff(fact_5854_acc_Osimps,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] :
( pp(member(A,A3,acc(A,R)))
<=> ? [X4: A] :
( ( A3 = X4 )
& ! [Xa2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),R))
=> pp(member(A,Xa2,acc(A,R))) ) ) ) ).
% acc.simps
tff(fact_5855_acc_Ointros,axiom,
! [A: $tType,X2: A,R: set(product_prod(A,A))] :
( ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X2),R))
=> pp(member(A,Y4,acc(A,R))) )
=> pp(member(A,X2,acc(A,R))) ) ).
% acc.intros
tff(fact_5856_acc__induct,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
( pp(member(A,A3,acc(A,R)))
=> ( ! [X3: A] :
( pp(member(A,X3,acc(A,R)))
=> ( ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R))
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) ) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% acc_induct
tff(fact_5857_acc__downward,axiom,
! [A: $tType,B2: A,R: set(product_prod(A,A)),A3: A] :
( pp(member(A,B2,acc(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2),R))
=> pp(member(A,A3,acc(A,R))) ) ) ).
% acc_downward
tff(fact_5858_not__acc__down,axiom,
! [A: $tType,X2: A,R4: set(product_prod(A,A))] :
( ~ pp(member(A,X2,acc(A,R4)))
=> ~ ! [Z3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X2),R4))
=> pp(member(A,Z3,acc(A,R4))) ) ) ).
% not_acc_down
tff(fact_5859_acc__induct__rule,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
( pp(member(A,A3,acc(A,R)))
=> ( ! [X3: A] :
( pp(member(A,X3,acc(A,R)))
=> ( ! [Y5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X3),R))
=> pp(aa(A,bool,P,Y5)) )
=> pp(aa(A,bool,P,X3)) ) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% acc_induct_rule
tff(fact_5860_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay))
=> ( ! [A5: A,B3: B,Aa2: A,Ba: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)))
=> ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).
% rtranclp_induct2
tff(fact_5861_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Xa: A,Xb: B,Za2: A,Zb: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
=> ~ ! [A5: A,B3: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)))
=> ~ pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))) ) ) ) ).
% converse_rtranclpE2
tff(fact_5862_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By))
=> ( ! [A5: A,B3: B,Aa2: A,Ba: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
=> ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,A5),B3)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay)) ) ) ) ).
% converse_rtranclp_induct2
tff(fact_5863_accp__downwards__aux,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,R),B2),A3))
=> ( pp(aa(A,bool,accp(A,R),A3))
=> pp(aa(A,bool,accp(A,R),B2)) ) ) ).
% accp_downwards_aux
tff(fact_5864_accp__downwards,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A,B2: A] :
( pp(aa(A,bool,accp(A,R),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,R),B2),A3))
=> pp(aa(A,bool,accp(A,R),B2)) ) ) ).
% accp_downwards
tff(fact_5865_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A,Xa3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X),Xa3))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa3),transitive_rtrancl(A,R))) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_5866_acc__subset__induct,axiom,
! [A: $tType,D4: set(A),R4: set(product_prod(A,A)),X2: A,P: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D4),acc(A,R4)))
=> ( ! [X3: A,Z3: A] :
( pp(member(A,X3,D4))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X3),R4))
=> pp(member(A,Z3,D4)) ) )
=> ( pp(member(A,X2,D4))
=> ( ! [X3: A] :
( pp(member(A,X3,D4))
=> ( ! [Z5: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z5),X3),R4))
=> pp(aa(A,bool,P,Z5)) )
=> pp(aa(A,bool,P,X3)) ) )
=> pp(aa(A,bool,P,X2)) ) ) ) ) ).
% acc_subset_induct
tff(fact_5867_acc__downwards,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A)),B2: A] :
( pp(member(A,A3,acc(A,R)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),transitive_rtrancl(A,R)))
=> pp(member(A,B2,acc(A,R))) ) ) ).
% acc_downwards
tff(fact_5868_acc__downwards__aux,axiom,
! [A: $tType,B2: A,A3: A,R: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3),transitive_rtrancl(A,R)))
=> ( pp(member(A,A3,acc(A,R)))
=> pp(member(A,B2,acc(A,R))) ) ) ).
% acc_downwards_aux
tff(fact_5869_accp__acc__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X: A] :
( pp(aa(A,bool,accp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X))
<=> pp(member(A,X,acc(A,R))) ) ).
% accp_acc_eq
tff(fact_5870_acc__def,axiom,
! [A: $tType,X: set(product_prod(A,A))] : acc(A,X) = aa(fun(A,bool),set(A),collect(A),accp(A,aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),X))) ).
% acc_def
tff(fact_5871_scomp__fcomp,axiom,
! [A: $tType,D: $tType,E: $tType,B: $tType,C: $tType,F: fun(A,product_prod(D,E)),G: fun(D,fun(E,C)),H: fun(C,B)] : fcomp(A,C,B,product_scomp(A,D,E,C,F,G),H) = product_scomp(A,D,E,B,F,aa(fun(C,B),fun(D,fun(E,B)),aTP_Lamp_ww(fun(D,fun(E,C)),fun(fun(C,B),fun(D,fun(E,B))),G),H)) ).
% scomp_fcomp
tff(fact_5872_successively_Opelims_I2_J,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A)] :
( successively(A,X2,Xa)
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ( ( ( Xa = nil(A) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),nil(A)))) )
=> ( ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) )
=> ~ ! [X3: A,Y4: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)))))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
& successively(A,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) ) ) ) ) ) ) ) ).
% successively.pelims(2)
tff(fact_5873_successively_Opelims_I1_J,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
( ( successively(A,X2,Xa)
<=> pp(Y) )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ( ( ( Xa = nil(A) )
=> ( pp(Y)
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),nil(A)))) ) )
=> ( ! [X3: A] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
=> ( pp(Y)
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) )
=> ~ ! [X3: A,Y4: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) )
=> ( ( pp(Y)
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
& successively(A,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) ) )
=> ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2))))) ) ) ) ) ) ) ).
% successively.pelims(1)
tff(fact_5874_fcomp__apply,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C),G: fun(C,A),X2: B] : aa(B,A,fcomp(B,C,A,F,G),X2) = aa(C,A,G,aa(B,C,F,X2)) ).
% fcomp_apply
tff(fact_5875_id__fcomp,axiom,
! [B: $tType,A: $tType,G: fun(A,B)] : fcomp(A,A,B,id(A),G) = G ).
% id_fcomp
tff(fact_5876_fcomp__id,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : fcomp(A,B,B,F,id(B)) = F ).
% fcomp_id
tff(fact_5877_fcomp__scomp,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F: fun(A,E),G: fun(E,product_prod(C,D)),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,fcomp(A,E,product_prod(C,D),F,G),H) = fcomp(A,E,B,F,product_scomp(E,C,D,B,G,H)) ).
% fcomp_scomp
tff(fact_5878_fcomp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,F: fun(A,D),G: fun(D,C),H: fun(C,B)] : fcomp(A,C,B,fcomp(A,D,C,F,G),H) = fcomp(A,D,B,F,fcomp(D,C,B,G,H)) ).
% fcomp_assoc
tff(fact_5879_fcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(B,C),X: A] : aa(A,C,fcomp(A,B,C,F,G),X) = aa(B,C,G,aa(A,B,F,X)) ).
% fcomp_def
tff(fact_5880_fcomp__comp,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C),G: fun(C,B)] : fcomp(A,C,B,F,G) = aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),F) ).
% fcomp_comp
tff(fact_5881_successively_Opelims_I3_J,axiom,
! [A: $tType,X2: fun(A,fun(A,bool)),Xa: list(A)] :
( ~ successively(A,X2,Xa)
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),Xa)))
=> ~ ! [X3: A,Y4: A,Xs2: list(A)] :
( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) )
=> ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),successively_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),aa(fun(A,fun(A,bool)),fun(list(A),product_prod(fun(A,fun(A,bool)),list(A))),product_Pair(fun(A,fun(A,bool)),list(A)),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)))))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),X2,X3),Y4))
& successively(A,X2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) ) ) ) ) ) ).
% successively.pelims(3)
tff(fact_5882_folding__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [X3: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,X3)) = aa(A,fun(B,B),F,X3)
=> finite7837460588564673216axioms(A,B,F) ) ).
% folding_idem_axioms.intro
tff(fact_5883_comp__fun__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [X3: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,X3)) = aa(A,fun(B,B),F,X3)
=> finite1981172056906289455axioms(A,B,F) ) ).
% comp_fun_idem_axioms.intro
tff(fact_5884_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
( ! [X3: A,Y4: A] :
( pp(member(A,X3,S2))
=> ( pp(member(A,Y4,S2))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,X3)) = aa(A,fun(B,B),F,X3) ) ) )
=> finite6916993218817215295axioms(A,B,S2,F) ) ).
% folding_idem_on_axioms.intro
tff(fact_5885_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
( finite6916993218817215295axioms(A,B,S2,F)
<=> ! [X4: A,Y3: A] :
( pp(member(A,X4,S2))
=> ( pp(member(A,Y3,S2))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,X4)) = aa(A,fun(B,B),F,X4) ) ) ) ) ).
% folding_idem_on_axioms_def
tff(fact_5886_comp__fun__idem__axioms__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite1981172056906289455axioms(A,B,F)
<=> ! [X4: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,X4)) = aa(A,fun(B,B),F,X4) ) ).
% comp_fun_idem_axioms_def
tff(fact_5887_folding__idem__axioms__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite7837460588564673216axioms(A,B,F)
<=> ! [X4: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,X4)) = aa(A,fun(B,B),F,X4) ) ).
% folding_idem_axioms_def
tff(fact_5888_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
( finite4980608107308702382axioms(A,B,S2,F)
<=> ! [X4: A] :
( pp(member(A,X4,S2))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,X4)) = aa(A,fun(B,B),F,X4) ) ) ) ).
% comp_fun_idem_on_axioms_def
tff(fact_5889_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
( ! [X3: A] :
( pp(member(A,X3,S2))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,X3)) = aa(A,fun(B,B),F,X3) ) )
=> finite4980608107308702382axioms(A,B,S2,F) ) ).
% comp_fun_idem_on_axioms.intro
tff(fact_5890_image__mset__If,axiom,
! [A: $tType,B: $tType,P: fun(B,bool),F: fun(B,A),G: fun(B,A),A4: multiset(B)] : aa(multiset(B),multiset(A),image_mset(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_wx(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F),G)),A4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(B),multiset(A),image_mset(B,A,F),filter_mset(B,P,A4))),aa(multiset(B),multiset(A),image_mset(B,A,G),filter_mset(B,aTP_Lamp_wy(fun(B,bool),fun(B,bool),P),A4))) ).
% image_mset_If
tff(fact_5891_union__filter__mset__complement,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool),M2: multiset(A)] :
( ! [X3: A] :
( pp(aa(A,bool,P,X3))
<=> ~ pp(aa(A,bool,Q,X3)) )
=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),filter_mset(A,P,M2)),filter_mset(A,Q,M2)) = M2 ) ) ).
% union_filter_mset_complement
tff(fact_5892_filter__union__mset,axiom,
! [A: $tType,P: fun(A,bool),M2: multiset(A),N5: multiset(A)] : filter_mset(A,P,aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M2),N5)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),filter_mset(A,P,M2)),filter_mset(A,P,N5)) ).
% filter_union_mset
tff(fact_5893_multiset__partition,axiom,
! [A: $tType,M2: multiset(A),P: fun(A,bool)] : M2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),filter_mset(A,P,M2)),filter_mset(A,aTP_Lamp_ol(fun(A,bool),fun(A,bool),P),M2)) ).
% multiset_partition
tff(fact_5894_heap_Olub__upper,axiom,
! [A: $tType,A4: set(heap_Time_Heap(A)),X2: heap_Time_Heap(A)] :
( comple1602240252501008431_chain(heap_Time_Heap(A),heap_Time_Heap_ord(A),A4)
=> ( pp(member(heap_Time_Heap(A),X2,A4))
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X2),aa(set(heap_Time_Heap(A)),heap_Time_Heap(A),heap_Time_Heap_lub(A),A4))) ) ) ).
% heap.lub_upper
tff(fact_5895_heap_Olub__least,axiom,
! [A: $tType,A4: set(heap_Time_Heap(A)),Z2: heap_Time_Heap(A)] :
( comple1602240252501008431_chain(heap_Time_Heap(A),heap_Time_Heap_ord(A),A4)
=> ( ! [X3: heap_Time_Heap(A)] :
( pp(member(heap_Time_Heap(A),X3,A4))
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X3),Z2)) )
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),aa(set(heap_Time_Heap(A)),heap_Time_Heap(A),heap_Time_Heap_lub(A),A4)),Z2)) ) ) ).
% heap.lub_least
tff(fact_5896_plus__int__code_I6_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M)),neg(N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% plus_int_code(6)
tff(fact_5897_ccpo__Sup__mono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A4: set(A),B5: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
=> ( comple1602240252501008431_chain(A,ord_less_eq(A),B5)
=> ( ! [X3: A] :
( pp(member(A,X3,A4))
=> ? [Xa3: A] :
( pp(member(A,Xa3,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ) ).
% ccpo_Sup_mono
tff(fact_5898_plus__int__code_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M)),pos(N)) = sub(N,M) ).
% plus_int_code(5)
tff(fact_5899_plus__int__code_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),pos(M)),neg(N)) = sub(M,N) ).
% plus_int_code(4)
tff(fact_5900_minus__int__code_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),pos(M)),neg(N)) = pos(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% minus_int_code(4)
tff(fact_5901_plus__int__code_I3_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),pos(M)),pos(N)) = pos(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% plus_int_code(3)
tff(fact_5902_minus__int__code_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M)),pos(N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% minus_int_code(5)
tff(fact_5903_Unity__def,axiom,
product_Unity = aa(bool,product_unit,product_Abs_unit,fTrue) ).
% Unity_def
tff(fact_5904_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X: A] :
( pp(aa(A,bool,domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X))
<=> pp(member(A,X,domain(A,B,R))) ) ).
% Domainp_Domain_eq
tff(fact_5905_Domain__def,axiom,
! [B: $tType,A: $tType,X: set(product_prod(A,B))] : domain(A,B,X) = aa(fun(A,bool),set(A),collect(A),domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),X))) ).
% Domain_def
tff(fact_5906_Abs__unit__inject,axiom,
! [X2: bool,Y: bool] :
( pp(member(bool,X2,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))))
=> ( pp(member(bool,Y,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))))
=> ( ( aa(bool,product_unit,product_Abs_unit,X2) = aa(bool,product_unit,product_Abs_unit,Y) )
<=> ( pp(X2)
<=> pp(Y) ) ) ) ) ).
% Abs_unit_inject
tff(fact_5907_Abs__unit__induct,axiom,
! [P: fun(product_unit,bool),X2: product_unit] :
( ! [Y4: bool] :
( pp(member(bool,Y4,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))))
=> pp(aa(product_unit,bool,P,aa(bool,product_unit,product_Abs_unit,Y4))) )
=> pp(aa(product_unit,bool,P,X2)) ) ).
% Abs_unit_induct
tff(fact_5908_Abs__unit__cases,axiom,
! [X2: product_unit] :
~ ! [Y4: bool] :
( ( X2 = aa(bool,product_unit,product_Abs_unit,Y4) )
=> ~ pp(member(bool,Y4,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool))))) ) ).
% Abs_unit_cases
tff(fact_5909_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : domainp(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) = aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),domainp(A,B,R4)) ).
% Heap.Domainp_rel
tff(fact_5910_integer_Odomain,axiom,
! [P: fun(int,bool)] :
( ( domainp(int,int,fequal(int)) = P )
=> ( domainp(int,code_integer,code_pcr_integer) = P ) ) ).
% integer.domain
tff(fact_5911_natural_Odomain,axiom,
! [P: fun(nat,bool)] :
( ( domainp(nat,nat,fequal(nat)) = P )
=> ( domainp(nat,code_natural,code_pcr_natural) = P ) ) ).
% natural.domain
tff(fact_5912_option_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : domainp(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) = aa(fun(A,bool),fun(option(A),bool),pred_option(A),domainp(A,B,R4)) ).
% option.Domainp_rel
tff(fact_5913_Abs__unit__inverse,axiom,
! [Y: bool] :
( pp(member(bool,Y,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))))
=> ( pp(aa(product_unit,bool,product_Rep_unit,aa(bool,product_unit,product_Abs_unit,Y)))
<=> pp(Y) ) ) ).
% Abs_unit_inverse
tff(fact_5914_type__definition__unit,axiom,
type_definition(product_unit,bool,product_Rep_unit,product_Abs_unit,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))) ).
% type_definition_unit
tff(fact_5915_Rep__unit__inverse,axiom,
! [X2: product_unit] : aa(bool,product_unit,product_Abs_unit,aa(product_unit,bool,product_Rep_unit,X2)) = X2 ).
% Rep_unit_inverse
tff(fact_5916_Rep__unit__inject,axiom,
! [X2: product_unit,Y: product_unit] :
( ( pp(aa(product_unit,bool,product_Rep_unit,X2))
<=> pp(aa(product_unit,bool,product_Rep_unit,Y)) )
<=> ( X2 = Y ) ) ).
% Rep_unit_inject
tff(fact_5917_Rep__unit,axiom,
! [X2: product_unit] : pp(member(bool,aa(product_unit,bool,product_Rep_unit,X2),aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool))))) ).
% Rep_unit
tff(fact_5918_Rep__unit__cases,axiom,
! [Y: bool] :
( pp(member(bool,Y,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))))
=> ~ ! [X3: product_unit] :
( pp(Y)
<=> ~ pp(aa(product_unit,bool,product_Rep_unit,X3)) ) ) ).
% Rep_unit_cases
tff(fact_5919_Rep__unit__induct,axiom,
! [Y: bool,P: fun(bool,bool)] :
( pp(member(bool,Y,aa(set(bool),set(bool),insert2(bool,fTrue),bot_bot(set(bool)))))
=> ( ! [X3: product_unit] : pp(aa(bool,bool,P,aa(product_unit,bool,product_Rep_unit,X3)))
=> pp(aa(bool,bool,P,Y)) ) ) ).
% Rep_unit_induct
tff(fact_5920_adm__wf__def,axiom,
! [B: $tType,A: $tType,R4: set(product_prod(A,A)),F5: fun(fun(A,B),fun(A,B))] :
( adm_wf(A,B,R4,F5)
<=> ! [F6: fun(A,B),G5: fun(A,B),X4: A] :
( ! [Z4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X4),R4))
=> ( aa(A,B,F6,Z4) = aa(A,B,G5,Z4) ) )
=> ( aa(A,B,aa(fun(A,B),fun(A,B),F5,F6),X4) = aa(A,B,aa(fun(A,B),fun(A,B),F5,G5),X4) ) ) ) ).
% adm_wf_def
tff(fact_5921_folding__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A] :
( finite_folding_idem(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,X2)) = aa(A,fun(B,B),F,X2) ) ) ).
% folding_idem.comp_fun_idem
tff(fact_5922_compat__def,axiom,
! [A: $tType,A2: $tType,R: set(product_prod(A,A)),R3: set(product_prod(A2,A2)),F: fun(A,A2)] :
( bNF_Wellorder_compat(A,A2,R,R3,F)
<=> ! [A7: A,B7: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),R))
=> pp(member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,F,A7)),aa(A,A2,F,B7)),R3)) ) ) ).
% compat_def
tff(fact_5923_positive__add,axiom,
! [X2: rat,Y: rat] :
( positive(X2)
=> ( positive(Y)
=> positive(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),X2),Y)) ) ) ).
% positive_add
tff(fact_5924_inj__on__iff__Uniq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A4: set(A)] :
( inj_on(A,B,F,A4)
<=> ! [X4: A] :
( pp(member(A,X4,A4))
=> uniq(A,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_wz(fun(A,B),fun(set(A),fun(A,fun(A,bool))),F),A4),X4)) ) ) ).
% inj_on_iff_Uniq
tff(fact_5925_mono__imp__mono__on,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( order_mono(A,B,F)
=> mono_on(A,B,F,A4) ) ) ).
% mono_imp_mono_on
tff(fact_5926_mono__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [F: fun(A,B),A4: set(A),B5: set(A)] :
( mono_on(A,B,F,A4)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A4))
=> mono_on(A,B,F,B5) ) ) ) ).
% mono_on_subset
tff(fact_5927_mono__on__greaterD,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [G: fun(A,B),A4: set(A),X2: A,Y: A] :
( mono_on(A,B,G,A4)
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,G,Y)),aa(A,B,G,X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X2)) ) ) ) ) ) ).
% mono_on_greaterD
tff(fact_5928_mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [F: fun(A,B),A4: set(A),R: A,S: A] :
( mono_on(A,B,F,A4)
=> ( pp(member(A,R,A4))
=> ( pp(member(A,S,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),S))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,R)),aa(A,B,F,S))) ) ) ) ) ) ).
% mono_onD
tff(fact_5929_mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [A4: set(A),F: fun(A,B)] :
( ! [R2: A,S4: A] :
( pp(member(A,R2,A4))
=> ( pp(member(A,S4,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R2),S4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,R2)),aa(A,B,F,S4))) ) ) )
=> mono_on(A,B,F,A4) ) ) ).
% mono_onI
tff(fact_5930_mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( mono_on(A,B,F,A4)
<=> ! [R5: A,S6: A] :
( ( pp(member(A,R5,A4))
& pp(member(A,S6,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R5),S6)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,R5)),aa(A,B,F,S6))) ) ) ) ).
% mono_on_def
tff(fact_5931_strict__mono__on__imp__mono__on,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& preorder(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( strict_mono_on(A,B,F,A4)
=> mono_on(A,B,F,A4) ) ) ).
% strict_mono_on_imp_mono_on
tff(fact_5932_folding_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [Y4: A,X3: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y4))
=> finite_folding(A,B,F) ) ).
% folding.intro
tff(fact_5933_folding_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Y: A,X2: A] :
( finite_folding(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X2)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,Y)) ) ) ).
% folding.comp_fun_commute
tff(fact_5934_strict__mono__on__eqD,axiom,
! [D: $tType,C: $tType] :
( ( linorder(C)
& preorder(D) )
=> ! [F: fun(C,D),A4: set(C),X2: C,Y: C] :
( strict_mono_on(C,D,F,A4)
=> ( ( aa(C,D,F,X2) = aa(C,D,F,Y) )
=> ( pp(member(C,X2,A4))
=> ( pp(member(C,Y,A4))
=> ( Y = X2 ) ) ) ) ) ) ).
% strict_mono_on_eqD
tff(fact_5935_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& preorder(B) )
=> ! [F: fun(A,B),A4: set(A),X2: A,Y: A] :
( strict_mono_on(A,B,F,A4)
=> ( pp(member(A,X2,A4))
=> ( pp(member(A,Y,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y))) ) ) ) ) ) ).
% strict_mono_on_leD
tff(fact_5936_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( strict_mono_on(A,B,F,A4)
<=> ! [R5: A,S6: A] :
( ( pp(member(A,R5,A4))
& pp(member(A,S6,A4))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R5),S6)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,R5)),aa(A,B,F,S6))) ) ) ) ).
% strict_mono_on_def
tff(fact_5937_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [A4: set(A),F: fun(A,B)] :
( ! [R2: A,S4: A] :
( pp(member(A,R2,A4))
=> ( pp(member(A,S4,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R2),S4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,R2)),aa(A,B,F,S4))) ) ) )
=> strict_mono_on(A,B,F,A4) ) ) ).
% strict_mono_onI
tff(fact_5938_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ord(A)
& ord(B) )
=> ! [F: fun(A,B),A4: set(A),R: A,S: A] :
( strict_mono_on(A,B,F,A4)
=> ( pp(member(A,R,A4))
=> ( pp(member(A,S,A4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R),S))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,R)),aa(A,B,F,S))) ) ) ) ) ) ).
% strict_mono_onD
tff(fact_5939_strict__mono__on__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& preorder(B) )
=> ! [F: fun(A,B),A4: set(A)] :
( strict_mono_on(A,B,F,A4)
=> inj_on(A,B,F,A4) ) ) ).
% strict_mono_on_imp_inj_on
tff(fact_5940_folding__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite_folding(A,B,F)
<=> ! [Y3: A,X4: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y3)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y3)) ) ).
% folding_def
tff(fact_5941_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B)),X2: A,Y: A] :
( finite1890593828518410140dem_on(A,B,S2,F)
=> ( pp(member(A,X2,S2))
=> ( pp(member(A,Y,S2))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,X2)) = aa(A,fun(B,B),F,X2) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
tff(fact_5942_sum_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [A4: set(B),B5: set(C),G: fun(sum_sum(B,C),A)] :
( finite_finite2(B,A4)
=> ( finite_finite2(C,B5)
=> ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups7311177749621191930dd_sum(sum_sum(B,C),A),G),sum_Plus(B,C,A4,B5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A4)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B5)) ) ) ) ) ).
% sum.Plus
tff(fact_5943_map__sum__Inl__conv,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,Fl: fun(C,A),Fr: fun(D,B),S: sum_sum(C,D),Y: A] :
( ( aa(sum_sum(C,D),sum_sum(A,B),sum_map_sum(C,A,D,B,Fl,Fr),S) = aa(A,sum_sum(A,B),sum_Inl(A,B),Y) )
<=> ? [X4: C] :
( ( S = aa(C,sum_sum(C,D),sum_Inl(C,D),X4) )
& ( Y = aa(C,A,Fl,X4) ) ) ) ).
% map_sum_Inl_conv
tff(fact_5944_map__sum__Inr__conv,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Fl: fun(C,A),Fr: fun(D,B),S: sum_sum(C,D),Y: B] :
( ( aa(sum_sum(C,D),sum_sum(A,B),sum_map_sum(C,A,D,B,Fl,Fr),S) = aa(B,sum_sum(A,B),sum_Inr(B,A),Y) )
<=> ? [X4: D] :
( ( S = aa(D,sum_sum(C,D),sum_Inr(D,C),X4) )
& ( Y = aa(D,B,Fr,X4) ) ) ) ).
% map_sum_Inr_conv
tff(fact_5945_Union__plus,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(sum_sum(B,C),set(A)),A4: set(B),B5: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(sum_sum(B,C)),set(set(A)),image2(sum_sum(B,C),set(A),F),sum_Plus(B,C,A4,B5))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_xa(fun(sum_sum(B,C),set(A)),fun(B,set(A)),F)),A4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aTP_Lamp_xb(fun(sum_sum(B,C),set(A)),fun(C,set(A)),F)),B5))) ).
% Union_plus
tff(fact_5946_Union__sum,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(sum_sum(A,B),set(C))] : aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(sum_sum(A,B)),set(set(C)),image2(sum_sum(A,B),set(C),F),top_top(set(sum_sum(A,B))))) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),sup_sup(set(C)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aTP_Lamp_xc(fun(sum_sum(A,B),set(C)),fun(A,set(C)),F)),top_top(set(A))))),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aTP_Lamp_xd(fun(sum_sum(A,B),set(C)),fun(B,set(C)),F)),top_top(set(B))))) ).
% Union_sum
tff(fact_5947_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa: fun(A,nat),X2: fun(B,nat),X1: A] : basic_BNF_size_sum(A,B,Xa,X2,aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X1)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(1)
tff(fact_5948_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa: fun(A,nat),X2: fun(B,nat),X22: B] : basic_BNF_size_sum(A,B,Xa,X2,aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,X2,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(2)
tff(fact_5949_Node__def,axiom,
! [A: $tType,B: $tType] : old_Node(B,A) = aa(fun(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),collect(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),aTP_Lamp_xe(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool)) ).
% Node_def
tff(fact_5950_Node__K0__I,axiom,
! [B: $tType,A: $tType,A3: sum_sum(B,nat)] : pp(member(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(fun(nat,sum_sum(A,nat)),fun(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aTP_Lamp_xf(nat,sum_sum(A,nat))),A3),old_Node(A,B))) ).
% Node_K0_I
tff(fact_5951_nth__item_Opinduct,axiom,
! [A0: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,accp(nat,nth_item_rel),A0))
=> ( ( pp(aa(nat,bool,accp(nat,nth_item_rel),zero_zero(nat)))
=> pp(aa(nat,bool,P,zero_zero(nat))) )
=> ( ! [N3: nat] :
( pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N3)))
=> ( ! [A14: nat,Aa3: nat] :
( ( nat_sum_decode(N3) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A14) )
=> ( ( nat_sum_decode(A14) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Aa3) )
=> pp(aa(nat,bool,P,Aa3)) ) )
=> ( ! [A14: nat,B6: nat] :
( ( nat_sum_decode(N3) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),A14) )
=> ( ( nat_sum_decode(A14) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B6) )
=> pp(aa(nat,bool,P,B6)) ) )
=> ( ! [B6: nat,Ba2: nat,X: nat,Y5: nat] :
( ( nat_sum_decode(N3) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B6) )
=> ( ( nat_sum_decode(B6) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
=> ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y5) = nat_prod_decode(Ba2) )
=> pp(aa(nat,bool,P,X)) ) ) )
=> ( ! [B6: nat,Ba2: nat,X: nat,Y5: nat] :
( ( nat_sum_decode(N3) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),B6) )
=> ( ( nat_sum_decode(B6) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Ba2) )
=> ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Y5) = nat_prod_decode(Ba2) )
=> pp(aa(nat,bool,P,Y5)) ) ) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) ) ) ) )
=> pp(aa(nat,bool,P,A0)) ) ) ) ).
% nth_item.pinduct
tff(fact_5952_ndepth__K0,axiom,
! [A: $tType,B: $tType,X2: sum_sum(A,nat)] : old_ndepth(A,B,old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_xg(nat,sum_sum(B,nat))),X2))) = zero_zero(nat) ).
% ndepth_K0
tff(fact_5953_Atom__def,axiom,
! [B: $tType,A: $tType,X: sum_sum(A,nat)] : old_Atom(A,B,X) = aa(set(old_node(A,B)),set(old_node(A,B)),insert2(old_node(A,B),old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_xg(nat,sum_sum(B,nat))),X))),bot_bot(set(old_node(A,B)))) ).
% Atom_def
tff(fact_5954_nth__item_Opsimps_I1_J,axiom,
! [A: $tType] :
( countable(A)
=> ( pp(aa(nat,bool,accp(nat,nth_item_rel),zero_zero(nat)))
=> ( nth_item(A,zero_zero(nat)) = undefined(set(old_node(A,product_unit))) ) ) ) ).
% nth_item.psimps(1)
tff(fact_5955_nth__item_Opelims,axiom,
! [A: $tType] :
( countable(A)
=> ! [X2: nat,Y: set(old_node(A,product_unit))] :
( ( nth_item(A,X2) = Y )
=> ( pp(aa(nat,bool,accp(nat,nth_item_rel),X2))
=> ( ( ( X2 = zero_zero(nat) )
=> ( ( Y = undefined(set(old_node(A,product_unit))) )
=> ~ pp(aa(nat,bool,accp(nat,nth_item_rel),zero_zero(nat))) ) )
=> ~ ! [N3: nat] :
( ( X2 = aa(nat,nat,suc,N3) )
=> ( ( Y = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_xj(nat,set(old_node(A,product_unit))),aTP_Lamp_xn(nat,set(old_node(A,product_unit)))),nat_sum_decode(N3)) )
=> ~ pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N3))) ) ) ) ) ) ) ).
% nth_item.pelims
tff(fact_5956_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( countable(A)
=> ! [N: nat] :
( pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N)))
=> ( nth_item(A,aa(nat,nat,suc,N)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_xj(nat,set(old_node(A,product_unit))),aTP_Lamp_xn(nat,set(old_node(A,product_unit)))),nat_sum_decode(N)) ) ) ) ).
% nth_item.psimps(2)
tff(fact_5957_Scons__def,axiom,
! [B: $tType,A: $tType,M2: set(old_node(A,B)),N5: set(old_node(A,B))] : old_Scons(A,B,M2,N5) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),sup_sup(set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),one_one(nat)))),M2)),aa(set(old_node(A,B)),set(old_node(A,B)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,one_one(nat))))),N5)) ).
% Scons_def
tff(fact_5958_In1__def,axiom,
! [B: $tType,A: $tType,M2: set(old_node(A,B))] : old_In1(A,B,M2) = old_Scons(A,B,old_Numb(A,B,one_one(nat)),M2) ).
% In1_def
tff(fact_5959_dsum__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dsum(A,B,R,S) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),sup_sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_xo(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),R))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_xp(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),S))) ).
% dsum_def
tff(fact_5960_dsumE,axiom,
! [B: $tType,A: $tType,W: product_prod(set(old_node(A,B)),set(old_node(A,B))),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),W,old_dsum(A,B,R,S)))
=> ( ! [X3: set(old_node(A,B)),X5: set(old_node(A,B))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X3),X5),R))
=> ( W != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,X3)),old_In0(A,B,X5)) ) )
=> ~ ! [Y4: set(old_node(A,B)),Y11: set(old_node(A,B))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y4),Y11),S))
=> ( W != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,Y4)),old_In1(A,B,Y11)) ) ) ) ) ).
% dsumE
tff(fact_5961_dsum__In0I,axiom,
! [B: $tType,A: $tType,M2: set(old_node(A,B)),M9: set(old_node(A,B)),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M2),M9),R))
=> pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,M2)),old_In0(A,B,M9)),old_dsum(A,B,R,S))) ) ).
% dsum_In0I
tff(fact_5962_dsum__In1I,axiom,
! [B: $tType,A: $tType,N5: set(old_node(A,B)),N10: set(old_node(A,B)),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N5),N10),S))
=> pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,N5)),old_In1(A,B,N10)),old_dsum(A,B,R,S))) ) ).
% dsum_In1I
tff(fact_5963_dprod__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dprod(A,B,R,S) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_xr(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),S))),R)) ).
% dprod_def
tff(fact_5964_dprodI,axiom,
! [B: $tType,A: $tType,M2: set(old_node(A,B)),M9: set(old_node(A,B)),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),N5: set(old_node(A,B)),N10: set(old_node(A,B)),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M2),M9),R))
=> ( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N5),N10),S))
=> pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,M2,N5)),old_Scons(A,B,M9,N10)),old_dprod(A,B,R,S))) ) ) ).
% dprodI
tff(fact_5965_dprodE,axiom,
! [B: $tType,A: $tType,C2: product_prod(set(old_node(A,B)),set(old_node(A,B))),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),C2,old_dprod(A,B,R,S)))
=> ~ ! [X3: set(old_node(A,B)),Y4: set(old_node(A,B)),X5: set(old_node(A,B))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X3),X5),R))
=> ! [Y11: set(old_node(A,B))] :
( pp(member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y4),Y11),S))
=> ( C2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,X3,Y4)),old_Scons(A,B,X5,Y11)) ) ) ) ) ).
% dprodE
tff(fact_5966_prop__matchD,axiom,
! [A: $tType,Al: list(A),E2: A,Bl: list(A),Al2: list(A),E5: A,Bl2: list(A),P: fun(A,bool)] :
( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),Bl2)) )
=> ( list_all(A,P,Al)
=> ( ~ pp(aa(A,bool,P,E2))
=> ( ~ pp(aa(A,bool,P,E5))
=> ( list_all(A,P,Bl)
=> ( ( Al = Al2 )
& ( E2 = E5 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
tff(fact_5967_prop__match,axiom,
! [A: $tType,P: fun(A,bool),Al: list(A),E2: A,E5: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
( list_all(A,P,Al)
=> ( ~ pp(aa(A,bool,P,E2))
=> ( ~ pp(aa(A,bool,P,E5))
=> ( list_all(A,P,Bl)
=> ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),Bl2)) )
<=> ( ( Al = Al2 )
& ( E2 = E5 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
tff(fact_5968_Heap__lub__def,axiom,
! [A: $tType] : heap_Time_Heap_lub(A) = partial_img_lub(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_execute(A),heap_Time_Heap2(A),partial_fun_lub(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),partial_flat_lub(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).
% Heap_lub_def
tff(fact_5969_heap__step__admissible,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,P: fun(E,fun(A,fun(C,fun(B,fun(D,bool))))),X2: E] : comple1908693960933563346ssible(fun(A,option(product_prod(B,product_prod(C,D)))),partial_fun_lub(option(product_prod(B,product_prod(C,D))),option(product_prod(B,product_prod(C,D))),A,partial_flat_lub(option(product_prod(B,product_prod(C,D))),none(product_prod(B,product_prod(C,D))))),partial_fun_ord(option(product_prod(B,product_prod(C,D))),option(product_prod(B,product_prod(C,D))),A,partial_flat_ord(option(product_prod(B,product_prod(C,D))),none(product_prod(B,product_prod(C,D))))),aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),bool),aTP_Lamp_xs(fun(E,fun(A,fun(C,fun(B,fun(D,bool))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),bool)),P),X2)) ).
% heap_step_admissible
tff(fact_5970_option__admissible,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : comple1908693960933563346ssible(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),aTP_Lamp_xt(fun(A,fun(B,bool)),fun(fun(A,option(B)),bool),P)) ).
% option_admissible
tff(fact_5971_option_Olub__upper,axiom,
! [A: $tType,A4: set(option(A)),X2: option(A)] :
( comple1602240252501008431_chain(option(A),partial_flat_ord(option(A),none(A)),A4)
=> ( pp(member(option(A),X2,A4))
=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),X2),aa(set(option(A)),option(A),partial_flat_lub(option(A),none(A)),A4))) ) ) ).
% option.lub_upper
tff(fact_5972_option_Oleq__antisym,axiom,
! [A: $tType,X2: option(A),Y: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),X2),Y))
=> ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),Y),X2))
=> ( X2 = Y ) ) ) ).
% option.leq_antisym
tff(fact_5973_option_Oleq__trans,axiom,
! [A: $tType,X2: option(A),Y: option(A),Z2: option(A)] :
( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),X2),Y))
=> ( pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),Y),Z2))
=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),X2),Z2)) ) ) ).
% option.leq_trans
tff(fact_5974_option_Oleq__refl,axiom,
! [A: $tType,X2: option(A)] : pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),X2),X2)) ).
% option.leq_refl
tff(fact_5975_option_Olub__least,axiom,
! [A: $tType,A4: set(option(A)),Z2: option(A)] :
( comple1602240252501008431_chain(option(A),partial_flat_ord(option(A),none(A)),A4)
=> ( ! [X3: option(A)] :
( pp(member(option(A),X3,A4))
=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),X3),Z2)) )
=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),partial_flat_ord(option(A),none(A)),aa(set(option(A)),option(A),partial_flat_lub(option(A),none(A)),A4)),Z2)) ) ) ).
% option.lub_least
tff(fact_5976_Heap__ord__def,axiom,
! [A: $tType] : heap_Time_Heap_ord(A) = partial_img_ord(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,heap_Time_execute(A),partial_fun_ord(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),partial_flat_ord(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).
% Heap_ord_def
tff(fact_5977_Partial__Function_Obind__mono,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,B5: fun(fun(A,option(B)),option(C)),C5: fun(C,fun(fun(A,option(B)),option(D)))] :
( comple7038119648293358887notone(fun(A,option(B)),option(C),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(C),none(C)),B5)
=> ( ! [Y4: C] : comple7038119648293358887notone(fun(A,option(B)),option(D),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(D),none(D)),aa(C,fun(fun(A,option(B)),option(D)),C5,Y4))
=> comple7038119648293358887notone(fun(A,option(B)),option(D),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(D),none(D)),aa(fun(C,fun(fun(A,option(B)),option(D))),fun(fun(A,option(B)),option(D)),aTP_Lamp_xv(fun(fun(A,option(B)),option(C)),fun(fun(C,fun(fun(A,option(B)),option(D))),fun(fun(A,option(B)),option(D))),B5),C5)) ) ) ).
% Partial_Function.bind_mono
tff(fact_5978_option_Oconst__mono,axiom,
! [B: $tType,A: $tType,Ord: fun(B,fun(B,bool)),C2: option(A)] : comple7038119648293358887notone(B,option(A),Ord,partial_flat_ord(option(A),none(A)),aTP_Lamp_xw(option(A),fun(B,option(A)),C2)) ).
% option.const_mono
tff(fact_5979_option_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U3: fun(C,fun(B,option(A))),F5: fun(C,C),C5: fun(fun(B,option(A)),C),F: C,P: fun(fun(B,option(A)),bool)] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,option(A)),option(A),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),partial_flat_ord(option(A),none(A)),aa(B,fun(fun(B,option(A)),option(A)),aa(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A)))),aTP_Lamp_xx(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))))),U3),F5),C5),X3))
=> ( ( F = aa(fun(B,option(A)),C,C5,comple187402453842119260l_fixp(fun(B,option(A)),partial_fun_lub(option(A),option(A),B,partial_flat_lub(option(A),none(A))),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),aa(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A)))),aTP_Lamp_xy(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))))),U3),F5),C5))) )
=> ( ! [F3: fun(B,option(A))] : aa(C,fun(B,option(A)),U3,aa(fun(B,option(A)),C,C5,F3)) = F3
=> ( comple1908693960933563346ssible(fun(B,option(A)),partial_fun_lub(option(A),option(A),B,partial_flat_lub(option(A),none(A))),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),P)
=> ( pp(aa(fun(B,option(A)),bool,P,aTP_Lamp_bd(B,option(A))))
=> ( ! [F3: C] :
( pp(aa(fun(B,option(A)),bool,P,aa(C,fun(B,option(A)),U3,F3)))
=> pp(aa(fun(B,option(A)),bool,P,aa(C,fun(B,option(A)),U3,aa(C,C,F5,F3)))) )
=> pp(aa(fun(B,option(A)),bool,P,aa(C,fun(B,option(A)),U3,F))) ) ) ) ) ) ) ).
% option.fixp_induct_uc
tff(fact_5980_fixp__induct__option,axiom,
! [C: $tType,B: $tType,A: $tType,U3: fun(C,fun(B,option(A))),F5: fun(C,C),C5: fun(fun(B,option(A)),C),F: C,P: fun(B,fun(A,bool)),X2: B,Y: A] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,option(A)),option(A),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),partial_flat_ord(option(A),none(A)),aa(B,fun(fun(B,option(A)),option(A)),aa(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A)))),aTP_Lamp_xx(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))))),U3),F5),C5),X3))
=> ( ( F = aa(fun(B,option(A)),C,C5,comple187402453842119260l_fixp(fun(B,option(A)),partial_fun_lub(option(A),option(A),B,partial_flat_lub(option(A),none(A))),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),aa(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A)))),aTP_Lamp_xy(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))))),U3),F5),C5))) )
=> ( ! [F3: fun(B,option(A))] : aa(C,fun(B,option(A)),U3,aa(fun(B,option(A)),C,C5,F3)) = F3
=> ( ! [F3: C,X3: B,Y4: A] :
( ! [Xa3: B,Ya3: A] :
( ( aa(B,option(A),aa(C,fun(B,option(A)),U3,F3),Xa3) = aa(A,option(A),some(A),Ya3) )
=> pp(aa(A,bool,aa(B,fun(A,bool),P,Xa3),Ya3)) )
=> ( ( aa(B,option(A),aa(C,fun(B,option(A)),U3,aa(C,C,F5,F3)),X3) = aa(A,option(A),some(A),Y4) )
=> pp(aa(A,bool,aa(B,fun(A,bool),P,X3),Y4)) ) )
=> ( ( aa(B,option(A),aa(C,fun(B,option(A)),U3,F),X2) = aa(A,option(A),some(A),Y) )
=> pp(aa(A,bool,aa(B,fun(A,bool),P,X2),Y)) ) ) ) ) ) ).
% fixp_induct_option
tff(fact_5981_option_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U3: fun(C,fun(B,option(A))),F5: fun(C,C),C5: fun(fun(B,option(A)),C),F: C] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,option(A)),option(A),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),partial_flat_ord(option(A),none(A)),aa(B,fun(fun(B,option(A)),option(A)),aa(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A)))),aTP_Lamp_xx(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))))),U3),F5),C5),X3))
=> ( ( F = aa(fun(B,option(A)),C,C5,comple187402453842119260l_fixp(fun(B,option(A)),partial_fun_lub(option(A),option(A),B,partial_flat_lub(option(A),none(A))),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),aa(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A)))),aTP_Lamp_xy(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))))),U3),F5),C5))) )
=> ( ! [F3: C] : aa(fun(B,option(A)),C,C5,aa(C,fun(B,option(A)),U3,F3)) = F3
=> ( F = aa(C,C,F5,F) ) ) ) ) ).
% option.fixp_rule_uc
tff(fact_5982_option_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F5: fun(fun(B,option(A)),fun(B,option(A)))] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,option(A)),option(A),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),partial_flat_ord(option(A),none(A)),aa(B,fun(fun(B,option(A)),option(A)),aTP_Lamp_xz(fun(fun(B,option(A)),fun(B,option(A))),fun(B,fun(fun(B,option(A)),option(A))),F5),X3))
=> ( comple187402453842119260l_fixp(fun(B,option(A)),partial_fun_lub(option(A),option(A),B,partial_flat_lub(option(A),none(A))),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),F5) = aa(fun(B,option(A)),fun(B,option(A)),F5,comple187402453842119260l_fixp(fun(B,option(A)),partial_fun_lub(option(A),option(A),B,partial_flat_lub(option(A),none(A))),partial_fun_ord(option(A),option(A),B,partial_flat_ord(option(A),none(A))),F5)) ) ) ).
% option.mono_body_fixp
tff(fact_5983_option_Opartial__function__definitions__axioms,axiom,
! [A: $tType] : partia7178651479351089652itions(option(A),partial_flat_ord(option(A),none(A)),partial_flat_lub(option(A),none(A))) ).
% option.partial_function_definitions_axioms
tff(fact_5984_fixp__mono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A),G: fun(A,A)] :
( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),G)
=> ( ! [Z9: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,Z9)),aa(A,A,G,Z9)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comple115746919287870866o_fixp(A,F)),comple115746919287870866o_fixp(A,G))) ) ) ) ) ).
% fixp_mono
tff(fact_5985_heap_Oconst__mono,axiom,
! [B: $tType,A: $tType,Ord: fun(B,fun(B,bool)),C2: heap_Time_Heap(A)] : comple7038119648293358887notone(B,heap_Time_Heap(A),Ord,heap_Time_Heap_ord(A),aTP_Lamp_ya(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),C2)) ).
% heap.const_mono
tff(fact_5986_Heap__Time__Monad_Obind__mono,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,B5: fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),C5: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)))] :
( comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(C),B5)
=> ( ! [Y4: C] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(D),aa(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),C5,Y4))
=> comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(D),aa(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),aTP_Lamp_yc(fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),fun(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),B5),C5)) ) ) ).
% Heap_Time_Monad.bind_mono
tff(fact_5987_fixp__induct__heap,axiom,
! [C: $tType,A: $tType,B: $tType,U3: fun(C,fun(B,heap_Time_Heap(A))),F5: fun(C,C),C5: fun(fun(B,heap_Time_Heap(A)),C),F: C,P: fun(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool))))),X2: B,H: heap_ext(product_unit),H3: heap_ext(product_unit),R: A,N: nat] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),heap_Time_Heap_ord(A),aa(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)))),aTP_Lamp_yd(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))))),U3),F5),C5),X3))
=> ( ( F = aa(fun(B,heap_Time_Heap(A)),C,C5,comple187402453842119260l_fixp(fun(B,heap_Time_Heap(A)),partial_fun_lub(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_lub(A)),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)))),aTP_Lamp_ye(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))))),U3),F5),C5))) )
=> ( ! [F3: fun(B,heap_Time_Heap(A))] : aa(C,fun(B,heap_Time_Heap(A)),U3,aa(fun(B,heap_Time_Heap(A)),C,C5,F3)) = F3
=> ( ! [F3: C,X3: B,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: A,N3: nat] :
( ! [Xa3: B,Ha: heap_ext(product_unit),H_a: heap_ext(product_unit),Ra3: A,Na: nat] :
( heap_Time_effect(A,aa(B,heap_Time_Heap(A),aa(C,fun(B,heap_Time_Heap(A)),U3,F3),Xa3),Ha,H_a,Ra3,Na)
=> pp(aa(nat,bool,aa(A,fun(nat,bool),aa(heap_ext(product_unit),fun(A,fun(nat,bool)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool))),aa(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool)))),P,Xa3),Ha),H_a),Ra3),Na)) )
=> ( heap_Time_effect(A,aa(B,heap_Time_Heap(A),aa(C,fun(B,heap_Time_Heap(A)),U3,aa(C,C,F5,F3)),X3),H2,H5,R2,N3)
=> pp(aa(nat,bool,aa(A,fun(nat,bool),aa(heap_ext(product_unit),fun(A,fun(nat,bool)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool))),aa(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool)))),P,X3),H2),H5),R2),N3)) ) )
=> ( heap_Time_effect(A,aa(B,heap_Time_Heap(A),aa(C,fun(B,heap_Time_Heap(A)),U3,F),X2),H,H3,R,N)
=> pp(aa(nat,bool,aa(A,fun(nat,bool),aa(heap_ext(product_unit),fun(A,fun(nat,bool)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool))),aa(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,fun(nat,bool)))),P,X2),H),H3),R),N)) ) ) ) ) ) ).
% fixp_induct_heap
tff(fact_5988_refines__adm,axiom,
! [B: $tType,A: $tType,T2: fun(A,heap_Time_Heap(B))] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_yf(fun(A,heap_Time_Heap(B)),fun(fun(A,heap_Time_Heap(B)),bool),T2)) ).
% refines_adm
tff(fact_5989_heap_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U3: fun(C,fun(B,heap_Time_Heap(A))),F5: fun(C,C),C5: fun(fun(B,heap_Time_Heap(A)),C),F: C,P: fun(fun(B,heap_Time_Heap(A)),bool)] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),heap_Time_Heap_ord(A),aa(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)))),aTP_Lamp_yd(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))))),U3),F5),C5),X3))
=> ( ( F = aa(fun(B,heap_Time_Heap(A)),C,C5,comple187402453842119260l_fixp(fun(B,heap_Time_Heap(A)),partial_fun_lub(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_lub(A)),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)))),aTP_Lamp_ye(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))))),U3),F5),C5))) )
=> ( ! [F3: fun(B,heap_Time_Heap(A))] : aa(C,fun(B,heap_Time_Heap(A)),U3,aa(fun(B,heap_Time_Heap(A)),C,C5,F3)) = F3
=> ( comple1908693960933563346ssible(fun(B,heap_Time_Heap(A)),partial_fun_lub(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_lub(A)),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),P)
=> ( pp(aa(fun(B,heap_Time_Heap(A)),bool,P,aTP_Lamp_yg(B,heap_Time_Heap(A))))
=> ( ! [F3: C] :
( pp(aa(fun(B,heap_Time_Heap(A)),bool,P,aa(C,fun(B,heap_Time_Heap(A)),U3,F3)))
=> pp(aa(fun(B,heap_Time_Heap(A)),bool,P,aa(C,fun(B,heap_Time_Heap(A)),U3,aa(C,C,F5,F3)))) )
=> pp(aa(fun(B,heap_Time_Heap(A)),bool,P,aa(C,fun(B,heap_Time_Heap(A)),U3,F))) ) ) ) ) ) ) ).
% heap.fixp_induct_uc
tff(fact_5990_heap_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F5: fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)))] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),heap_Time_Heap_ord(A),aa(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)),aTP_Lamp_yh(fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))),F5),X3))
=> ( comple187402453842119260l_fixp(fun(B,heap_Time_Heap(A)),partial_fun_lub(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_lub(A)),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),F5) = aa(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)),F5,comple187402453842119260l_fixp(fun(B,heap_Time_Heap(A)),partial_fun_lub(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_lub(A)),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),F5)) ) ) ).
% heap.mono_body_fixp
tff(fact_5991_heap_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U3: fun(C,fun(B,heap_Time_Heap(A))),F5: fun(C,C),C5: fun(fun(B,heap_Time_Heap(A)),C),F: C] :
( ! [X3: B] : comple7038119648293358887notone(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),heap_Time_Heap_ord(A),aa(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)))),aTP_Lamp_yd(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))))),U3),F5),C5),X3))
=> ( ( F = aa(fun(B,heap_Time_Heap(A)),C,C5,comple187402453842119260l_fixp(fun(B,heap_Time_Heap(A)),partial_fun_lub(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_lub(A)),partial_fun_ord(heap_Time_Heap(A),heap_Time_Heap(A),B,heap_Time_Heap_ord(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)))),aTP_Lamp_ye(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))))),U3),F5),C5))) )
=> ( ! [F3: C] : aa(fun(B,heap_Time_Heap(A)),C,C5,aa(C,fun(B,heap_Time_Heap(A)),U3,F3)) = F3
=> ( F = aa(C,C,F5,F) ) ) ) ) ).
% heap.fixp_rule_uc
tff(fact_5992_admissible__heap,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,bool)))))] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_yi(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,bool))))),fun(fun(A,heap_Time_Heap(B)),bool),P)) ).
% admissible_heap
tff(fact_5993_ATP_Olambda__1,axiom,
! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_kk(product_prod(int,int),product_prod(int,int)),Uu) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uu))) ).
% ATP.lambda_1
tff(fact_5994_ATP_Olambda__2,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_xn(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_xk(nat,set(old_node(A,product_unit))),aTP_Lamp_xm(nat,set(old_node(A,product_unit)))),nat_sum_decode(Uu)) ) ).
% ATP.lambda_2
tff(fact_5995_ATP_Olambda__3,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_xj(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_xh(nat,set(old_node(A,product_unit))),aTP_Lamp_xi(nat,set(old_node(A,product_unit)))),nat_sum_decode(Uu)) ) ).
% ATP.lambda_3
tff(fact_5996_ATP_Olambda__4,axiom,
! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_io(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),finite_card(set(A),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),finite_card(A,complete_Inf_Inf(set(A),Uu)))) ).
% ATP.lambda_4
tff(fact_5997_ATP_Olambda__5,axiom,
! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_kl(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).
% ATP.lambda_5
tff(fact_5998_ATP_Olambda__6,axiom,
! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_kn(product_prod(int,int),product_prod(int,int)),Uu) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(product_prod(int,int),int,product_fst(int,int),Uu))),aa(product_prod(int,int),int,product_snd(int,int),Uu)) ).
% ATP.lambda_6
tff(fact_5999_ATP_Olambda__7,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_pq(list(A),product_prod(nat,list(A))),Uu) = aa(list(A),product_prod(nat,list(A)),aa(nat,fun(list(A),product_prod(nat,list(A))),product_Pair(nat,list(A)),aa(list(A),nat,size_size(list(A)),Uu)),Uu) ).
% ATP.lambda_7
tff(fact_6000_ATP_Olambda__8,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_bt(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).
% ATP.lambda_8
tff(fact_6001_ATP_Olambda__9,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_hf(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).
% ATP.lambda_9
tff(fact_6002_ATP_Olambda__10,axiom,
! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_jf(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).
% ATP.lambda_10
tff(fact_6003_ATP_Olambda__11,axiom,
! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_je(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).
% ATP.lambda_11
tff(fact_6004_ATP_Olambda__12,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_di(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).
% ATP.lambda_12
tff(fact_6005_ATP_Olambda__13,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] :
( pp(aa(A,bool,aTP_Lamp_uq(A,bool),Uu))
<=> pp(dvd_dvd(A,Uu,one_one(A))) ) ) ).
% ATP.lambda_13
tff(fact_6006_ATP_Olambda__14,axiom,
! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_ki(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),zero_zero(nat)) ).
% ATP.lambda_14
tff(fact_6007_ATP_Olambda__15,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,list(A),aTP_Lamp_pv(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ) ).
% ATP.lambda_15
tff(fact_6008_ATP_Olambda__16,axiom,
! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_nj(A,set(A)),Uu) = aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A))) ).
% ATP.lambda_16
tff(fact_6009_ATP_Olambda__17,axiom,
! [B: $tType,Uu: option(B)] :
( pp(aa(option(B),bool,aTP_Lamp_uc(option(B),bool),Uu))
<=> ( Uu = none(B) ) ) ).
% ATP.lambda_17
tff(fact_6010_ATP_Olambda__18,axiom,
! [A: $tType,Uu: option(A)] :
( pp(aa(option(A),bool,aTP_Lamp_ub(option(A),bool),Uu))
<=> ( Uu = none(A) ) ) ).
% ATP.lambda_18
tff(fact_6011_ATP_Olambda__19,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_xm(nat,set(old_node(A,product_unit))),Uu) = aa(product_prod(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,fun(nat,set(old_node(A,product_unit)))),fun(product_prod(nat,nat),set(old_node(A,product_unit))),product_case_prod(nat,nat,set(old_node(A,product_unit))),aTP_Lamp_xl(nat,fun(nat,set(old_node(A,product_unit))))),nat_prod_decode(Uu)) ) ).
% ATP.lambda_19
tff(fact_6012_ATP_Olambda__20,axiom,
! [Uu: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_se(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,aTP_Lamp_sd(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu)) ).
% ATP.lambda_20
tff(fact_6013_ATP_Olambda__21,axiom,
! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_la(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).
% ATP.lambda_21
tff(fact_6014_ATP_Olambda__22,axiom,
! [B: $tType,Uu: option(B)] :
( pp(aa(option(B),bool,aTP_Lamp_ua(option(B),bool),Uu))
<=> ( Uu != none(B) ) ) ).
% ATP.lambda_22
tff(fact_6015_ATP_Olambda__23,axiom,
! [A: $tType,Uu: option(A)] :
( pp(aa(option(A),bool,aTP_Lamp_tz(option(A),bool),Uu))
<=> ( Uu != none(A) ) ) ).
% ATP.lambda_23
tff(fact_6016_ATP_Olambda__24,axiom,
! [A: $tType,Uu: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_pm(list(A),bool),Uu))
<=> ( Uu != nil(A) ) ) ).
% ATP.lambda_24
tff(fact_6017_ATP_Olambda__25,axiom,
! [A: $tType,Uu: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A),aTP_Lamp_vh(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A)),Uu) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(set(A)),image2(product_prod(A,product_prod(heap_ext(product_unit),nat)),set(A),basic_fsts(A,product_prod(heap_ext(product_unit),nat))),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu))) ).
% ATP.lambda_25
tff(fact_6018_ATP_Olambda__26,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_qo(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_qn(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).
% ATP.lambda_26
tff(fact_6019_ATP_Olambda__27,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_ql(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_qk(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).
% ATP.lambda_27
tff(fact_6020_ATP_Olambda__28,axiom,
! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),A),aTP_Lamp_dm(A,fun(product_prod(heap_ext(product_unit),nat),A)),Uu) = aa(fun(heap_ext(product_unit),fun(nat,A)),fun(product_prod(heap_ext(product_unit),nat),A),product_case_prod(heap_ext(product_unit),nat,A),aTP_Lamp_dl(A,fun(heap_ext(product_unit),fun(nat,A)),Uu)) ).
% ATP.lambda_28
tff(fact_6021_ATP_Olambda__29,axiom,
! [A: $tType,Uu: fun(A,nat)] :
( pp(aa(fun(A,nat),bool,aTP_Lamp_ts(fun(A,nat),bool),Uu))
<=> finite_finite2(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_tr(fun(A,nat),fun(A,bool),Uu))) ) ).
% ATP.lambda_29
tff(fact_6022_ATP_Olambda__30,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: nat] : aa(nat,heap_Time_Heap(A),aTP_Lamp_rx(nat,heap_Time_Heap(A)),Uu) = aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).
% ATP.lambda_30
tff(fact_6023_ATP_Olambda__31,axiom,
! [Uu: num] : aa(num,option(num),aTP_Lamp_dy(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).
% ATP.lambda_31
tff(fact_6024_ATP_Olambda__32,axiom,
! [Uu: num] : aa(num,option(num),aTP_Lamp_dk(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).
% ATP.lambda_32
tff(fact_6025_ATP_Olambda__33,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_gv(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).
% ATP.lambda_33
tff(fact_6026_ATP_Olambda__34,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_gw(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).
% ATP.lambda_34
tff(fact_6027_ATP_Olambda__35,axiom,
! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bi(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).
% ATP.lambda_35
tff(fact_6028_ATP_Olambda__36,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_xk(nat,set(old_node(A,product_unit))),Uu) = old_Leaf(A,product_unit,aa(nat,A,from_nat(A),Uu)) ) ).
% ATP.lambda_36
tff(fact_6029_ATP_Olambda__37,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_xi(nat,set(old_node(A,product_unit))),Uu) = old_In1(A,product_unit,nth_item(A,Uu)) ) ).
% ATP.lambda_37
tff(fact_6030_ATP_Olambda__38,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_xh(nat,set(old_node(A,product_unit))),Uu) = old_In0(A,product_unit,nth_item(A,Uu)) ) ).
% ATP.lambda_38
tff(fact_6031_ATP_Olambda__39,axiom,
! [A: $tType,Uu: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aTP_Lamp_mg(product_prod(A,A),bool),Uu))
<=> ? [X4: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).
% ATP.lambda_39
tff(fact_6032_ATP_Olambda__40,axiom,
! [A: $tType,B: $tType,Uu: product_prod(product_prod(bool,A),product_prod(bool,B))] :
( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_ly(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu))
<=> ? [X4: A,Y3: B] : Uu = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fTrue),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y3)) ) ).
% ATP.lambda_40
tff(fact_6033_ATP_Olambda__41,axiom,
! [A: $tType,B: $tType,Uu: product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))] :
( pp(aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool,aTP_Lamp_xe(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),Uu))
<=> ? [F6: fun(nat,sum_sum(B,nat)),X4: sum_sum(A,nat),K4: nat] :
( ( Uu = aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),F6),X4) )
& ( aa(nat,sum_sum(B,nat),F6,K4) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ) ).
% ATP.lambda_41
tff(fact_6034_ATP_Olambda__42,axiom,
! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),nat),aTP_Lamp_fk(A,fun(product_prod(heap_ext(product_unit),nat),nat)),Uu) = aa(fun(heap_ext(product_unit),fun(nat,nat)),fun(product_prod(heap_ext(product_unit),nat),nat),product_case_prod(heap_ext(product_unit),nat,nat),aTP_Lamp_fj(heap_ext(product_unit),fun(nat,nat))) ).
% ATP.lambda_42
tff(fact_6035_ATP_Olambda__43,axiom,
! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),aTP_Lamp_dq(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit))),Uu) = aa(fun(heap_ext(product_unit),fun(nat,heap_ext(product_unit))),fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),product_case_prod(heap_ext(product_unit),nat,heap_ext(product_unit)),aTP_Lamp_dp(heap_ext(product_unit),fun(nat,heap_ext(product_unit)))) ).
% ATP.lambda_43
tff(fact_6036_ATP_Olambda__44,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_yg(B,heap_Time_Heap(A)),Uu) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_ao(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% ATP.lambda_44
tff(fact_6037_ATP_Olambda__45,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A)] : aa(array(A),heap_Time_Heap(product_unit),aTP_Lamp_rv(array(A),heap_Time_Heap(product_unit)),Uu) = aa(product_unit,heap_Time_Heap(product_unit),heap_Time_ureturn(product_unit),product_Unity) ) ).
% ATP.lambda_45
tff(fact_6038_ATP_Olambda__46,axiom,
! [B: $tType,Uu: nat] : aa(nat,sum_sum(B,nat),aTP_Lamp_xg(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ).
% ATP.lambda_46
tff(fact_6039_ATP_Olambda__47,axiom,
! [A: $tType,Uu: nat] : aa(nat,sum_sum(A,nat),aTP_Lamp_xf(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).
% ATP.lambda_47
tff(fact_6040_ATP_Olambda__48,axiom,
! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_gh(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_gf(nat,fun(num,option(num)),Uua),aTP_Lamp_gg(nat,fun(num,option(num)),Uua),Uu) ).
% ATP.lambda_48
tff(fact_6041_ATP_Olambda__49,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] : aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_pk(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).
% ATP.lambda_49
tff(fact_6042_ATP_Olambda__50,axiom,
! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,option(A),aTP_Lamp_oy(list(A),fun(nat,option(A)),Uu),Uua) = if(option(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),aa(list(A),nat,size_size(list(A)),Uu)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu),Uua)),none(A)) ).
% ATP.lambda_50
tff(fact_6043_ATP_Olambda__51,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_fh(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).
% ATP.lambda_51
tff(fact_6044_ATP_Olambda__52,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_fa(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(code_integer,num,code_num_of_integer,Uu)),aa(code_integer,num,code_num_of_integer,Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(code_integer,num,code_num_of_integer,Uu)),aa(code_integer,num,code_num_of_integer,Uu))),one2)) ).
% ATP.lambda_52
tff(fact_6045_ATP_Olambda__53,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_ff(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(code_integer,nat,code_nat_of_integer,Uu))),one_one(nat))) ).
% ATP.lambda_53
tff(fact_6046_ATP_Olambda__54,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_he(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).
% ATP.lambda_54
tff(fact_6047_ATP_Olambda__55,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kx(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Uu),Uua)) ).
% ATP.lambda_55
tff(fact_6048_ATP_Olambda__56,axiom,
! [A: $tType] :
( inf(A)
=> ! [Uu: option(A),Uua: A] : aa(A,option(A),aTP_Lamp_ms(option(A),fun(A,option(A)),Uu),Uua) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),none(A)),aTP_Lamp_mr(A,fun(A,option(A)),Uua)),Uu) ) ).
% ATP.lambda_56
tff(fact_6049_ATP_Olambda__57,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: option(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_al(option(A),fun(A,bool),Uu),Uua))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fTrue),aTP_Lamp_ak(A,fun(A,bool),Uua)),Uu)) ) ) ).
% ATP.lambda_57
tff(fact_6050_ATP_Olambda__58,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_gi(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_gh(num,fun(nat,option(num)),Uua),Uu) ).
% ATP.lambda_58
tff(fact_6051_ATP_Olambda__59,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: option(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bf(option(A),fun(A,bool),Uu),Uua))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fFalse),aa(A,fun(A,bool),ord_less_eq(A),Uua)),Uu)) ) ) ).
% ATP.lambda_59
tff(fact_6052_ATP_Olambda__60,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_gf(nat,fun(num,option(num)),Uu),Uua) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_dk(num,option(num))),bit_take_bit_num(Uu,Uua)) ).
% ATP.lambda_60
tff(fact_6053_ATP_Olambda__61,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_nm(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua))) ) ).
% ATP.lambda_61
tff(fact_6054_ATP_Olambda__62,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gk(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).
% ATP.lambda_62
tff(fact_6055_ATP_Olambda__63,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_du(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_dt(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).
% ATP.lambda_63
tff(fact_6056_ATP_Olambda__64,axiom,
! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_kj(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uu)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).
% ATP.lambda_64
tff(fact_6057_ATP_Olambda__65,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ey(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).
% ATP.lambda_65
tff(fact_6058_ATP_Olambda__66,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ef(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).
% ATP.lambda_66
tff(fact_6059_ATP_Olambda__67,axiom,
! [Uu: product_prod(int,int),Uua: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aTP_Lamp_km(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_fst(int,int),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uua))) ).
% ATP.lambda_67
tff(fact_6060_ATP_Olambda__68,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_xp(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,Uu)),old_In1(A,B,Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_68
tff(fact_6061_ATP_Olambda__69,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_xo(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,Uu)),old_In0(A,B,Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_69
tff(fact_6062_ATP_Olambda__70,axiom,
! [Uu: rat,Uua: int] :
( pp(aa(int,bool,aTP_Lamp_nb(rat,fun(int,bool),Uu),Uua))
<=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu))
& pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).
% ATP.lambda_70
tff(fact_6063_ATP_Olambda__71,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_of(set(A),fun(list(A),bool),Uu),Uua))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu))
& distinct(A,Uua) ) ) ).
% ATP.lambda_71
tff(fact_6064_ATP_Olambda__72,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_oe(set(A),fun(list(A),bool),Uu),Uua))
<=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu )
& distinct(A,Uua) ) ) ).
% ATP.lambda_72
tff(fact_6065_ATP_Olambda__73,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_eh(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).
% ATP.lambda_73
tff(fact_6066_ATP_Olambda__74,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dz(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).
% ATP.lambda_74
tff(fact_6067_ATP_Olambda__75,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dx(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_75
tff(fact_6068_ATP_Olambda__76,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ea(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).
% ATP.lambda_76
tff(fact_6069_ATP_Olambda__77,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_rp(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat)),aa(list(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),product_Pair(list(A),product_prod(heap_ext(product_unit),nat)),array_get(A,Uua,Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(array(A),nat,array_length(A,Uua),Uu)))) ) ).
% ATP.lambda_77
tff(fact_6070_ATP_Olambda__78,axiom,
! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
( pp(aa(set(set(A)),bool,aTP_Lamp_ip(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
<=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
& ( Uua != bot_bot(set(set(A))) ) ) ) ).
% ATP.lambda_78
tff(fact_6071_ATP_Olambda__79,axiom,
! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
( pp(aa(option(A),bool,aTP_Lamp_hq(set(option(A)),fun(option(A),bool),Uu),Uua))
<=> ( pp(member(option(A),Uua,Uu))
& ( Uua != none(A) ) ) ) ).
% ATP.lambda_79
tff(fact_6072_ATP_Olambda__80,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_sm(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
<=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
& ! [A7: A,B7: A,C4: A] :
( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),Uua))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4),Uu)) )
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),Uu)) ) ) ) ).
% ATP.lambda_80
tff(fact_6073_ATP_Olambda__81,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_er(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).
% ATP.lambda_81
tff(fact_6074_ATP_Olambda__82,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_eq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).
% ATP.lambda_82
tff(fact_6075_ATP_Olambda__83,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_ku(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).
% ATP.lambda_83
tff(fact_6076_ATP_Olambda__84,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_kf(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).
% ATP.lambda_84
tff(fact_6077_ATP_Olambda__85,axiom,
! [D: $tType,B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(D,product_prod(A,B))] :
( pp(aa(fun(D,product_prod(A,B)),bool,aTP_Lamp_ji(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(D),set(product_prod(A,B)),image2(D,product_prod(A,B),Uua),top_top(set(D)))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_85
tff(fact_6078_ATP_Olambda__86,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
( pp(aa(fun(A,B),bool,aTP_Lamp_vc(set(B),fun(fun(A,B),bool),Uu),Uua))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,Uua),top_top(set(A)))),Uu)) ) ).
% ATP.lambda_86
tff(fact_6079_ATP_Olambda__87,axiom,
! [D: $tType,A: $tType,Uu: set(A),Uua: fun(D,A)] :
( pp(aa(fun(D,A),bool,aTP_Lamp_un(set(A),fun(fun(D,A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,Uua),top_top(set(D)))),Uu)) ) ).
% ATP.lambda_87
tff(fact_6080_ATP_Olambda__88,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_jh(fun(C,product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(C),set(B),image2(C,B,aa(fun(C,product_prod(A,B)),fun(C,B),aa(fun(product_prod(A,B),B),fun(fun(C,product_prod(A,B)),fun(C,B)),comp(product_prod(A,B),B,C),product_snd(A,B)),Uu)),top_top(set(C))) ).
% ATP.lambda_88
tff(fact_6081_ATP_Olambda__89,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).
% ATP.lambda_89
tff(fact_6082_ATP_Olambda__90,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).
% ATP.lambda_90
tff(fact_6083_ATP_Olambda__91,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_br(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).
% ATP.lambda_91
tff(fact_6084_ATP_Olambda__92,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fr(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).
% ATP.lambda_92
tff(fact_6085_ATP_Olambda__93,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fc(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).
% ATP.lambda_93
tff(fact_6086_ATP_Olambda__94,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bj(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).
% ATP.lambda_94
tff(fact_6087_ATP_Olambda__95,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).
% ATP.lambda_95
tff(fact_6088_ATP_Olambda__96,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_do(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).
% ATP.lambda_96
tff(fact_6089_ATP_Olambda__97,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fv(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).
% ATP.lambda_97
tff(fact_6090_ATP_Olambda__98,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_en(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).
% ATP.lambda_98
tff(fact_6091_ATP_Olambda__99,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_gp(fun(A,nat),fun(A,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(A,nat,Uu,Uua)),Uua) ).
% ATP.lambda_99
tff(fact_6092_ATP_Olambda__100,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_nh(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).
% ATP.lambda_100
tff(fact_6093_ATP_Olambda__101,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_if(fun(B,A),fun(B,bool),Uu),Uua))
<=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).
% ATP.lambda_101
tff(fact_6094_ATP_Olambda__102,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_or(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_oq(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,Uu,Uua)) ) ).
% ATP.lambda_102
tff(fact_6095_ATP_Olambda__103,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_im(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).
% ATP.lambda_103
tff(fact_6096_ATP_Olambda__104,axiom,
! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_dj(num,fun(num,int),Uu),Uua) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).
% ATP.lambda_104
tff(fact_6097_ATP_Olambda__105,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: heap_Time_Heap(product_prod(A,B))] :
( pp(aa(heap_Time_Heap(product_prod(A,B)),bool,aTP_Lamp_vj(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),heap_Time_set_Heap(product_prod(A,B),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_105
tff(fact_6098_ATP_Olambda__106,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(product_prod(A,B))] :
( pp(aa(option(product_prod(A,B)),bool,aTP_Lamp_tv(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(option(product_prod(A,B)),set(product_prod(A,B)),set_option(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_106
tff(fact_6099_ATP_Olambda__107,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_pb(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu)),aa(product_prod(A,B),A,product_fst(A,B),Uua))) ) ) ).
% ATP.lambda_107
tff(fact_6100_ATP_Olambda__108,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: product_prod(A,B),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),aTP_Lamp_qg(product_prod(A,B),fun(product_prod(A,B),bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu))) ) ) ).
% ATP.lambda_108
tff(fact_6101_ATP_Olambda__109,axiom,
! [A: $tType,Uu: list(option(A)),Uua: A] : aa(A,option(list(A)),aTP_Lamp_vk(list(option(A)),fun(A,option(list(A))),Uu),Uua) = aa(option(list(A)),option(list(A)),aa(fun(list(A),list(A)),fun(option(list(A)),option(list(A))),map_option(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Uua)),those(A,Uu)) ).
% ATP.lambda_109
tff(fact_6102_ATP_Olambda__110,axiom,
! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_hn(num,fun(num,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu)),aa(num,int,numeral_numeral(int),Uua)) ).
% ATP.lambda_110
tff(fact_6103_ATP_Olambda__111,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,set(old_node(A,product_unit)),aa(nat,fun(nat,set(old_node(A,product_unit))),aTP_Lamp_xl(nat,fun(nat,set(old_node(A,product_unit)))),Uu),Uua) = old_Scons(A,product_unit,nth_item(A,Uu),nth_item(A,Uua)) ) ).
% ATP.lambda_111
tff(fact_6104_ATP_Olambda__112,axiom,
! [A: $tType,Uu: set(A),Uua: heap_Time_Heap(A)] :
( pp(aa(heap_Time_Heap(A),bool,aTP_Lamp_vi(set(A),fun(heap_Time_Heap(A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),heap_Time_set_Heap(A,Uua)),Uu)) ) ).
% ATP.lambda_112
tff(fact_6105_ATP_Olambda__113,axiom,
! [A: $tType,Uu: set(A),Uua: option(A)] :
( pp(aa(option(A),bool,aTP_Lamp_um(set(A),fun(option(A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(option(A),set(A),set_option(A),Uua)),Uu)) ) ).
% ATP.lambda_113
tff(fact_6106_ATP_Olambda__114,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_eb(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).
% ATP.lambda_114
tff(fact_6107_ATP_Olambda__115,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_pn(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).
% ATP.lambda_115
tff(fact_6108_ATP_Olambda__116,axiom,
! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_qv(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
<=> pp(member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu)) ) ).
% ATP.lambda_116
tff(fact_6109_ATP_Olambda__117,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_ue(nat,fun(list(A),bool),Uu),Uua))
<=> ( aa(list(A),nat,size_size(list(A)),Uua) = Uu ) ) ).
% ATP.lambda_117
tff(fact_6110_ATP_Olambda__118,axiom,
! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aTP_Lamp_wd(heap_ext(product_unit),fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),lim(product_unit,Uu)),one_one(nat)) ).
% ATP.lambda_118
tff(fact_6111_ATP_Olambda__119,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_dh(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).
% ATP.lambda_119
tff(fact_6112_ATP_Olambda__120,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_dg(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).
% ATP.lambda_120
tff(fact_6113_ATP_Olambda__121,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fg(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).
% ATP.lambda_121
tff(fact_6114_ATP_Olambda__122,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ei(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).
% ATP.lambda_122
tff(fact_6115_ATP_Olambda__123,axiom,
! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_an(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),zero_zero(nat))) ).
% ATP.lambda_123
tff(fact_6116_ATP_Olambda__124,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_we(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(ref(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_Pair(ref(A),product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ) ).
% ATP.lambda_124
tff(fact_6117_ATP_Olambda__125,axiom,
! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_aa(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ).
% ATP.lambda_125
tff(fact_6118_ATP_Olambda__126,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_oz(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).
% ATP.lambda_126
tff(fact_6119_ATP_Olambda__127,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_ng(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).
% ATP.lambda_127
tff(fact_6120_ATP_Olambda__128,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_ny(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).
% ATP.lambda_128
tff(fact_6121_ATP_Olambda__129,axiom,
! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_dr(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).
% ATP.lambda_129
tff(fact_6122_ATP_Olambda__130,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fx(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_130
tff(fact_6123_ATP_Olambda__131,axiom,
! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_pf(nat,fun(nat,list(nat))),Uu),Uua) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Uu),nat_list_decode(Uua)) ).
% ATP.lambda_131
tff(fact_6124_ATP_Olambda__132,axiom,
! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_po(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua) = map_add(A,B,Uua,map_of(A,B,Uu)) ).
% ATP.lambda_132
tff(fact_6125_ATP_Olambda__133,axiom,
! [A: $tType,Uu: list(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_oo(list(set(A)),fun(set(A),bool),Uu),Uua))
<=> pp(member(set(A),Uua,aa(list(set(A)),set(set(A)),set2(set(A)),Uu))) ) ).
% ATP.lambda_133
tff(fact_6126_ATP_Olambda__134,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_uf(nat,fun(list(A),bool),Uu),Uua))
<=> ( Uu = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).
% ATP.lambda_134
tff(fact_6127_ATP_Olambda__135,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qi(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_135
tff(fact_6128_ATP_Olambda__136,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_dn(A,fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_136
tff(fact_6129_ATP_Olambda__137,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).
% ATP.lambda_137
tff(fact_6130_ATP_Olambda__138,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_gn(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).
% ATP.lambda_138
tff(fact_6131_ATP_Olambda__139,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_cw(nat,fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).
% ATP.lambda_139
tff(fact_6132_ATP_Olambda__140,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ak(A,fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_140
tff(fact_6133_ATP_Olambda__141,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ek(A,fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_141
tff(fact_6134_ATP_Olambda__142,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_hk(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_142
tff(fact_6135_ATP_Olambda__143,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ho(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).
% ATP.lambda_143
tff(fact_6136_ATP_Olambda__144,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_gm(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).
% ATP.lambda_144
tff(fact_6137_ATP_Olambda__145,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_go(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).
% ATP.lambda_145
tff(fact_6138_ATP_Olambda__146,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_gy(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu) ).
% ATP.lambda_146
tff(fact_6139_ATP_Olambda__147,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ow(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).
% ATP.lambda_147
tff(fact_6140_ATP_Olambda__148,axiom,
! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_gx(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).
% ATP.lambda_148
tff(fact_6141_ATP_Olambda__149,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ni(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).
% ATP.lambda_149
tff(fact_6142_ATP_Olambda__150,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_gl(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).
% ATP.lambda_150
tff(fact_6143_ATP_Olambda__151,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aTP_Lamp_wa(list(A),fun(array(A),assn),Uu),Uua) = snga_assn(A,Uua,Uu) ) ).
% ATP.lambda_151
tff(fact_6144_ATP_Olambda__152,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_jv(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).
% ATP.lambda_152
tff(fact_6145_ATP_Olambda__153,axiom,
! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).
% ATP.lambda_153
tff(fact_6146_ATP_Olambda__154,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_is(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).
% ATP.lambda_154
tff(fact_6147_ATP_Olambda__155,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ox(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).
% ATP.lambda_155
tff(fact_6148_ATP_Olambda__156,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_it(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).
% ATP.lambda_156
tff(fact_6149_ATP_Olambda__157,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),nat,aTP_Lamp_rw(array(A),fun(heap_ext(product_unit),nat),Uu),Uua) = aa(array(A),nat,array_length(A,Uua),Uu) ) ).
% ATP.lambda_157
tff(fact_6150_ATP_Olambda__158,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),A,aTP_Lamp_bb(ref(A),fun(heap_ext(product_unit),A),Uu),Uua) = ref_get(A,Uua,Uu) ) ).
% ATP.lambda_158
tff(fact_6151_ATP_Olambda__159,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
<=> pp(member(A,Uua,Uu)) ) ).
% ATP.lambda_159
tff(fact_6152_ATP_Olambda__160,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ac(A,fun(A,bool),Uu),Uua))
<=> ( Uua = Uu ) ) ).
% ATP.lambda_160
tff(fact_6153_ATP_Olambda__161,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_rf(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).
% ATP.lambda_161
tff(fact_6154_ATP_Olambda__162,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_rc(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).
% ATP.lambda_162
tff(fact_6155_ATP_Olambda__163,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_ns(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A))) ).
% ATP.lambda_163
tff(fact_6156_ATP_Olambda__164,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_wf(A,fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)),aa(fun(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(ref(A),heap_ext(product_unit)),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(ref(A),heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_we(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),product_prod(heap_ext(product_unit),nat))))),ref_alloc(A,Uu,Uua)) ) ).
% ATP.lambda_164
tff(fact_6157_ATP_Olambda__165,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_tr(fun(A,nat),fun(A,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu,Uua))) ) ).
% ATP.lambda_165
tff(fact_6158_ATP_Olambda__166,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_kh(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).
% ATP.lambda_166
tff(fact_6159_ATP_Olambda__167,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_iu(fun(B,C),fun(product_prod(A,B),C),Uu),Uua) = aa(B,C,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).
% ATP.lambda_167
tff(fact_6160_ATP_Olambda__168,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ix(fun(A,C),fun(product_prod(A,B),C),Uu),Uua) = aa(A,C,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).
% ATP.lambda_168
tff(fact_6161_ATP_Olambda__169,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(sum_sum(B,C),set(A)),Uua: C] : aa(C,set(A),aTP_Lamp_xb(fun(sum_sum(B,C),set(A)),fun(C,set(A)),Uu),Uua) = aa(sum_sum(B,C),set(A),Uu,aa(C,sum_sum(B,C),sum_Inr(C,B),Uua)) ).
% ATP.lambda_169
tff(fact_6162_ATP_Olambda__170,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(sum_sum(A,B),set(C)),Uua: B] : aa(B,set(C),aTP_Lamp_xd(fun(sum_sum(A,B),set(C)),fun(B,set(C)),Uu),Uua) = aa(sum_sum(A,B),set(C),Uu,aa(B,sum_sum(A,B),sum_Inr(B,A),Uua)) ).
% ATP.lambda_170
tff(fact_6163_ATP_Olambda__171,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(sum_sum(B,C),set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_xa(fun(sum_sum(B,C),set(A)),fun(B,set(A)),Uu),Uua) = aa(sum_sum(B,C),set(A),Uu,aa(B,sum_sum(B,C),sum_Inl(B,C),Uua)) ).
% ATP.lambda_171
tff(fact_6164_ATP_Olambda__172,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(sum_sum(A,B),set(C)),Uua: A] : aa(A,set(C),aTP_Lamp_xc(fun(sum_sum(A,B),set(C)),fun(A,set(C)),Uu),Uua) = aa(sum_sum(A,B),set(C),Uu,aa(A,sum_sum(A,B),sum_Inl(A,B),Uua)) ).
% ATP.lambda_172
tff(fact_6165_ATP_Olambda__173,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_173
tff(fact_6166_ATP_Olambda__174,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bn(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_174
tff(fact_6167_ATP_Olambda__175,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).
% ATP.lambda_175
tff(fact_6168_ATP_Olambda__176,axiom,
! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_ux(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).
% ATP.lambda_176
tff(fact_6169_ATP_Olambda__177,axiom,
! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_mo(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).
% ATP.lambda_177
tff(fact_6170_ATP_Olambda__178,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_lk(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),ord_min(A,Uu)),Uua)) ) ).
% ATP.lambda_178
tff(fact_6171_ATP_Olambda__179,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_lj(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),ord_max(A,Uu)),Uua)) ) ).
% ATP.lambda_179
tff(fact_6172_ATP_Olambda__180,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_li(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),sup_sup(A),Uu)),Uua)) ) ).
% ATP.lambda_180
tff(fact_6173_ATP_Olambda__181,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_mw(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),inf_inf(A,Uu)),Uua)) ) ).
% ATP.lambda_181
tff(fact_6174_ATP_Olambda__182,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_gg(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(Uu,Uua))) ).
% ATP.lambda_182
tff(fact_6175_ATP_Olambda__183,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C] : aa(C,fun(product_prod(B,A),C),aTP_Lamp_qs(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu),Uua) = aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_qr(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua)) ).
% ATP.lambda_183
tff(fact_6176_ATP_Olambda__184,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(bool)),aTP_Lamp_qb(list(A),fun(list(A),fun(product_prod(A,list(A)),option(bool))),Uu),Uua) = aa(fun(A,fun(list(A),option(bool))),fun(product_prod(A,list(A)),option(bool)),product_case_prod(A,list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_qa(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua)) ).
% ATP.lambda_184
tff(fact_6177_ATP_Olambda__185,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_pz(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_py(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).
% ATP.lambda_185
tff(fact_6178_ATP_Olambda__186,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_kt(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_ks(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_186
tff(fact_6179_ATP_Olambda__187,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cv(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_cu(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_187
tff(fact_6180_ATP_Olambda__188,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ct(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_cs(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_188
tff(fact_6181_ATP_Olambda__189,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_cr(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_cq(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_189
tff(fact_6182_ATP_Olambda__190,axiom,
! [B: $tType,A: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cm(fun(A,heap_Time_Heap(B)),fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = aa(fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(heap_ext(product_unit),nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_cl(fun(A,heap_Time_Heap(B)),fun(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).
% ATP.lambda_190
tff(fact_6183_ATP_Olambda__191,axiom,
! [A: $tType,B: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cj(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = aa(fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),product_case_prod(heap_ext(product_unit),nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_ci(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).
% ATP.lambda_191
tff(fact_6184_ATP_Olambda__192,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ch(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_cg(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_192
tff(fact_6185_ATP_Olambda__193,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_cf(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_ce(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_193
tff(fact_6186_ATP_Olambda__194,axiom,
! [A: $tType,Uu: fun(heap_ext(product_unit),A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ar(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,Uu,Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat)))) ).
% ATP.lambda_194
tff(fact_6187_ATP_Olambda__195,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_nw(fun(A,option(B)),fun(A,bool),Uu),Uua))
<=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).
% ATP.lambda_195
tff(fact_6188_ATP_Olambda__196,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_kg(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),image2(B,set(product_prod(A,B)),aTP_Lamp_kf(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).
% ATP.lambda_196
tff(fact_6189_ATP_Olambda__197,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_pl(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_pk(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).
% ATP.lambda_197
tff(fact_6190_ATP_Olambda__198,axiom,
! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_sd(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),times_times(code_natural),Uu),aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))) ).
% ATP.lambda_198
tff(fact_6191_ATP_Olambda__199,axiom,
! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_jc(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).
% ATP.lambda_199
tff(fact_6192_ATP_Olambda__200,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_eg(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_200
tff(fact_6193_ATP_Olambda__201,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ez(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_201
tff(fact_6194_ATP_Olambda__202,axiom,
! [A: $tType,Uu: list(A),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_ta(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(nat,A,nth(A,Uu),aa(code_natural,nat,code_nat_of_natural,Uua))) ).
% ATP.lambda_202
tff(fact_6195_ATP_Olambda__203,axiom,
! [A: $tType] :
( sup(A)
=> ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_kq(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).
% ATP.lambda_203
tff(fact_6196_ATP_Olambda__204,axiom,
! [A: $tType] :
( inf(A)
=> ! [Uu: A,Uua: A] : aa(A,option(A),aTP_Lamp_mr(A,fun(A,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(A,A,inf_inf(A,Uu),Uua)) ) ).
% ATP.lambda_204
tff(fact_6197_ATP_Olambda__205,axiom,
! [Uu: code_natural,Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),aTP_Lamp_sf(code_natural,fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))),product_Pair(code_natural,product_prod(code_natural,code_natural)),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),modulo_modulo(code_natural),Uua),Uu)) ).
% ATP.lambda_205
tff(fact_6198_ATP_Olambda__206,axiom,
! [A: $tType,Uu: pred(A),Uua: pred(A)] : aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_ug(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uu),Uua)) ).
% ATP.lambda_206
tff(fact_6199_ATP_Olambda__207,axiom,
! [A: $tType,Uu: list(product_prod(code_natural,A)),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_sy(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),pick(A,Uu,Uua)) ).
% ATP.lambda_207
tff(fact_6200_ATP_Olambda__208,axiom,
! [B: $tType,A: $tType,Uu: list(product_prod(A,B)),Uua: A] : aa(A,B,aTP_Lamp_ry(list(product_prod(A,B)),fun(A,B),Uu),Uua) = aa(option(B),B,the2(B),aa(A,option(B),map_of(A,B,Uu),Uua)) ).
% ATP.lambda_208
tff(fact_6201_ATP_Olambda__209,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_kv(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).
% ATP.lambda_209
tff(fact_6202_ATP_Olambda__210,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_ky(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_210
tff(fact_6203_ATP_Olambda__211,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_te(A,fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uua)) ) ) ).
% ATP.lambda_211
tff(fact_6204_ATP_Olambda__212,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_qx(A,fun(A,bool),Uu),Uua))
<=> ( Uua != Uu ) ) ).
% ATP.lambda_212
tff(fact_6205_ATP_Olambda__213,axiom,
! [A: $tType,Uu: A,Uua: A] : aa(A,set(A),aTP_Lamp_na(A,fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A)))) ).
% ATP.lambda_213
tff(fact_6206_ATP_Olambda__214,axiom,
! [Uu: nat,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mt(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),some(product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),Uu))) ).
% ATP.lambda_214
tff(fact_6207_ATP_Olambda__215,axiom,
! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_va(fun(B,set(C)),fun(B,set(product_prod(C,C))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Uu,Uua)) ).
% ATP.lambda_215
tff(fact_6208_ATP_Olambda__216,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_vb(fun(A,set(B)),fun(A,set(product_prod(B,B))),Uu),Uua) = bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_216
tff(fact_6209_ATP_Olambda__217,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_nt(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),Uu),Uua) = transitive_trancl(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).
% ATP.lambda_217
tff(fact_6210_ATP_Olambda__218,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fw(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).
% ATP.lambda_218
tff(fact_6211_ATP_Olambda__219,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_ap(fun(B,A),fun(B,heap_Time_Heap(A)),Uu),Uua) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,Uu,Uua)) ).
% ATP.lambda_219
tff(fact_6212_ATP_Olambda__220,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_bu(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).
% ATP.lambda_220
tff(fact_6213_ATP_Olambda__221,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_ja(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ) ).
% ATP.lambda_221
tff(fact_6214_ATP_Olambda__222,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_ax(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).
% ATP.lambda_222
tff(fact_6215_ATP_Olambda__223,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(B)
=> ! [Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_jb(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ) ).
% ATP.lambda_223
tff(fact_6216_ATP_Olambda__224,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_nv(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).
% ATP.lambda_224
tff(fact_6217_ATP_Olambda__225,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_qt(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).
% ATP.lambda_225
tff(fact_6218_ATP_Olambda__226,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_oc(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).
% ATP.lambda_226
tff(fact_6219_ATP_Olambda__227,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: A] : aa(A,set(B),aTP_Lamp_jx(fun(A,fun(B,bool)),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),Uu,Uua)) ).
% ATP.lambda_227
tff(fact_6220_ATP_Olambda__228,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ie(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).
% ATP.lambda_228
tff(fact_6221_ATP_Olambda__229,axiom,
! [B: $tType,Uu: fun(B,bool),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_wy(fun(B,bool),fun(B,bool),Uu),Uua))
<=> ~ pp(aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_229
tff(fact_6222_ATP_Olambda__230,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ol(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,Uu,Uua)) ) ).
% ATP.lambda_230
tff(fact_6223_ATP_Olambda__231,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_tc(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
<=> ? [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ).
% ATP.lambda_231
tff(fact_6224_ATP_Olambda__232,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_td(fun(A,fun(B,bool)),fun(B,bool),Uu),Uua))
<=> ? [A7: A] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,A7),Uua)) ) ).
% ATP.lambda_232
tff(fact_6225_ATP_Olambda__233,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_sq(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
<=> ! [X4: A] :
( pp(member(A,X4,Uua))
=> ! [Xa2: A] :
( pp(member(A,Xa2,Uua))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu))
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu)) ) ) ) ) ).
% ATP.lambda_233
tff(fact_6226_ATP_Olambda__234,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_tt(set(product_prod(A,B)),fun(A,bool),Uu),Uua))
<=> ? [Y3: B] : pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y3),Uu)) ) ).
% ATP.lambda_234
tff(fact_6227_ATP_Olambda__235,axiom,
! [B: $tType,A: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: fun(A,heap_Time_Heap(B))] :
( pp(aa(fun(A,heap_Time_Heap(B)),bool,aTP_Lamp_yf(fun(A,heap_Time_Heap(B)),fun(fun(A,heap_Time_Heap(B)),bool),Uu),Uua))
<=> ! [X4: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),Uu,X4),aa(A,heap_Time_Heap(B),Uua,X4)) ) ).
% ATP.lambda_235
tff(fact_6228_ATP_Olambda__236,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_lp(fun(A,option(B)),fun(B,bool),Uu),Uua))
<=> ? [A7: A] : aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),Uua) ) ).
% ATP.lambda_236
tff(fact_6229_ATP_Olambda__237,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_lw(A,fun(product_prod(A,B),bool),Uu),Uua))
<=> ? [V7: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),V7) ) ).
% ATP.lambda_237
tff(fact_6230_ATP_Olambda__238,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,option(B))] :
( pp(aa(fun(A,option(B)),bool,aTP_Lamp_xt(fun(A,fun(B,bool)),fun(fun(A,option(B)),bool),Uu),Uua))
<=> ! [X4: A,Y3: B] :
( ( aa(A,option(B),Uua,X4) = aa(B,option(B),some(B),Y3) )
=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,X4),Y3)) ) ) ).
% ATP.lambda_238
tff(fact_6231_ATP_Olambda__239,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: product_prod(product_prod(bool,A),product_prod(bool,B))] :
( pp(aa(product_prod(product_prod(bool,A),product_prod(bool,B)),bool,aTP_Lamp_lx(set(product_prod(A,B)),fun(product_prod(product_prod(bool,A),product_prod(bool,B)),bool),Uu),Uua))
<=> ? [X4: A,Y3: B] :
( ( Uua = aa(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B)),aa(product_prod(bool,A),fun(product_prod(bool,B),product_prod(product_prod(bool,A),product_prod(bool,B))),product_Pair(product_prod(bool,A),product_prod(bool,B)),aa(A,product_prod(bool,A),aa(bool,fun(A,product_prod(bool,A)),product_Pair(bool,A),fFalse),X4)),aa(B,product_prod(bool,B),aa(bool,fun(B,product_prod(bool,B)),product_Pair(bool,B),fFalse),Y3)) )
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y3),Uu)) ) ) ).
% ATP.lambda_239
tff(fact_6232_ATP_Olambda__240,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_lu(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
<=> ? [A7: A,B7: B] :
( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
& ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B7) ) ) ) ).
% ATP.lambda_240
tff(fact_6233_ATP_Olambda__241,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_so(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
<=> ? [X10: set(A),Y8: set(A)] :
( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X10),Y8) )
& ( X10 != bot_bot(set(A)) )
& ! [X4: A] :
( pp(member(A,X4,Y8))
=> ? [Xa2: A] :
( pp(member(A,Xa2,X10))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X4),Uu)) ) ) ) ) ).
% ATP.lambda_241
tff(fact_6234_ATP_Olambda__242,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,bool))))),Uua: fun(A,heap_Time_Heap(B))] :
( pp(aa(fun(A,heap_Time_Heap(B)),bool,aTP_Lamp_yi(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,bool))))),fun(fun(A,heap_Time_Heap(B)),bool),Uu),Uua))
<=> ! [X4: A,H7: heap_ext(product_unit),H8: heap_ext(product_unit),R5: B,N7: nat] :
( heap_Time_effect(B,aa(A,heap_Time_Heap(B),Uua,X4),H7,H8,R5,N7)
=> pp(aa(nat,bool,aa(B,fun(nat,bool),aa(heap_ext(product_unit),fun(B,fun(nat,bool)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,bool))),aa(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,bool)))),Uu,X4),H7),H8),R5),N7)) ) ) ).
% ATP.lambda_242
tff(fact_6235_ATP_Olambda__243,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_jo(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).
% ATP.lambda_243
tff(fact_6236_ATP_Olambda__244,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_np(set(A),fun(A,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),Uu) ).
% ATP.lambda_244
tff(fact_6237_ATP_Olambda__245,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: array(A)] : aa(array(A),heap_Time_Heap(array(A)),aTP_Lamp_ru(array(A),fun(array(A),heap_Time_Heap(array(A))),Uu),Uua) = aa(array(A),heap_Time_Heap(array(A)),heap_Time_ureturn(array(A)),Uu) ) ).
% ATP.lambda_245
tff(fact_6238_ATP_Olambda__246,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_rz(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).
% ATP.lambda_246
tff(fact_6239_ATP_Olambda__247,axiom,
! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_uw(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = field2(B,Uu) ).
% ATP.lambda_247
tff(fact_6240_ATP_Olambda__248,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_uz(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).
% ATP.lambda_248
tff(fact_6241_ATP_Olambda__249,axiom,
! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_uv(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_249
tff(fact_6242_ATP_Olambda__250,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_uy(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_250
tff(fact_6243_ATP_Olambda__251,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_kw(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_251
tff(fact_6244_ATP_Olambda__252,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_qm(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).
% ATP.lambda_252
tff(fact_6245_ATP_Olambda__253,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,set(B),aTP_Lamp_jk(fun(B,bool),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),Uu) ).
% ATP.lambda_253
tff(fact_6246_ATP_Olambda__254,axiom,
! [A: $tType,Uu: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_ok(nat,fun(A,nat)),Uu),Uua) = aa(nat,nat,suc,Uu) ).
% ATP.lambda_254
tff(fact_6247_ATP_Olambda__255,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(A),Uub: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aTP_Lamp_ty(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),Uu),Uua),Uub))
<=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fTrue),aTP_Lamp_tw(B,bool)),Uub)),aa(option(B),fun(A,bool),aTP_Lamp_tx(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uub)),Uua)) ) ).
% ATP.lambda_255
tff(fact_6248_ATP_Olambda__256,axiom,
! [Uu: num,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_de(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2))),Uua)),Uub)) ).
% ATP.lambda_256
tff(fact_6249_ATP_Olambda__257,axiom,
! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_cz(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).
% ATP.lambda_257
tff(fact_6250_ATP_Olambda__258,axiom,
! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_da(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).
% ATP.lambda_258
tff(fact_6251_ATP_Olambda__259,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: num,Uua: A,Uub: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_cy(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).
% ATP.lambda_259
tff(fact_6252_ATP_Olambda__260,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_mu(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,member(B,Uub,Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).
% ATP.lambda_260
tff(fact_6253_ATP_Olambda__261,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hs(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).
% ATP.lambda_261
tff(fact_6254_ATP_Olambda__262,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_hr(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).
% ATP.lambda_262
tff(fact_6255_ATP_Olambda__263,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_tj(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),Uua,one_one(A)) ) ).
% ATP.lambda_263
tff(fact_6256_ATP_Olambda__264,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_tk(B,fun(A,fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),Uua,one_one(A)) ) ).
% ATP.lambda_264
tff(fact_6257_ATP_Olambda__265,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_gb(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).
% ATP.lambda_265
tff(fact_6258_ATP_Olambda__266,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_fi(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).
% ATP.lambda_266
tff(fact_6259_ATP_Olambda__267,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_ga(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).
% ATP.lambda_267
tff(fact_6260_ATP_Olambda__268,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_pd(fun(A,bool),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).
% ATP.lambda_268
tff(fact_6261_ATP_Olambda__269,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_ic(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).
% ATP.lambda_269
tff(fact_6262_ATP_Olambda__270,axiom,
! [A: $tType,Uu: fun(heap_ext(product_unit),bool),Uua: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_aq(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = if(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),bool,Uu,Uub),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Uua,Uub)),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ).
% ATP.lambda_270
tff(fact_6263_ATP_Olambda__271,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_op(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).
% ATP.lambda_271
tff(fact_6264_ATP_Olambda__272,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_kz(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_ky(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).
% ATP.lambda_272
tff(fact_6265_ATP_Olambda__273,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_lh(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_kv(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).
% ATP.lambda_273
tff(fact_6266_ATP_Olambda__274,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_aj(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),aa(A,option(A),some(A),Uub)),aa(A,fun(A,option(A)),aTP_Lamp_ai(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub)),Uua) ).
% ATP.lambda_274
tff(fact_6267_ATP_Olambda__275,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_pa(fun(A,fun(A,bool)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,aa(list(A),list(A),mergesort_by_rel(A,Uu),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu),Uub)) ).
% ATP.lambda_275
tff(fact_6268_ATP_Olambda__276,axiom,
! [A: $tType] :
( sup(A)
=> ! [Uu: option(A),Uua: option(A),Uub: A] : aa(A,option(A),aa(option(A),fun(A,option(A)),aTP_Lamp_kr(option(A),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),Uu),aTP_Lamp_kq(A,fun(A,option(A)),Uub)),Uua) ) ).
% ATP.lambda_276
tff(fact_6269_ATP_Olambda__277,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: list(A),Uub: fun(typerep,fun(nat,list(nat)))] : aa(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat))),aa(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat)))),aTP_Lamp_wg(array(A),fun(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat))))),Uu),Uua),Uub) = fun_upd(typerep,fun(nat,list(nat)),Uub,typerep_typerep(A,type2(A)),fun_upd(nat,list(nat),aa(typerep,fun(nat,list(nat)),Uub,typerep_typerep(A,type2(A))),addr_of_array(A,Uu),aa(list(A),list(nat),map(A,nat,to_nat(A)),Uua))) ) ).
% ATP.lambda_277
tff(fact_6270_ATP_Olambda__278,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_qn(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).
% ATP.lambda_278
tff(fact_6271_ATP_Olambda__279,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_oj(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).
% ATP.lambda_279
tff(fact_6272_ATP_Olambda__280,axiom,
! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_tp(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,compow(fun(A,A),Uub,Uu),Uua) ).
% ATP.lambda_280
tff(fact_6273_ATP_Olambda__281,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: option(B),Uub: A] :
( pp(aa(A,bool,aa(option(B),fun(A,bool),aTP_Lamp_tx(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fFalse),aa(A,fun(B,bool),Uu,Uub)),Uua)) ) ).
% ATP.lambda_281
tff(fact_6274_ATP_Olambda__282,axiom,
! [B: $tType,A: $tType,Uu: heap_Time_Heap(A),Uua: fun(A,heap_Time_Heap(B)),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cn(heap_Time_Heap(A),fun(fun(A,heap_Time_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),case_option(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat))),none(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),product_case_prod(A,product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_cm(fun(A,heap_Time_Heap(B)),fun(A,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),Uua))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Uu),Uub)) ).
% ATP.lambda_282
tff(fact_6275_ATP_Olambda__283,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wt(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),Uu,aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub))),aa(nat,A,Uua,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uub)))) ).
% ATP.lambda_283
tff(fact_6276_ATP_Olambda__284,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: product_prod(D,B)] : aa(product_prod(D,B),C,aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_iq(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,aa(product_prod(D,B),D,product_fst(D,B),Uub))),aa(product_prod(D,B),B,product_snd(D,B),Uub)) ).
% ATP.lambda_284
tff(fact_6277_ATP_Olambda__285,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).
% ATP.lambda_285
tff(fact_6278_ATP_Olambda__286,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dt(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).
% ATP.lambda_286
tff(fact_6279_ATP_Olambda__287,axiom,
! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ).
% ATP.lambda_287
tff(fact_6280_ATP_Olambda__288,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).
% ATP.lambda_288
tff(fact_6281_ATP_Olambda__289,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_eo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).
% ATP.lambda_289
tff(fact_6282_ATP_Olambda__290,axiom,
! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ).
% ATP.lambda_290
tff(fact_6283_ATP_Olambda__291,axiom,
! [A: $tType,B: $tType,Uu: fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),Uua: B,Uub: fun(B,heap_Time_Heap(A))] : aa(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A),aa(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)),aTP_Lamp_yh(fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)),Uu,Uub),Uua) ).
% ATP.lambda_291
tff(fact_6284_ATP_Olambda__292,axiom,
! [A: $tType,B: $tType,Uu: fun(fun(B,option(A)),fun(B,option(A))),Uua: B,Uub: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(B,fun(fun(B,option(A)),option(A)),aTP_Lamp_xz(fun(fun(B,option(A)),fun(B,option(A))),fun(B,fun(fun(B,option(A)),option(A))),Uu),Uua),Uub) = aa(B,option(A),aa(fun(B,option(A)),fun(B,option(A)),Uu,Uub),Uua) ).
% ATP.lambda_292
tff(fact_6285_ATP_Olambda__293,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),Uua: fun(A,heap_Time_Heap(B)),Uub: C] : aa(C,heap_Time_Heap(D),aa(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D)),aTP_Lamp_yb(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D))),Uu),Uua),Uub) = aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),aa(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),Uu,Uub),Uua) ).
% ATP.lambda_293
tff(fact_6286_ATP_Olambda__294,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(C,fun(fun(A,option(B)),option(D))),Uua: fun(A,option(B)),Uub: C] : aa(C,option(D),aa(fun(A,option(B)),fun(C,option(D)),aTP_Lamp_xu(fun(C,fun(fun(A,option(B)),option(D))),fun(fun(A,option(B)),fun(C,option(D))),Uu),Uua),Uub) = aa(fun(A,option(B)),option(D),aa(C,fun(fun(A,option(B)),option(D)),Uu,Uub),Uua) ).
% ATP.lambda_294
tff(fact_6287_ATP_Olambda__295,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_mf(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ).
% ATP.lambda_295
tff(fact_6288_ATP_Olambda__296,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tf(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).
% ATP.lambda_296
tff(fact_6289_ATP_Olambda__297,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,B)),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_om(fun(A,fun(B,B)),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(A,fun(B,B),Uu,Uub),Uua) ).
% ATP.lambda_297
tff(fact_6290_ATP_Olambda__298,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_re(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).
% ATP.lambda_298
tff(fact_6291_ATP_Olambda__299,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_gc(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).
% ATP.lambda_299
tff(fact_6292_ATP_Olambda__300,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).
% ATP.lambda_300
tff(fact_6293_ATP_Olambda__301,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ep(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_eo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).
% ATP.lambda_301
tff(fact_6294_ATP_Olambda__302,axiom,
! [Uu: product_prod(code_natural,code_natural),Uua: code_natural,Uub: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aTP_Lamp_sb(product_prod(code_natural,code_natural),fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_case_prod(code_natural,code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_sa(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uua),Uub)),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),product_prod(code_natural,code_natural),product_snd(code_natural,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural)),next,Uu))) ).
% ATP.lambda_302
tff(fact_6295_ATP_Olambda__303,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aTP_Lamp_uk(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural))),product_case_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aa(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aTP_Lamp_uj(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),Uu),Uua)),split_seed(Uub)) ).
% ATP.lambda_303
tff(fact_6296_ATP_Olambda__304,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hc(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_hb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_304
tff(fact_6297_ATP_Olambda__305,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ha(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_gz(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_305
tff(fact_6298_ATP_Olambda__306,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_gt(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_gs(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_306
tff(fact_6299_ATP_Olambda__307,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_gr(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_gq(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_307
tff(fact_6300_ATP_Olambda__308,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ro(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uub,Uu)),Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),one_one(nat))) ) ).
% ATP.lambda_308
tff(fact_6301_ATP_Olambda__309,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_sw(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
& ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).
% ATP.lambda_309
tff(fact_6302_ATP_Olambda__310,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pp(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
| ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu))) ) ) ) ).
% ATP.lambda_310
tff(fact_6303_ATP_Olambda__311,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pj(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& ? [Xys2: list(A),X4: A,Y3: A,Xs6: list(A),Ys5: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Uu)) ) ) ) ).
% ATP.lambda_311
tff(fact_6304_ATP_Olambda__312,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_sx(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
& ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).
% ATP.lambda_312
tff(fact_6305_ATP_Olambda__313,axiom,
! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_oi(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
& distinct(A,Uub)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).
% ATP.lambda_313
tff(fact_6306_ATP_Olambda__314,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_oh(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& distinct(A,Uub)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).
% ATP.lambda_314
tff(fact_6307_ATP_Olambda__315,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_sv(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
& ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).
% ATP.lambda_315
tff(fact_6308_ATP_Olambda__316,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ii(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(member(nat,aa(nat,nat,suc,Uub),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).
% ATP.lambda_316
tff(fact_6309_ATP_Olambda__317,axiom,
! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_qw(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
<=> pp(member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu)) ) ).
% ATP.lambda_317
tff(fact_6310_ATP_Olambda__318,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: fun(A,A),Uua: ref(A),Uub: A] : aa(A,heap_Time_Heap(A),aa(ref(A),fun(A,heap_Time_Heap(A)),aTP_Lamp_ba(fun(A,A),fun(ref(A),fun(A,heap_Time_Heap(A))),Uu),Uua),Uub) = heap_Time_bind(product_unit,A,ref_update(A,Uua,aa(A,A,Uu,Uub)),aa(A,fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_az(fun(A,A),fun(A,fun(product_unit,heap_Time_Heap(A))),Uu),Uub)) ) ).
% ATP.lambda_318
tff(fact_6311_ATP_Olambda__319,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_pr(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,aa(list(A),set(A),set2(A),Uu)))
& pp(aa(A,bool,Uua,Uub)) ) ) ) ).
% ATP.lambda_319
tff(fact_6312_ATP_Olambda__320,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_le(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ( pp(member(product_prod(A,B),Uub,graph(A,B,Uu)))
& ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).
% ATP.lambda_320
tff(fact_6313_ATP_Olambda__321,axiom,
! [Uu: heap_ext(product_unit),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_wc(heap_ext(product_unit),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),lim(product_unit,Uu)))
& ~ pp(member(nat,Uub,Uua)) ) ) ).
% ATP.lambda_321
tff(fact_6314_ATP_Olambda__322,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_rk(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,field2(A,Uu)))
& ! [X4: A] :
( pp(member(A,X4,Uua))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4),Uu)) ) ) ) ).
% ATP.lambda_322
tff(fact_6315_ATP_Olambda__323,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_rm(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,field2(A,Uu)))
& ! [X4: A] :
( pp(member(A,X4,Uua))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub),Uu)) ) ) ) ).
% ATP.lambda_323
tff(fact_6316_ATP_Olambda__324,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_rl(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,field2(A,Uu)))
& ! [X4: A] :
( pp(member(A,X4,Uua))
=> ( ( Uub != X4 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X4),Uu)) ) ) ) ) ).
% ATP.lambda_324
tff(fact_6317_ATP_Olambda__325,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_sr(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,field2(A,Uu)))
& ! [X4: A] :
( pp(member(A,X4,Uua))
=> ( ( Uub != X4 )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Uub),Uu)) ) ) ) ) ).
% ATP.lambda_325
tff(fact_6318_ATP_Olambda__326,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_tu(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> pp(member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)),Uua)) ) ).
% ATP.lambda_326
tff(fact_6319_ATP_Olambda__327,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ij(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(member(nat,Uub,Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).
% ATP.lambda_327
tff(fact_6320_ATP_Olambda__328,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A)),Uub: fun(B,A)] : aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_hd(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),Uu),Uua),Uub) = aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),inv_image(A,B,Uu,Uub)),inv_image(A,B,Uua,Uub)) ).
% ATP.lambda_328
tff(fact_6321_ATP_Olambda__329,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_mj(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uua))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).
% ATP.lambda_329
tff(fact_6322_ATP_Olambda__330,axiom,
! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_rj(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uua,Uu))
& pp(member(A,Uub,Uu)) ) ) ).
% ATP.lambda_330
tff(fact_6323_ATP_Olambda__331,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ht(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),Uu) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),Uu) ) ) ).
% ATP.lambda_331
tff(fact_6324_ATP_Olambda__332,axiom,
! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_ib(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member(B,Uub,Uu))
& pp(aa(B,bool,Uua,Uub)) ) ) ).
% ATP.lambda_332
tff(fact_6325_ATP_Olambda__333,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_ke(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,Uu))
& ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).
% ATP.lambda_333
tff(fact_6326_ATP_Olambda__334,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member(B,Uub,Uu))
& ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).
% ATP.lambda_334
tff(fact_6327_ATP_Olambda__335,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_hy(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member(B,Uub,Uu))
& ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).
% ATP.lambda_335
tff(fact_6328_ATP_Olambda__336,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_ik(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member(B,Uub,Uua))
& ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).
% ATP.lambda_336
tff(fact_6329_ATP_Olambda__337,axiom,
! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_og(list(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).
% ATP.lambda_337
tff(fact_6330_ATP_Olambda__338,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ds(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).
% ATP.lambda_338
tff(fact_6331_ATP_Olambda__339,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ej(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).
% ATP.lambda_339
tff(fact_6332_ATP_Olambda__340,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hg(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).
% ATP.lambda_340
tff(fact_6333_ATP_Olambda__341,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hi(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).
% ATP.lambda_341
tff(fact_6334_ATP_Olambda__342,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hh(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).
% ATP.lambda_342
tff(fact_6335_ATP_Olambda__343,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_hl(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).
% ATP.lambda_343
tff(fact_6336_ATP_Olambda__344,axiom,
! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_ul(set(product_prod(B,C)),fun(B,fun(C,bool)),Uu),Uua),Uub))
<=> pp(member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub),Uu)) ) ).
% ATP.lambda_344
tff(fact_6337_ATP_Olambda__345,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_au(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu)) ) ).
% ATP.lambda_345
tff(fact_6338_ATP_Olambda__346,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_be(set(product_prod(A,B)),fun(A,fun(B,bool))),Uu),Uua),Uub))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu)) ) ).
% ATP.lambda_346
tff(fact_6339_ATP_Olambda__347,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_in(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu)) ) ).
% ATP.lambda_347
tff(fact_6340_ATP_Olambda__348,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tg(set(product_prod(B,A)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua),Uu)) ) ).
% ATP.lambda_348
tff(fact_6341_ATP_Olambda__349,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_sk(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu)) ) ).
% ATP.lambda_349
tff(fact_6342_ATP_Olambda__350,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),Uua: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),Uub: fun(A,heap_Time_Heap(B))] : aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),aa(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),aTP_Lamp_yc(fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),fun(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),Uu),Uua),Uub) = heap_Time_bind(C,D,aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C),Uu,Uub),aa(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D)),aTP_Lamp_yb(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D))),Uua),Uub)) ).
% ATP.lambda_350
tff(fact_6343_ATP_Olambda__351,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(fun(A,option(B)),option(C)),Uua: fun(C,fun(fun(A,option(B)),option(D))),Uub: fun(A,option(B))] : aa(fun(A,option(B)),option(D),aa(fun(C,fun(fun(A,option(B)),option(D))),fun(fun(A,option(B)),option(D)),aTP_Lamp_xv(fun(fun(A,option(B)),option(C)),fun(fun(C,fun(fun(A,option(B)),option(D))),fun(fun(A,option(B)),option(D))),Uu),Uua),Uub) = aa(fun(C,option(D)),option(D),aa(option(C),fun(fun(C,option(D)),option(D)),bind(C,D),aa(fun(A,option(B)),option(C),Uu,Uub)),aa(fun(A,option(B)),fun(C,option(D)),aTP_Lamp_xu(fun(C,fun(fun(A,option(B)),option(D))),fun(fun(A,option(B)),fun(C,option(D))),Uua),Uub)) ).
% ATP.lambda_351
tff(fact_6344_ATP_Olambda__352,axiom,
! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hu(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,Uu,Uub))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).
% ATP.lambda_352
tff(fact_6345_ATP_Olambda__353,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_il(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_353
tff(fact_6346_ATP_Olambda__354,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bw(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_354
tff(fact_6347_ATP_Olambda__355,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_ju(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_355
tff(fact_6348_ATP_Olambda__356,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_em(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).
% ATP.lambda_356
tff(fact_6349_ATP_Olambda__357,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_kp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_357
tff(fact_6350_ATP_Olambda__358,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_mp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),inf_inf(set(B),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_358
tff(fact_6351_ATP_Olambda__359,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bz(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_359
tff(fact_6352_ATP_Olambda__360,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aa(fun(A,nat),fun(fun(A,nat),fun(A,nat)),aTP_Lamp_tq(fun(A,nat),fun(fun(A,nat),fun(A,nat))),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub)) ).
% ATP.lambda_360
tff(fact_6353_ATP_Olambda__361,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_mx(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).
% ATP.lambda_361
tff(fact_6354_ATP_Olambda__362,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ud(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).
% ATP.lambda_362
tff(fact_6355_ATP_Olambda__363,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mb(fun(A,option(B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(A,option(B),Uu,Uua) = aa(B,option(B),some(B),Uub) ) ) ).
% ATP.lambda_363
tff(fact_6356_ATP_Olambda__364,axiom,
! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hv(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua)) ) ).
% ATP.lambda_364
tff(fact_6357_ATP_Olambda__365,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ml(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_365
tff(fact_6358_ATP_Olambda__366,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ps(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).
% ATP.lambda_366
tff(fact_6359_ATP_Olambda__367,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_lv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_367
tff(fact_6360_ATP_Olambda__368,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,heap_Time_Heap(C)),Uua: fun(C,heap_Time_Heap(B)),Uub: A] : aa(A,heap_Time_Heap(B),aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_ab(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),Uu),Uua),Uub) = heap_Time_bind(C,B,aa(A,heap_Time_Heap(C),Uu,Uub),Uua) ).
% ATP.lambda_368
tff(fact_6361_ATP_Olambda__369,axiom,
! [E: $tType,F4: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F4,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F4,B),aa(fun(C,fun(D,B)),fun(E,fun(F4,B)),aTP_Lamp_sg(fun(E,fun(F4,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F4,B))),Uu),Uua),Uub) = product_scomp(F4,C,D,B,aa(E,fun(F4,product_prod(C,D)),Uu,Uub),Uua) ).
% ATP.lambda_369
tff(fact_6362_ATP_Olambda__370,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,option(B)),Uua: fun(B,option(A)),Uub: C] : aa(C,option(A),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_bc(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),Uu),Uua),Uub) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind(B,A),aa(C,option(B),Uu,Uub)),Uua) ).
% ATP.lambda_370
tff(fact_6363_ATP_Olambda__371,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_nk(set(A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> pp(member(A,aa(B,A,Uua,Uub),Uu)) ) ).
% ATP.lambda_371
tff(fact_6364_ATP_Olambda__372,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,Uu: fun(D,fun(E,C)),Uua: fun(C,B),Uub: D] : aa(D,fun(E,B),aa(fun(C,B),fun(D,fun(E,B)),aTP_Lamp_ww(fun(D,fun(E,C)),fun(fun(C,B),fun(D,fun(E,B))),Uu),Uua),Uub) = fcomp(E,C,B,aa(D,fun(E,C),Uu,Uub),Uua) ).
% ATP.lambda_372
tff(fact_6365_ATP_Olambda__373,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_pt(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).
% ATP.lambda_373
tff(fact_6366_ATP_Olambda__374,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_nq(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = Uua ) ) ).
% ATP.lambda_374
tff(fact_6367_ATP_Olambda__375,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ou(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ot(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu)),Uub)) ) ).
% ATP.lambda_375
tff(fact_6368_ATP_Olambda__376,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ov(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),fun(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ot(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,replicate(A,Uu,Uua),Uub)) ) ).
% ATP.lambda_376
tff(fact_6369_ATP_Olambda__377,axiom,
! [A: $tType,Uu: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_uh(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_ug(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),Uu,Uub)) ).
% ATP.lambda_377
tff(fact_6370_ATP_Olambda__378,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_th(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub != Uua )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu)) ) ) ).
% ATP.lambda_378
tff(fact_6371_ATP_Olambda__379,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_st(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub != Uua )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua),Uu)) ) ) ).
% ATP.lambda_379
tff(fact_6372_ATP_Olambda__380,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ri(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uua != Uu )
& ( Uub != Uu ) ) ) ).
% ATP.lambda_380
tff(fact_6373_ATP_Olambda__381,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_sl(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( finite_finite2(A,Uua)
& finite_finite2(A,Uub)
& ( Uub != bot_bot(set(A)) )
& ! [X4: A] :
( pp(member(A,X4,Uua))
=> ? [Xa2: A] :
( pp(member(A,Xa2,Uub))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa2),Uu)) ) ) ) ) ).
% ATP.lambda_381
tff(fact_6374_ATP_Olambda__382,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).
% ATP.lambda_382
tff(fact_6375_ATP_Olambda__383,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fd(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).
% ATP.lambda_383
tff(fact_6376_ATP_Olambda__384,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: list(B),Uub: A] : aa(A,list(B),aa(list(B),fun(A,list(B)),aTP_Lamp_ph(fun(A,B),fun(list(B),fun(A,list(B))),Uu),Uua),Uub) = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Uua),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,Uu,Uub)),nil(B))) ).
% ATP.lambda_384
tff(fact_6377_ATP_Olambda__385,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cp(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).
% ATP.lambda_385
tff(fact_6378_ATP_Olambda__386,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_co(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).
% ATP.lambda_386
tff(fact_6379_ATP_Olambda__387,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ot(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)))) ) ).
% ATP.lambda_387
tff(fact_6380_ATP_Olambda__388,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: list(A),Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_oq(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Uu)))) ) ).
% ATP.lambda_388
tff(fact_6381_ATP_Olambda__389,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: array(A)] : aa(array(A),assn,aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_vy(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu))) ) ).
% ATP.lambda_389
tff(fact_6382_ATP_Olambda__390,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ew(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).
% ATP.lambda_390
tff(fact_6383_ATP_Olambda__391,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_rq(nat,fun(array(A),fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(array(A),nat,array_length(A,Uub),Uua))) ) ) ).
% ATP.lambda_391
tff(fact_6384_ATP_Olambda__392,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_rn(array(A),fun(nat,fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),aa(array(A),nat,array_length(A,Uub),Uu))) ) ) ).
% ATP.lambda_392
tff(fact_6385_ATP_Olambda__393,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bs(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).
% ATP.lambda_393
tff(fact_6386_ATP_Olambda__394,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_by(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).
% ATP.lambda_394
tff(fact_6387_ATP_Olambda__395,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_wb(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = snga_assn(A,Uub,replicate(A,Uu,Uua)) ) ).
% ATP.lambda_395
tff(fact_6388_ATP_Olambda__396,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_qj(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).
% ATP.lambda_396
tff(fact_6389_ATP_Olambda__397,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_qk(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).
% ATP.lambda_397
tff(fact_6390_ATP_Olambda__398,axiom,
! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_pc(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).
% ATP.lambda_398
tff(fact_6391_ATP_Olambda__399,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_pu(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_399
tff(fact_6392_ATP_Olambda__400,axiom,
! [B: $tType,Uu: assn,Uua: fun(B,assn),Uub: B] : aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_vz(assn,fun(fun(B,assn),fun(B,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(B,assn,Uua,Uub)) ).
% ATP.lambda_400
tff(fact_6393_ATP_Olambda__401,axiom,
! [A: $tType,B: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mm(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_401
tff(fact_6394_ATP_Olambda__402,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_od(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).
% ATP.lambda_402
tff(fact_6395_ATP_Olambda__403,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fy(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).
% ATP.lambda_403
tff(fact_6396_ATP_Olambda__404,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: set(A),Uua: set(C),Uub: fun(A,B)] : aa(fun(A,B),set(fun(A,C)),aa(set(C),fun(fun(A,B),set(fun(A,C))),aTP_Lamp_vd(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),Uu),Uua),Uub) = bNF_Wellorder_Func(A,C,Uu,Uua) ).
% ATP.lambda_404
tff(fact_6397_ATP_Olambda__405,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] : aa(A,set(A),aa(set(A),fun(A,set(A)),aTP_Lamp_no(set(A),fun(set(A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu),Uua) ).
% ATP.lambda_405
tff(fact_6398_ATP_Olambda__406,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_mq(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),inf_inf(set(B),Uu),Uua) ).
% ATP.lambda_406
tff(fact_6399_ATP_Olambda__407,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_ss(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).
% ATP.lambda_407
tff(fact_6400_ATP_Olambda__408,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(B),Uua: set(C),Uub: A] : aa(A,set(sum_sum(B,C)),aa(set(C),fun(A,set(sum_sum(B,C))),aTP_Lamp_uu(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).
% ATP.lambda_408
tff(fact_6401_ATP_Olambda__409,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_ko(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert2(B,Uu),Uua) ).
% ATP.lambda_409
tff(fact_6402_ATP_Olambda__410,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_kd(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uu),Uua) ).
% ATP.lambda_410
tff(fact_6403_ATP_Olambda__411,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_jq(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),Uua) ).
% ATP.lambda_411
tff(fact_6404_ATP_Olambda__412,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mk(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat)),aa(product_unit,fun(product_prod(heap_ext(product_unit),nat),product_prod(product_unit,product_prod(heap_ext(product_unit),nat))),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat)),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),ref_set(A,Uu,Uua,Uub)),one_one(nat))) ) ).
% ATP.lambda_412
tff(fact_6405_ATP_Olambda__413,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),Uua: fun(A,B),Uub: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,aa(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_vg(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C,Uu,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),map_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_map_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),Uua,id(product_prod(heap_ext(product_unit),nat))))),Uub)) ).
% ATP.lambda_413
tff(fact_6406_ATP_Olambda__414,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fp(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).
% ATP.lambda_414
tff(fact_6407_ATP_Olambda__415,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_el(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).
% ATP.lambda_415
tff(fact_6408_ATP_Olambda__416,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wp(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ).
% ATP.lambda_416
tff(fact_6409_ATP_Olambda__417,axiom,
! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_jd(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).
% ATP.lambda_417
tff(fact_6410_ATP_Olambda__418,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fn(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).
% ATP.lambda_418
tff(fact_6411_ATP_Olambda__419,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bo(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).
% ATP.lambda_419
tff(fact_6412_ATP_Olambda__420,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).
% ATP.lambda_420
tff(fact_6413_ATP_Olambda__421,axiom,
! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tb(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).
% ATP.lambda_421
tff(fact_6414_ATP_Olambda__422,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_bp(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).
% ATP.lambda_422
tff(fact_6415_ATP_Olambda__423,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_pg(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ).
% ATP.lambda_423
tff(fact_6416_ATP_Olambda__424,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_js(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).
% ATP.lambda_424
tff(fact_6417_ATP_Olambda__425,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),Uub: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_vl(fun(B,C),fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B,Uua,Uub)) ).
% ATP.lambda_425
tff(fact_6418_ATP_Olambda__426,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_ag(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).
% ATP.lambda_426
tff(fact_6419_ATP_Olambda__427,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,bool),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_vv(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),Uu),Uua),Uub) = aa(A,fun(B,bool),Uu,aa(C,A,Uua,Uub)) ).
% ATP.lambda_427
tff(fact_6420_ATP_Olambda__428,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: D] : aa(D,fun(B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_ir(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),Uu),Uua),Uub) = aa(A,fun(B,C),Uu,aa(D,A,Uua,Uub)) ).
% ATP.lambda_428
tff(fact_6421_ATP_Olambda__429,axiom,
! [C: $tType,D: $tType,Uu: fun(D,C),Uua: fun(C,bool),Uub: D] :
( pp(aa(D,bool,aa(fun(C,bool),fun(D,bool),aTP_Lamp_uo(fun(D,C),fun(fun(C,bool),fun(D,bool)),Uu),Uua),Uub))
<=> pp(aa(C,bool,Uua,aa(D,C,Uu,Uub))) ) ).
% ATP.lambda_429
tff(fact_6422_ATP_Olambda__430,axiom,
! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(D,bool),Uub: C] :
( pp(aa(C,bool,aa(fun(D,bool),fun(C,bool),aTP_Lamp_up(fun(C,D),fun(fun(D,bool),fun(C,bool)),Uu),Uua),Uub))
<=> pp(aa(D,bool,Uua,aa(C,D,Uu,Uub))) ) ).
% ATP.lambda_430
tff(fact_6423_ATP_Olambda__431,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_tl(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).
% ATP.lambda_431
tff(fact_6424_ATP_Olambda__432,axiom,
! [T: $tType,Uu: fun(code_natural,T),Uua: code_natural,Uub: T] : aa(T,T,aa(code_natural,fun(T,T),aTP_Lamp_rh(fun(code_natural,T),fun(code_natural,fun(T,T)),Uu),Uua),Uub) = aa(code_natural,T,Uu,Uua) ).
% ATP.lambda_432
tff(fact_6425_ATP_Olambda__433,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_jy(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).
% ATP.lambda_433
tff(fact_6426_ATP_Olambda__434,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,fun(A,C)),Uua: C,Uub: B] : aa(B,fun(A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_qr(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu),Uua),Uub) = aa(C,fun(A,C),Uu,Uua) ).
% ATP.lambda_434
tff(fact_6427_ATP_Olambda__435,axiom,
! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vq(fun(A,bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,Uu,Uua)) ) ).
% ATP.lambda_435
tff(fact_6428_ATP_Olambda__436,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_iy(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).
% ATP.lambda_436
tff(fact_6429_ATP_Olambda__437,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_nf(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).
% ATP.lambda_437
tff(fact_6430_ATP_Olambda__438,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: fun(B,heap_Time_Heap(A)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_as(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Time_Heap(A),Uua,Uu) ).
% ATP.lambda_438
tff(fact_6431_ATP_Olambda__439,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_mi(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_mh(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).
% ATP.lambda_439
tff(fact_6432_ATP_Olambda__440,axiom,
! [A: $tType,Uu: heap_Time_Heap(A),Uua: heap_ext(product_unit),Uub: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),bool),aa(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),bool)),aTP_Lamp_md(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),bool))),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),fun(nat,bool)),fun(product_prod(heap_ext(product_unit),nat),bool),product_case_prod(heap_ext(product_unit),nat,bool),aa(A,fun(heap_ext(product_unit),fun(nat,bool)),aa(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,bool))),aTP_Lamp_mc(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,bool)))),Uu),Uua),Uub)) ).
% ATP.lambda_440
tff(fact_6433_ATP_Olambda__441,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_vw(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uua),aa(A,fun(A,A),Uu,Uua)),Uub)) ).
% ATP.lambda_441
tff(fact_6434_ATP_Olambda__442,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: product_prod(code_natural,code_natural),Uub: A] : aa(A,pred(B),aa(product_prod(code_natural,code_natural),fun(A,pred(B)),aTP_Lamp_ui(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(A,pred(B))),Uu),Uua),Uub) = aa(product_prod(pred(B),product_prod(code_natural,code_natural)),pred(B),product_fst(pred(B),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),Uu,Uub),Uua)) ).
% ATP.lambda_442
tff(fact_6435_ATP_Olambda__443,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: A] : aa(A,option(A),aa(A,fun(A,option(A)),aTP_Lamp_ai(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Uu,Uua),Uub)) ).
% ATP.lambda_443
tff(fact_6436_ATP_Olambda__444,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,bool))),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_vr(fun(A,fun(B,fun(C,bool))),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ? [X_1: C] : pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Uu,Uua),Uub),X_1)) ) ).
% ATP.lambda_444
tff(fact_6437_ATP_Olambda__445,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),Uua: set(old_node(A,B)),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_xr(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua),Uub) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_xq(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uua),Uub))),Uu)) ).
% ATP.lambda_445
tff(fact_6438_ATP_Olambda__446,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aa(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aTP_Lamp_uj(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),Uu),Uua),Uub) = aa(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),product_Pair(pred(B),product_prod(code_natural,code_natural)),bind2(A,B,Uua,aa(product_prod(code_natural,code_natural),fun(A,pred(B)),aTP_Lamp_ui(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(A,pred(B))),Uu),Uub))) ).
% ATP.lambda_446
tff(fact_6439_ATP_Olambda__447,axiom,
! [A: $tType,B: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mn(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ~ pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_447
tff(fact_6440_ATP_Olambda__448,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_kb(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),image2(D,set(B),Uu),Uua)) ).
% ATP.lambda_448
tff(fact_6441_ATP_Olambda__449,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: fun(A,A),Uua: A,Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(A,fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_az(fun(A,A),fun(A,fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(A,A,Uu,Uua)) ) ).
% ATP.lambda_449
tff(fact_6442_ATP_Olambda__450,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,bool))),Uua: A,Uub: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aTP_Lamp_vs(fun(A,fun(B,fun(C,bool))),fun(A,fun(C,bool)),Uu),Uua),Uub))
<=> ? [B7: B] : pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Uu,Uua),B7),Uub)) ) ).
% ATP.lambda_450
tff(fact_6443_ATP_Olambda__451,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,bool))),Uua: B,Uub: C] :
( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_vt(fun(A,fun(B,fun(C,bool))),fun(B,fun(C,bool)),Uu),Uua),Uub))
<=> ? [A7: A] : pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Uu,A7),Uua),Uub)) ) ).
% ATP.lambda_451
tff(fact_6444_ATP_Olambda__452,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
( pp(aa(A,bool,aa(set(C),fun(A,bool),aTP_Lamp_lr(fun(A,fun(B,C)),fun(set(C),fun(A,bool)),Uu),Uua),Uub))
<=> ? [B7: B] : pp(member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B7),Uua)) ) ).
% ATP.lambda_452
tff(fact_6445_ATP_Olambda__453,axiom,
! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_qh(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [I3: nat] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu),I3)),aa(nat,B,nth(B,Uua),I3)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,ord_min(nat,aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua)))) ) ) ).
% ATP.lambda_453
tff(fact_6446_ATP_Olambda__454,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_vf(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [A7: A] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
& pp(member(A,A7,Uu)) ) ) ).
% ATP.lambda_454
tff(fact_6447_ATP_Olambda__455,axiom,
! [A: $tType,Uu: nat,Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_pe(nat,fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [I3: nat] :
( ( Uub = aa(nat,A,nth(A,Uua),I3) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uu),I3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uua))) ) ) ).
% ATP.lambda_455
tff(fact_6448_ATP_Olambda__456,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_qd(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
<=> ? [I3: nat] :
( ( Uub = aa(B,set(A),Uu,aa(nat,B,nth(B,Uua),I3)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).
% ATP.lambda_456
tff(fact_6449_ATP_Olambda__457,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
( pp(aa(set(A),bool,aa(list(B),fun(set(A),bool),aTP_Lamp_qe(fun(B,set(A)),fun(list(B),fun(set(A),bool)),Uu),Uua),Uub))
<=> ? [A7: B] :
( ( Uub = aa(B,set(A),Uu,A7) )
& pp(member(B,A7,aa(list(B),set(B),set2(B),Uua))) ) ) ).
% ATP.lambda_457
tff(fact_6450_ATP_Olambda__458,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_ln(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [L3: B] :
( ( Uub = aa(B,A,Uu,L3) )
& pp(member(B,L3,Uua)) ) ) ).
% ATP.lambda_458
tff(fact_6451_ATP_Olambda__459,axiom,
! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
( pp(aa(A,bool,aa(list(B),fun(A,bool),aTP_Lamp_ob(fun(B,option(A)),fun(list(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X4: B] :
( pp(member(B,X4,aa(list(B),set(B),set2(B),Uua)))
& ( aa(B,option(A),Uu,X4) = aa(A,option(A),some(A),Uub) ) ) ) ).
% ATP.lambda_459
tff(fact_6452_ATP_Olambda__460,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_sj(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ! [X4: product_prod(A,A)] :
( pp(member(product_prod(A,A),X4,Uu))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_si(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)),X4)) ) ) ).
% ATP.lambda_460
tff(fact_6453_ATP_Olambda__461,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(set(C),fun(product_prod(A,B),bool),aTP_Lamp_lq(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [P6: product_prod(A,B)] :
( ( Uub = P6 )
& pp(member(C,aa(B,C,aa(A,fun(B,C),Uu,aa(product_prod(A,B),A,product_fst(A,B),P6)),aa(product_prod(A,B),B,product_snd(A,B),P6)),Uua)) ) ) ).
% ATP.lambda_461
tff(fact_6454_ATP_Olambda__462,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_sn(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X4: A] :
( pp(member(A,X4,Uua))
& pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Uub),Uu)) ) ) ).
% ATP.lambda_462
tff(fact_6455_ATP_Olambda__463,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_sp(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X4: A] :
( pp(member(A,X4,Uua))
& ( aa(A,option(B),Uu,X4) = aa(B,option(B),some(B),Uub) ) ) ) ).
% ATP.lambda_463
tff(fact_6456_ATP_Olambda__464,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,bool)),Uub: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aa(fun(B,fun(B,bool)),fun(product_prod(A,A),bool),aTP_Lamp_ll(fun(B,A),fun(fun(B,fun(B,bool)),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
<=> ? [A7: B,B7: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A7)),aa(B,A,Uu,B7)) )
& pp(aa(B,bool,aa(B,fun(B,bool),Uua,A7),B7)) ) ) ).
% ATP.lambda_464
tff(fact_6457_ATP_Olambda__465,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_oa(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
<=> ? [A18: B,A26: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A18)),aa(B,A,Uua,A26)) )
& pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A18),A26),Uu)) ) ) ).
% ATP.lambda_465
tff(fact_6458_ATP_Olambda__466,axiom,
! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: fun(A,A2),Uub: product_prod(A2,A2)] :
( pp(aa(product_prod(A2,A2),bool,aa(fun(A,A2),fun(product_prod(A2,A2),bool),aTP_Lamp_sz(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),Uu),Uua),Uub))
<=> ? [A7: A,B7: A] :
( ( Uub = aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,Uua,A7)),aa(A,A2,Uua,B7)) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7),Uu)) ) ) ).
% ATP.lambda_466
tff(fact_6459_ATP_Olambda__467,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_qc(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [A7: A,V3: list(A)] :
( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),V3)) )
| ? [U5: list(A),Aa4: A,B7: A,Va3: list(A),W4: list(A)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa4),B7),Uu))
& ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa4),Va3)) )
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),W4)) ) ) ) ) ).
% ATP.lambda_467
tff(fact_6460_ATP_Olambda__468,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: multiset(A),Uub: multiset(A)] :
( pp(aa(multiset(A),bool,aa(multiset(A),fun(multiset(A),bool),aTP_Lamp_sh(set(product_prod(A,A)),fun(multiset(A),fun(multiset(A),bool)),Uu),Uua),Uub))
<=> ? [A7: A,M03: multiset(A),K10: multiset(A)] :
( ( Uub = aa(multiset(A),multiset(A),aa(A,fun(multiset(A),multiset(A)),add_mset(A),A7),M03) )
& ( Uua = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M03),K10) )
& ! [B7: A] :
( pp(member(A,B7,set_mset(A,K10)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),A7),Uu)) ) ) ) ).
% ATP.lambda_468
tff(fact_6461_ATP_Olambda__469,axiom,
! [B: $tType,A: $tType,E: $tType,C: $tType,D: $tType,Uu: fun(E,fun(A,fun(C,fun(B,fun(D,bool))))),Uua: E,Uub: fun(A,option(product_prod(B,product_prod(C,D))))] :
( pp(aa(fun(A,option(product_prod(B,product_prod(C,D)))),bool,aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),bool),aTP_Lamp_xs(fun(E,fun(A,fun(C,fun(B,fun(D,bool))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),bool)),Uu),Uua),Uub))
<=> ! [H7: A,H8: C,R5: B,N7: D] :
( ( aa(A,option(product_prod(B,product_prod(C,D))),Uub,H7) = aa(product_prod(B,product_prod(C,D)),option(product_prod(B,product_prod(C,D))),some(product_prod(B,product_prod(C,D))),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),R5),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),H8),N7))) )
=> pp(aa(D,bool,aa(B,fun(D,bool),aa(C,fun(B,fun(D,bool)),aa(A,fun(C,fun(B,fun(D,bool))),aa(E,fun(A,fun(C,fun(B,fun(D,bool)))),Uu,Uua),H7),H8),R5),N7)) ) ) ).
% ATP.lambda_469
tff(fact_6462_ATP_Olambda__470,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_qf(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [Us3: list(A),Z4: A,Z10: A,Vs3: list(A)] :
( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z4),Vs3)) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Z10),Uu))
& ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z10),Vs3)) ) ) ) ).
% ATP.lambda_470
tff(fact_6463_ATP_Olambda__471,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_wv(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_wu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc)) ).
% ATP.lambda_471
tff(fact_6464_ATP_Olambda__472,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_wm(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uuc)) ).
% ATP.lambda_472
tff(fact_6465_ATP_Olambda__473,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ws(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),aa(nat,set(nat),set_ord_lessThan(nat),Uuc)) ).
% ATP.lambda_473
tff(fact_6466_ATP_Olambda__474,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_bk(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).
% ATP.lambda_474
tff(fact_6467_ATP_Olambda__475,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_lb(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(product_prod(C,A),A,product_snd(C,A),Uu)),Uua),aa(set(product_prod(C,B)),set(product_prod(C,B)),insert2(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).
% ATP.lambda_475
tff(fact_6468_ATP_Olambda__476,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_nr(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,member(B,Uuc,aa(set(A),set(B),image2(A,B,Uua),Uu)),aa(B,A,the_inv_into(A,B,Uu,Uua),Uuc),Uub) ).
% ATP.lambda_476
tff(fact_6469_ATP_Olambda__477,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).
% ATP.lambda_477
tff(fact_6470_ATP_Olambda__478,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).
% ATP.lambda_478
tff(fact_6471_ATP_Olambda__479,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ge(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).
% ATP.lambda_479
tff(fact_6472_ATP_Olambda__480,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dw(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).
% ATP.lambda_480
tff(fact_6473_ATP_Olambda__481,axiom,
! [A: $tType,Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_wi(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ).
% ATP.lambda_481
tff(fact_6474_ATP_Olambda__482,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_nn(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_482
tff(fact_6475_ATP_Olambda__483,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_ig(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).
% ATP.lambda_483
tff(fact_6476_ATP_Olambda__484,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_id(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).
% ATP.lambda_484
tff(fact_6477_ATP_Olambda__485,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B,Uub: fun(A,option(B)),Uuc: A] : aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_nu(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uu),aa(B,option(B),some(B),Uua),aa(A,option(B),Uub,Uuc)) ).
% ATP.lambda_485
tff(fact_6478_ATP_Olambda__486,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B,Uuc: A] : aa(A,option(B),aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_lo(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub),Uuc) = if(option(B),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uuc),Uua),aa(B,option(B),some(B),Uub),aa(A,option(B),Uu,Uuc)) ).
% ATP.lambda_486
tff(fact_6479_ATP_Olambda__487,axiom,
! [A: $tType,Uu: fun(code_natural,A),Uua: code_natural,Uub: code_natural,Uuc: product_unit] : aa(product_unit,seq(A),aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_vx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),Uu),Uua),Uub),Uuc) = if(seq(A),aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Uub),Uua),empty(A),insert(A,aa(code_natural,A,Uu,Uua),iterate_upto(A,Uu,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),one_one(code_natural)),Uub))) ).
% ATP.lambda_487
tff(fact_6480_ATP_Olambda__488,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A),Uuc: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_rb(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uuc),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).
% ATP.lambda_488
tff(fact_6481_ATP_Olambda__489,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mv(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).
% ATP.lambda_489
tff(fact_6482_ATP_Olambda__490,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_wx(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_490
tff(fact_6483_ATP_Olambda__491,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_lg(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_lf(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_491
tff(fact_6484_ATP_Olambda__492,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_ld(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_lc(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_492
tff(fact_6485_ATP_Olambda__493,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_jt(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).
% ATP.lambda_493
tff(fact_6486_ATP_Olambda__494,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,B),Uub: A,Uuc: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_vu(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(C,B,Uua,Uuc))) ) ).
% ATP.lambda_494
tff(fact_6487_ATP_Olambda__495,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_vo(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uuc))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_495
tff(fact_6488_ATP_Olambda__496,axiom,
! [Uu: code_natural,Uua: code_natural,Uub: code_natural,Uuc: code_natural] : aa(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),aa(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)))),aTP_Lamp_sa(code_natural,fun(code_natural,fun(code_natural,fun(code_natural,product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))))),Uu),Uua),Uub),Uuc) = aa(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural))),product_Pair(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),Uu)),Uuc)),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Uub),inc_shift(aa(num,code_natural,numeral_numeral(code_natural),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))),Uua))) ).
% ATP.lambda_496
tff(fact_6489_ATP_Olambda__497,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_me(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),product_Sigma(A,A,Uua,aTP_Lamp_ma(set(A),fun(A,set(A)),Uua))))
& pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).
% ATP.lambda_497
tff(fact_6490_ATP_Olambda__498,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_cu(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).
% ATP.lambda_498
tff(fact_6491_ATP_Olambda__499,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B)),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_xq(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uu),Uua),Uub),Uuc) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,Uu,Uub)),old_Scons(A,B,Uua,Uuc))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_499
tff(fact_6492_ATP_Olambda__500,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rs(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).
% ATP.lambda_500
tff(fact_6493_ATP_Olambda__501,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_nl(set(product_prod(A,A)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uu))
& ~ pp(member(A,Uub,Uua))
& ~ pp(member(A,Uuc,Uua)) ) ) ).
% ATP.lambda_501
tff(fact_6494_ATP_Olambda__502,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_db(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uua)) ) ) ) ).
% ATP.lambda_502
tff(fact_6495_ATP_Olambda__503,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_gu(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu)) ) ).
% ATP.lambda_503
tff(fact_6496_ATP_Olambda__504,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),bool),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),aTP_Lamp_df(fun(A,option(B)),fun(fun(product_prod(A,B),bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
& pp(aa(product_prod(A,B),bool,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc))) ) ) ).
% ATP.lambda_504
tff(fact_6497_ATP_Olambda__505,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fe(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).
% ATP.lambda_505
tff(fact_6498_ATP_Olambda__506,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ev(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).
% ATP.lambda_506
tff(fact_6499_ATP_Olambda__507,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ex(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).
% ATP.lambda_507
tff(fact_6500_ATP_Olambda__508,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_pi(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
& ? [Xys2: list(A),X4: A,Y3: A,Xs6: list(A),Ys5: list(A)] :
( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs6)) )
& ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y3),Uu)) ) ) ) ).
% ATP.lambda_508
tff(fact_6501_ATP_Olambda__509,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_et(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).
% ATP.lambda_509
tff(fact_6502_ATP_Olambda__510,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_wz(fun(A,B),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(A,Uuc,Uua))
& ( aa(A,B,Uu,Uub) = aa(A,B,Uu,Uuc) ) ) ) ).
% ATP.lambda_510
tff(fact_6503_ATP_Olambda__511,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_nx(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(A,Uuc,Uu))
& ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).
% ATP.lambda_511
tff(fact_6504_ATP_Olambda__512,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_nd(fun(A,B),fun(set(A),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(A,Uuc,Uua))
& ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).
% ATP.lambda_512
tff(fact_6505_ATP_Olambda__513,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_eu(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uuc))) ) ).
% ATP.lambda_513
tff(fact_6506_ATP_Olambda__514,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_es(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).
% ATP.lambda_514
tff(fact_6507_ATP_Olambda__515,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_cg(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).
% ATP.lambda_515
tff(fact_6508_ATP_Olambda__516,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_ce(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).
% ATP.lambda_516
tff(fact_6509_ATP_Olambda__517,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_cq(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).
% ATP.lambda_517
tff(fact_6510_ATP_Olambda__518,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_cs(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).
% ATP.lambda_518
tff(fact_6511_ATP_Olambda__519,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_qy(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).
% ATP.lambda_519
tff(fact_6512_ATP_Olambda__520,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_qz(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).
% ATP.lambda_520
tff(fact_6513_ATP_Olambda__521,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ne(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( Uu = Uub )
& ( Uua = Uuc ) ) ) ).
% ATP.lambda_521
tff(fact_6514_ATP_Olambda__522,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_ks(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua) ) ) ).
% ATP.lambda_522
tff(fact_6515_ATP_Olambda__523,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),aTP_Lamp_jg(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc),aa(C,set(product_prod(A,B)),Uu,Uua))) ) ).
% ATP.lambda_523
tff(fact_6516_ATP_Olambda__524,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_hz(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(B,Uuc,Uu))
& ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).
% ATP.lambda_524
tff(fact_6517_ATP_Olambda__525,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_hx(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(B,Uuc,Uu))
& ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).
% ATP.lambda_525
tff(fact_6518_ATP_Olambda__526,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_jw(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(A,bool,Uu,Uub))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_526
tff(fact_6519_ATP_Olambda__527,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_jz(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_jy(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).
% ATP.lambda_527
tff(fact_6520_ATP_Olambda__528,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_su(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).
% ATP.lambda_528
tff(fact_6521_ATP_Olambda__529,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_kc(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).
% ATP.lambda_529
tff(fact_6522_ATP_Olambda__530,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_tm(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).
% ATP.lambda_530
tff(fact_6523_ATP_Olambda__531,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_jj(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(A,bool,Uu,Uub))
& pp(aa(B,bool,Uua,Uuc)) ) ) ).
% ATP.lambda_531
tff(fact_6524_ATP_Olambda__532,axiom,
! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_dc(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(Uu)
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_532
tff(fact_6525_ATP_Olambda__533,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(A,A),Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rr(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,aa(A,A,Uua,aa(nat,A,nth(A,array_get(A,Uuc,Uub)),Uu)),Uuc)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).
% ATP.lambda_533
tff(fact_6526_ATP_Olambda__534,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_rt(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub),Uuc) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),array_update(A,Uub,Uu,Uua,Uuc)),one_one(nat))) ) ).
% ATP.lambda_534
tff(fact_6527_ATP_Olambda__535,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_my(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),inf_inf(set(C),aa(set(A),set(C),vimage(C,A,Uu),aa(set(A),set(A),insert2(A,Uuc),bot_bot(set(A))))),Uub)) ).
% ATP.lambda_535
tff(fact_6528_ATP_Olambda__536,axiom,
! [B: $tType,A: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_ci(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(A,Uuc,aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),aa(B,heap_Time_Heap(A),Uu,Uua)),Uub)) ).
% ATP.lambda_536
tff(fact_6529_ATP_Olambda__537,axiom,
! [A: $tType,B: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: A,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_cl(fun(A,heap_Time_Heap(B)),fun(A,fun(heap_ext(product_unit),fun(nat,option(product_prod(B,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(B,Uuc,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),aa(A,heap_Time_Heap(B),Uu,Uua)),Uub)) ).
% ATP.lambda_537
tff(fact_6530_ATP_Olambda__538,axiom,
! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(bool),aa(A,fun(list(A),option(bool)),aa(list(A),fun(A,fun(list(A),option(bool))),aTP_Lamp_qa(list(A),fun(list(A),fun(A,fun(list(A),option(bool)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(A,Uu,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),Uuc)) ).
% ATP.lambda_538
tff(fact_6531_ATP_Olambda__539,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ra(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = aa(A,A,ord_min(A,Uu),min_list(A,Uua)) ) ).
% ATP.lambda_539
tff(fact_6532_ATP_Olambda__540,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: fun(A,fun(B,C)),Uub: A,Uuc: B] : aa(B,D,aa(A,fun(B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_bl(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),Uu),Uua),Uub),Uuc) = aa(C,D,Uu,aa(B,C,aa(A,fun(B,C),Uua,Uub),Uuc)) ).
% ATP.lambda_540
tff(fact_6533_ATP_Olambda__541,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).
% ATP.lambda_541
tff(fact_6534_ATP_Olambda__542,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fb(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).
% ATP.lambda_542
tff(fact_6535_ATP_Olambda__543,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_wl(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ).
% ATP.lambda_543
tff(fact_6536_ATP_Olambda__544,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fo(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).
% ATP.lambda_544
tff(fact_6537_ATP_Olambda__545,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_bq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).
% ATP.lambda_545
tff(fact_6538_ATP_Olambda__546,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_wq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ).
% ATP.lambda_546
tff(fact_6539_ATP_Olambda__547,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: nat,Uub: list(A),Uuc: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aa(nat,fun(list(A),fun(nat,bool)),aTP_Lamp_rg(fun(A,bool),fun(nat,fun(list(A),fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(A,bool,Uu,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc))) ) ).
% ATP.lambda_547
tff(fact_6540_ATP_Olambda__548,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_qp(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).
% ATP.lambda_548
tff(fact_6541_ATP_Olambda__549,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_qq(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).
% ATP.lambda_549
tff(fact_6542_ATP_Olambda__550,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,heap_Time_Heap(A))),Uua: fun(C,C),Uub: fun(fun(B,heap_Time_Heap(A)),C),Uuc: fun(B,heap_Time_Heap(A))] : aa(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)))),aTP_Lamp_ye(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A))))),Uu),Uua),Uub),Uuc) = aa(C,fun(B,heap_Time_Heap(A)),Uu,aa(C,C,Uua,aa(fun(B,heap_Time_Heap(A)),C,Uub,Uuc))) ).
% ATP.lambda_550
tff(fact_6543_ATP_Olambda__551,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,option(A))),Uua: fun(C,C),Uub: fun(fun(B,option(A)),C),Uuc: fun(B,option(A))] : aa(fun(B,option(A)),fun(B,option(A)),aa(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A)))),aTP_Lamp_xy(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(fun(B,option(A)),fun(B,option(A))))),Uu),Uua),Uub),Uuc) = aa(C,fun(B,option(A)),Uu,aa(C,C,Uua,aa(fun(B,option(A)),C,Uub,Uuc))) ).
% ATP.lambda_551
tff(fact_6544_ATP_Olambda__552,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B)] : aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),aa(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))),aa(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))))),aTP_Lamp_px(fun(B,A),fun(A,fun(B,fun(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))))))),Uu),Uua),Uub),Uuc) = aa(fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_case_prod(list(B),list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_pw(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc)) ) ).
% ATP.lambda_552
tff(fact_6545_ATP_Olambda__553,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_cd(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_cc(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_553
tff(fact_6546_ATP_Olambda__554,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_cb(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ca(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_554
tff(fact_6547_ATP_Olambda__555,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_gq(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_555
tff(fact_6548_ATP_Olambda__556,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_gs(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_556
tff(fact_6549_ATP_Olambda__557,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_sc(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).
% ATP.lambda_557
tff(fact_6550_ATP_Olambda__558,axiom,
! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_py(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).
% ATP.lambda_558
tff(fact_6551_ATP_Olambda__559,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_gz(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_559
tff(fact_6552_ATP_Olambda__560,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_hb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).
% ATP.lambda_560
tff(fact_6553_ATP_Olambda__561,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_lt(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ? [Y3: C] :
( pp(member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y3),Uu))
& pp(member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Uuc),Uua)) ) ) ).
% ATP.lambda_561
tff(fact_6554_ATP_Olambda__562,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_lz(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
<=> ? [A7: C] :
( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A7)),aa(C,B,Uub,A7)) )
& pp(member(C,A7,Uu)) ) ) ).
% ATP.lambda_562
tff(fact_6555_ATP_Olambda__563,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(B,fun(set(B),fun(A,bool)),aTP_Lamp_lm(fun(B,A),fun(B,fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ? [L3: B] :
( ( Uuc = aa(B,A,Uu,L3) )
& ( ( L3 = Uua )
| pp(member(B,L3,Uub)) ) ) ) ).
% ATP.lambda_563
tff(fact_6556_ATP_Olambda__564,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A,Uub: fun(A,B),Uuc: B] :
( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aa(A,fun(fun(A,B),fun(B,bool)),aTP_Lamp_ls(A,fun(A,fun(fun(A,B),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ? [I3: A] :
( ( Uuc = aa(A,B,Uub,I3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),I3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I3),Uua)) ) ) ) ).
% ATP.lambda_564
tff(fact_6557_ATP_Olambda__565,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_ti(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
<=> ? [A18: A,A26: A] :
( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A18),A26) )
& pp(member(A,A18,Uu))
& pp(member(A,A26,Uu))
& pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A18)),aa(A,B,Uub,A26)),Uua)) ) ) ).
% ATP.lambda_565
tff(fact_6558_ATP_Olambda__566,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_wo(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_wn(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uuc)) ).
% ATP.lambda_566
tff(fact_6559_ATP_Olambda__567,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,A),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_wj(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),Uub),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc)),Uuc))) ).
% ATP.lambda_567
tff(fact_6560_ATP_Olambda__568,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_lf(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uua),Uub),aa(set(product_prod(C,B)),set(product_prod(C,B)),insert2(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_568
tff(fact_6561_ATP_Olambda__569,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_lc(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(A,C)),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uua),Uub),aa(set(product_prod(A,C)),set(product_prod(A,C)),insert2(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_569
tff(fact_6562_ATP_Olambda__570,axiom,
! [A: $tType,Uu: A,Uua: nat,Uub: fun(nat,A),Uuc: fun(nat,A),Uud: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_wh(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),Uua),aa(nat,A,Uub,Uud),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uud),Uua),Uu,aa(nat,A,Uuc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uud),aa(nat,nat,suc,zero_zero(nat)))))) ).
% ATP.lambda_570
tff(fact_6563_ATP_Olambda__571,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: list(A),Uub: A,Uuc: list(A),Uud: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(A,fun(list(A),fun(list(A),list(A))),aa(list(A),fun(A,fun(list(A),fun(list(A),list(A)))),aTP_Lamp_rd(fun(A,fun(A,bool)),fun(list(A),fun(A,fun(list(A),fun(list(A),list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),quicksort_by_rel(A,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),aa(list(A),list(A),quicksort_by_rel(A,Uu,Uua),Uud))),Uuc) ).
% ATP.lambda_571
tff(fact_6564_ATP_Olambda__572,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,fun(B,heap_Time_Heap(A))),Uua: fun(C,C),Uub: fun(fun(B,heap_Time_Heap(A)),C),Uuc: B,Uud: fun(B,heap_Time_Heap(A))] : aa(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A),aa(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)),aa(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A)))),aTP_Lamp_yd(fun(C,fun(B,heap_Time_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Time_Heap(A)),C),fun(B,fun(fun(B,heap_Time_Heap(A)),heap_Time_Heap(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,heap_Time_Heap(A),aa(C,fun(B,heap_Time_Heap(A)),Uu,aa(C,C,Uua,aa(fun(B,heap_Time_Heap(A)),C,Uub,Uud))),Uuc) ).
% ATP.lambda_572
tff(fact_6565_ATP_Olambda__573,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,fun(B,option(A))),Uua: fun(C,C),Uub: fun(fun(B,option(A)),C),Uuc: B,Uud: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(B,fun(fun(B,option(A)),option(A)),aa(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))),aa(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A)))),aTP_Lamp_xx(fun(C,fun(B,option(A))),fun(fun(C,C),fun(fun(fun(B,option(A)),C),fun(B,fun(fun(B,option(A)),option(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,option(A),aa(C,fun(B,option(A)),Uu,aa(C,C,Uua,aa(fun(B,option(A)),C,Uub,Uud))),Uuc) ).
% ATP.lambda_573
tff(fact_6566_ATP_Olambda__574,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_on(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).
% ATP.lambda_574
tff(fact_6567_ATP_Olambda__575,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ee(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).
% ATP.lambda_575
tff(fact_6568_ATP_Olambda__576,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ec(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).
% ATP.lambda_576
tff(fact_6569_ATP_Olambda__577,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_nz(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_trancl(A,Uub)))
| ( Uuc = Uu ) )
& ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_trancl(A,Uub)))
| ( Uud = Uua ) ) ) ) ).
% ATP.lambda_577
tff(fact_6570_ATP_Olambda__578,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_ed(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uud))) ) ).
% ATP.lambda_578
tff(fact_6571_ATP_Olambda__579,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_mh(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,Uua),aa(set(A),set(A),insert2(A,Uub),aa(set(A),set(A),insert2(A,Uuc),aa(set(A),set(A),insert2(A,Uud),bot_bot(set(A))))))),field2(A,Uu)))
& ( ( ( Uua = Uuc )
& ( Uub = Uud ) )
| pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))))
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) )
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& ( Uua = Uuc )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) ) ) ) ) ).
% ATP.lambda_579
tff(fact_6572_ATP_Olambda__580,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_nc(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu),transitive_rtrancl(A,Uub)))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),transitive_rtrancl(A,Uub))) ) ) ).
% ATP.lambda_580
tff(fact_6573_ATP_Olambda__581,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_si(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( ( Uub = Uuc )
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud),Uu)) ) ) ).
% ATP.lambda_581
tff(fact_6574_ATP_Olambda__582,axiom,
! [A: $tType,Uu: heap_Time_Heap(A),Uua: heap_ext(product_unit),Uub: A,Uuc: heap_ext(product_unit),Uud: nat] :
( pp(aa(nat,bool,aa(heap_ext(product_unit),fun(nat,bool),aa(A,fun(heap_ext(product_unit),fun(nat,bool)),aa(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,bool))),aTP_Lamp_mc(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uuc),Uud))) ) ) ).
% ATP.lambda_582
tff(fact_6575_ATP_Olambda__583,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: list(B),Uud: list(B),Uue: list(B)] : aa(list(B),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))),aa(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))),aa(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))),aa(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B))))))),aTP_Lamp_pw(fun(B,A),fun(A,fun(B,fun(list(B),fun(list(B),fun(list(B),product_prod(list(B),product_prod(list(B),list(B)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uuc)),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),Uue)),if(product_prod(list(B),product_prod(list(B),list(B))),aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub)),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),Uud),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uue))),aa(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B))),aa(list(B),fun(product_prod(list(B),list(B)),product_prod(list(B),product_prod(list(B),list(B)))),product_Pair(list(B),product_prod(list(B),list(B))),Uuc),aa(list(B),product_prod(list(B),list(B)),aa(list(B),fun(list(B),product_prod(list(B),list(B))),product_Pair(list(B),list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uub),Uud)),Uue)))) ) ).
% ATP.lambda_583
tff(fact_6576_ATP_Olambda__584,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ca(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
<=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu))
| ( ( Uub = Uud )
& pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),Uua)) ) ) ) ).
% ATP.lambda_584
tff(fact_6577_ATP_Olambda__585,axiom,
! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_cc(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
<=> ( ( Uub = Uud )
& pp(aa(A,bool,Uu,Uud))
& pp(member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),aa(A,set(product_prod(B,B)),Uua,Uud))) ) ) ).
% ATP.lambda_585
tff(fact_6578_ATP_Olambda__586,axiom,
! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ih(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(Uu) ) ).
% ATP.lambda_586
tff(fact_6579_ATP_Olambda__587,axiom,
! [A: $tType,Uu: A,Uua: heap_ext(product_unit),Uub: nat] : aa(nat,A,aa(heap_ext(product_unit),fun(nat,A),aTP_Lamp_dl(A,fun(heap_ext(product_unit),fun(nat,A)),Uu),Uua),Uub) = Uu ).
% ATP.lambda_587
tff(fact_6580_ATP_Olambda__588,axiom,
! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_at(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).
% ATP.lambda_588
tff(fact_6581_ATP_Olambda__589,axiom,
! [B: $tType,A: $tType] :
( heap(B)
=> ! [Uu: heap_Time_Heap(A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_aw(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),Uu),Uua) = Uu ) ).
% ATP.lambda_589
tff(fact_6582_ATP_Olambda__590,axiom,
! [B: $tType,A: $tType,Uu: heap_Time_Heap(A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_ya(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),Uu),Uua) = Uu ).
% ATP.lambda_590
tff(fact_6583_ATP_Olambda__591,axiom,
! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,heap_ext(product_unit),aa(heap_ext(product_unit),fun(nat,heap_ext(product_unit)),aTP_Lamp_dp(heap_ext(product_unit),fun(nat,heap_ext(product_unit))),Uu),Uua) = Uu ).
% ATP.lambda_591
tff(fact_6584_ATP_Olambda__592,axiom,
! [B: $tType,A: $tType,Uu: option(A),Uua: B] : aa(B,option(A),aTP_Lamp_xw(option(A),fun(B,option(A)),Uu),Uua) = Uu ).
% ATP.lambda_592
tff(fact_6585_ATP_Olambda__593,axiom,
! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_ka(set(D),fun(C,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_593
tff(fact_6586_ATP_Olambda__594,axiom,
! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_tn(set(D),fun(B,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_594
tff(fact_6587_ATP_Olambda__595,axiom,
! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_mz(set(C),fun(B,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_595
tff(fact_6588_ATP_Olambda__596,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_to(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_596
tff(fact_6589_ATP_Olambda__597,axiom,
! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_us(set(B),fun(C,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_597
tff(fact_6590_ATP_Olambda__598,axiom,
! [B: $tType,Uu: set(B),Uua: B] : aa(B,set(B),aTP_Lamp_jp(set(B),fun(B,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_598
tff(fact_6591_ATP_Olambda__599,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_jm(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_599
tff(fact_6592_ATP_Olambda__600,axiom,
! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_ur(set(A),fun(C,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_600
tff(fact_6593_ATP_Olambda__601,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_jr(set(A),fun(B,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_601
tff(fact_6594_ATP_Olambda__602,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_ma(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_602
tff(fact_6595_ATP_Olambda__603,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,fun(B,bool),aTP_Lamp_vp(fun(B,bool),fun(A,fun(B,bool)),Uu),Uua) = Uu ).
% ATP.lambda_603
tff(fact_6596_ATP_Olambda__604,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_iv(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).
% ATP.lambda_604
tff(fact_6597_ATP_Olambda__605,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_qu(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu),Uua) = Uu ).
% ATP.lambda_605
tff(fact_6598_ATP_Olambda__606,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ah(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).
% ATP.lambda_606
tff(fact_6599_ATP_Olambda__607,axiom,
! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,B,aTP_Lamp_av(B,fun(C,B),Uu),Uua) = Uu ).
% ATP.lambda_607
tff(fact_6600_ATP_Olambda__608,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_am(A,fun(nat,A),Uu),Uua) = Uu ) ).
% ATP.lambda_608
tff(fact_6601_ATP_Olambda__609,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_iz(A,fun(B,A)),Uu),Uua) = Uu ).
% ATP.lambda_609
tff(fact_6602_ATP_Olambda__610,axiom,
! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aa(heap_ext(product_unit),fun(nat,nat),aTP_Lamp_fj(heap_ext(product_unit),fun(nat,nat)),Uu),Uua) = Uua ).
% ATP.lambda_610
tff(fact_6603_ATP_Olambda__611,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_iw(A,fun(B,B)),Uu),Uua) = Uua ).
% ATP.lambda_611
tff(fact_6604_ATP_Olambda__612,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_dd(A,fun(B,bool)),Uu),Uua))
<=> $true ) ).
% ATP.lambda_612
tff(fact_6605_ATP_Olambda__613,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_bx(nat,nat),Uu) = Uu ).
% ATP.lambda_613
tff(fact_6606_ATP_Olambda__614,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_ck(int,int),Uu) = Uu ).
% ATP.lambda_614
tff(fact_6607_ATP_Olambda__615,axiom,
! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_hm(B,B),Uu) = Uu ).
% ATP.lambda_615
tff(fact_6608_ATP_Olambda__616,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_ve(A,A),Uu) = Uu ) ).
% ATP.lambda_616
tff(fact_6609_ATP_Olambda__617,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_bh(A,A),Uu) = Uu ) ).
% ATP.lambda_617
tff(fact_6610_ATP_Olambda__618,axiom,
! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ae(A,A),Uu) = Uu ).
% ATP.lambda_618
tff(fact_6611_ATP_Olambda__619,axiom,
! [A: $tType,Uu: A] : aa(A,set(bool),aTP_Lamp_ut(A,set(bool)),Uu) = top_top(set(bool)) ).
% ATP.lambda_619
tff(fact_6612_ATP_Olambda__620,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_jn(A,set(B)),Uu) = top_top(set(B)) ).
% ATP.lambda_620
tff(fact_6613_ATP_Olambda__621,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_jl(A,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_621
tff(fact_6614_ATP_Olambda__622,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_fl(B,A),Uu) = one_one(A) ) ).
% ATP.lambda_622
tff(fact_6615_ATP_Olambda__623,axiom,
! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_ia(A,nat),Uu) = one_one(nat) ).
% ATP.lambda_623
tff(fact_6616_ATP_Olambda__624,axiom,
! [A: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ao(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).
% ATP.lambda_624
tff(fact_6617_ATP_Olambda__625,axiom,
! [B: $tType,D: $tType,Uu: B] : aa(B,option(D),aTP_Lamp_bg(B,option(D)),Uu) = none(D) ).
% ATP.lambda_625
tff(fact_6618_ATP_Olambda__626,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_bd(B,option(A)),Uu) = none(A) ).
% ATP.lambda_626
tff(fact_6619_ATP_Olambda__627,axiom,
! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_ay(A,option(C)),Uu) = none(C) ).
% ATP.lambda_627
tff(fact_6620_ATP_Olambda__628,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_hj(A,option(B)),Uu) = none(B) ).
% ATP.lambda_628
tff(fact_6621_ATP_Olambda__629,axiom,
! [B: $tType,Uu: B] :
( pp(aa(B,bool,aTP_Lamp_tw(B,bool),Uu))
<=> $false ) ).
% ATP.lambda_629
tff(fact_6622_ATP_Olambda__630,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_ad(A,bool),Uu))
<=> $false ) ).
% ATP.lambda_630
tff(fact_6623_ATP_Olambda__631,axiom,
! [Uu: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aTP_Lamp_vn(heap_ext(product_unit),bool),Uu))
<=> $true ) ).
% ATP.lambda_631
tff(fact_6624_ATP_Olambda__632,axiom,
! [D: $tType,Uu: D] :
( pp(aa(D,bool,aTP_Lamp_vm(D,bool),Uu))
<=> $true ) ).
% ATP.lambda_632
tff(fact_6625_ATP_Olambda__633,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_af(A,bool),Uu))
<=> $true ) ).
% ATP.lambda_633
tff(fact_6626_ATP_Olambda__634,axiom,
! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_os(A,fun(nat,nat)),Uu) = suc ).
% ATP.lambda_634
% Subclasses (3)
tff(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
! [A: $tType] :
( heap(A)
=> typerep2(A) ) ).
tff(subcl_Heap_Oheap___HOL_Otype,axiom,
! [A: $tType] :
( heap(A)
=> type(A) ) ).
tff(subcl_Heap_Oheap___Countable_Ocountable,axiom,
! [A: $tType] :
( heap(A)
=> countable(A) ) ).
% Type constructors (697)
tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(heap_Time_Heap(A19)) ) ).
tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(heap_Time_Heap(A19)) ) ).
tff(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
code_term_of(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
typerep2(code_natural) ).
tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
code_term_of(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
typerep2(code_integer) ).
tff(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
code_term_of(code_term) ).
tff(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
typerep2(code_term) ).
tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(heap_ext(A19)) ) ).
tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(heap_ext(A19)) ) ).
tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
code_term_of(product_unit) ).
tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum(product_unit) ).
tff(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
typerep2(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top(product_unit) ).
tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> code_term_of(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
! [A19: $tType,A20: $tType] :
( ( enum(A19)
& enum(A20) )
=> enum(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(product_prod(A19,A20)) ) ).
tff(tcon_Old__Datatype_Onode___Typerep_Otyperep_14,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(old_node(A19,A20)) ) ).
tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Typerep_Otyperep_16,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(multiset(A19)) ) ).
tff(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17,axiom,
code_term_of(typerep) ).
tff(tcon_Typerep_Otyperep___Typerep_Otyperep_18,axiom,
typerep2(typerep) ).
tff(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
typerep2(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_20,axiom,
bounded_lattice_top(assn) ).
tff(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_21,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(pred(A19)) ) ).
tff(tcon_Predicate_Opred___Typerep_Otyperep_22,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(pred(A19)) ) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_23,axiom,
! [A19: $tType] : bounded_lattice_top(pred(A19)) ).
tff(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_24,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(seq(A19)) ) ).
tff(tcon_Predicate_Oseq___Typerep_Otyperep_25,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(seq(A19)) ) ).
tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_26,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(option(A19)) ) ).
tff(tcon_Option_Ooption___Enum_Oenum_27,axiom,
! [A19: $tType] :
( enum(A19)
=> enum(option(A19)) ) ).
tff(tcon_Option_Ooption___Typerep_Otyperep_28,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(option(A19)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_29,axiom,
! [A19: $tType] :
( bounded_lattice_top(A19)
=> bounded_lattice_top(option(A19)) ) ).
tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_30,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> code_term_of(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Enum_Oenum_31,axiom,
! [A19: $tType,A20: $tType] :
( ( enum(A19)
& enum(A20) )
=> enum(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Typerep_Otyperep_32,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(sum_sum(A19,A20)) ) ).
tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_33,axiom,
code_term_of(char) ).
tff(tcon_String_Ochar___Enum_Oenum_34,axiom,
enum(char) ).
tff(tcon_String_Ochar___Typerep_Otyperep_35,axiom,
typerep2(char) ).
tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_36,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(array(A19)) ) ).
tff(tcon_Heap_Oarray___Typerep_Otyperep_37,axiom,
! [A19: $tType] : typerep2(array(A19)) ).
tff(tcon_List_Olist___Code__Evaluation_Oterm__of_38,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(list(A19)) ) ).
tff(tcon_List_Olist___Typerep_Otyperep_39,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(list(A19)) ) ).
tff(tcon_Heap_Oref___Code__Evaluation_Oterm__of_40,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(ref(A19)) ) ).
tff(tcon_Heap_Oref___Typerep_Otyperep_41,axiom,
! [A19: $tType] : typerep2(ref(A19)) ).
tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_42,axiom,
code_term_of(bool) ).
tff(tcon_HOL_Obool___Enum_Oenum_43,axiom,
enum(bool) ).
tff(tcon_HOL_Obool___Typerep_Otyperep_44,axiom,
typerep2(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_45,axiom,
bounded_lattice_top(bool) ).
tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_46,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(set(A19)) ) ).
tff(tcon_Set_Oset___Enum_Oenum_47,axiom,
! [A19: $tType] :
( enum(A19)
=> enum(set(A19)) ) ).
tff(tcon_Set_Oset___Typerep_Otyperep_48,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(set(A19)) ) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_49,axiom,
! [A19: $tType] : bounded_lattice_top(set(A19)) ).
tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_50,axiom,
code_term_of(rat) ).
tff(tcon_Rat_Orat___Typerep_Otyperep_51,axiom,
typerep2(rat) ).
tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_52,axiom,
code_term_of(num) ).
tff(tcon_Num_Onum___Typerep_Otyperep_53,axiom,
typerep2(num) ).
tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_54,axiom,
code_term_of(nat) ).
tff(tcon_Nat_Onat___Typerep_Otyperep_55,axiom,
typerep2(nat) ).
tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_56,axiom,
code_term_of(int) ).
tff(tcon_Int_Oint___Typerep_Otyperep_57,axiom,
typerep2(int) ).
tff(tcon_itself___Code__Evaluation_Oterm__of_58,axiom,
! [A19: $tType] :
( typerep2(A19)
=> code_term_of(itself(A19)) ) ).
tff(tcon_itself___Typerep_Otyperep_59,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(itself(A19)) ) ).
tff(tcon_fun___Code__Evaluation_Oterm__of_60,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> code_term_of(fun(A19,A20)) ) ).
tff(tcon_fun___Enum_Oenum_61,axiom,
! [A19: $tType,A20: $tType] :
( ( enum(A19)
& enum(A20) )
=> enum(fun(A19,A20)) ) ).
tff(tcon_fun___Typerep_Otyperep_62,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__top_63,axiom,
! [A19: $tType,A20: $tType] :
( bounded_lattice(A20)
=> bounded_lattice_top(fun(A19,A20)) ) ).
tff(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A19: $tType,A20: $tType] :
( ( cl_HOL_Oequal(A19)
& quickc3360725361186068524ustive(A19)
& quickc3360725361186068524ustive(A20) )
=> quickc3360725361186068524ustive(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( comple6319245703460814977attice(A20)
=> comple6319245703460814977attice(fun(A19,A20)) ) ).
tff(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A19: $tType,A20: $tType] :
( ( cl_HOL_Oequal(A19)
& quickc658316121487927005ustive(A19)
& quickc658316121487927005ustive(A20) )
=> quickc658316121487927005ustive(fun(A19,A20)) ) ).
tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A19: $tType,A20: $tType] :
( boolea8198339166811842893lgebra(A20)
=> boolea8198339166811842893lgebra(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A19: $tType,A20: $tType] :
( comple6319245703460814977attice(A20)
=> comple9053668089753744459l_ccpo(fun(A19,A20)) ) ).
tff(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A19: $tType,A20: $tType] :
( ( code_term_of(A19)
& cl_HOL_Oequal(A19)
& quickcheck_random(A20) )
=> quickcheck_random(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A19: $tType,A20: $tType] :
( semilattice_sup(A20)
=> semilattice_sup(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A19: $tType,A20: $tType] :
( semilattice_inf(A20)
=> semilattice_inf(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A19: $tType,A20: $tType] :
( bounded_lattice(A20)
=> bounded_lattice(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_OSup,axiom,
! [A19: $tType,A20: $tType] :
( complete_Sup(A20)
=> complete_Sup(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_OInf,axiom,
! [A19: $tType,A20: $tType] :
( complete_Inf(A20)
=> complete_Inf(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Oorder__top,axiom,
! [A19: $tType,A20: $tType] :
( order_top(A20)
=> order_top(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Oorder__bot,axiom,
! [A19: $tType,A20: $tType] :
( order_bot(A20)
=> order_bot(fun(A19,A20)) ) ).
tff(tcon_fun___Countable_Ocountable,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& countable(A20) )
=> countable(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Opreorder,axiom,
! [A19: $tType,A20: $tType] :
( preorder(A20)
=> preorder(fun(A19,A20)) ) ).
tff(tcon_fun___Finite__Set_Ofinite,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& finite_finite(A20) )
=> finite_finite(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Oorder,axiom,
! [A19: $tType,A20: $tType] :
( order(A20)
=> order(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Oord,axiom,
! [A19: $tType,A20: $tType] :
( ord(A20)
=> ord(fun(A19,A20)) ) ).
tff(tcon_fun___Groups_Ouminus,axiom,
! [A19: $tType,A20: $tType] :
( uminus(A20)
=> uminus(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Osup,axiom,
! [A19: $tType,A20: $tType] :
( semilattice_sup(A20)
=> sup(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Oinf,axiom,
! [A19: $tType,A20: $tType] :
( semilattice_inf(A20)
=> inf(fun(A19,A20)) ) ).
tff(tcon_fun___HOL_Oequal,axiom,
! [A19: $tType,A20: $tType] :
( ( enum(A19)
& cl_HOL_Oequal(A20) )
=> cl_HOL_Oequal(fun(A19,A20)) ) ).
tff(tcon_itself___Quickcheck__Random_Orandom_64,axiom,
! [A19: $tType] :
( typerep2(A19)
=> quickcheck_random(itself(A19)) ) ).
tff(tcon_itself___HOL_Oequal_65,axiom,
! [A19: $tType] : cl_HOL_Oequal(itself(A19)) ).
tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations(int) ).
tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel(int) ).
tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel(int) ).
tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le(int) ).
tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations(int) ).
tff(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_66,axiom,
quickc3360725361186068524ustive(int) ).
tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict(int) ).
tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add(int) ).
tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor(int) ).
tff(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_67,axiom,
quickc658316121487927005ustive(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs(int) ).
tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add(int) ).
tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add(int) ).
tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel(int) ).
tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits(int) ).
tff(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(tcon_Int_Oint___Quickcheck__Random_Orandom_68,axiom,
quickcheck_random(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__sup_69,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_70,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom(int) ).
tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity(int) ).
tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(int) ).
tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult(int) ).
tff(tcon_Int_Oint___Complete__Lattices_OSup_71,axiom,
complete_Sup(int) ).
tff(tcon_Int_Oint___Complete__Lattices_OInf_72,axiom,
complete_Inf(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide(int) ).
tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral(int) ).
tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
tff(tcon_Int_Oint___Countable_Ocountable_73,axiom,
countable(int) ).
tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn(int) ).
tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity(int) ).
tff(tcon_Int_Oint___Orderings_Opreorder_74,axiom,
preorder(int) ).
tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top(int) ).
tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot(int) ).
tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring(int) ).
tff(tcon_Int_Oint___Orderings_Oorder_75,axiom,
order(int) ).
tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral(int) ).
tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0(int) ).
tff(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(tcon_Int_Oint___Orderings_Oord_76,axiom,
ord(int) ).
tff(tcon_Int_Oint___Groups_Ouminus_77,axiom,
uminus(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(tcon_Int_Oint___Lattices_Osup_78,axiom,
sup(int) ).
tff(tcon_Int_Oint___Lattices_Oinf_79,axiom,
inf(int) ).
tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd(int) ).
tff(tcon_Int_Oint___Power_Opower,axiom,
power(int) ).
tff(tcon_Int_Oint___Num_Onumeral,axiom,
numeral(int) ).
tff(tcon_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(tcon_Int_Oint___Groups_Oplus,axiom,
plus(int) ).
tff(tcon_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(tcon_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(tcon_Int_Oint___Rings_Odvd,axiom,
dvd(int) ).
tff(tcon_Int_Oint___Heap_Oheap,axiom,
heap(int) ).
tff(tcon_Int_Oint___HOL_Oequal_80,axiom,
cl_HOL_Oequal(int) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_81,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_82,axiom,
semiri1453513574482234551roduct(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_83,axiom,
euclid5411537665997757685th_nat(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_84,axiom,
ordere1937475149494474687imp_le(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_85,axiom,
euclid3128863361964157862miring(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_86,axiom,
euclid4440199948858584721cancel(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_87,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_88,axiom,
strict9044650504122735259up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
ordere1170586879665033532d_diff(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_89,axiom,
ordere580206878836729694up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_90,axiom,
ordere2412721322843649153imp_le(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_91,axiom,
bit_se359711467146920520ations(nat) ).
tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_92,axiom,
quickc3360725361186068524ustive(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_93,axiom,
strict7427464778891057005id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_94,axiom,
ordere8940638589300402666id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni5634975068530333245id_add(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_95,axiom,
euclid3725896446679973847miring(nat) ).
tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_96,axiom,
linord4140545234300271783up_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_97,axiom,
linord181362715937106298miring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_98,axiom,
semido2269285787275462019factor(nat) ).
tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_99,axiom,
quickc658316121487927005ustive(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_100,axiom,
semiri3467727345109120633visors(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_101,axiom,
ordere6658533253407199908up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_102,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_103,axiom,
cancel2418104881723323429up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_104,axiom,
cancel1802427076303600483id_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_105,axiom,
comm_s4317794764714335236cancel(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_106,axiom,
bit_semiring_bits(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom_107,axiom,
normal8620421768224518004emidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_108,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_109,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Quickcheck__Random_Orandom_110,axiom,
quickcheck_random(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_111,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_112,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_113,axiom,
ab_semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_114,axiom,
algebraic_semidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_115,axiom,
comm_monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__add_116,axiom,
ab_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Parity_Osemiring__parity_117,axiom,
semiring_parity(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_118,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__modulo_119,axiom,
semiring_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_120,axiom,
comm_semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_121,axiom,
comm_semiring_0(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__mult_122,axiom,
semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Complete__Lattices_OSup_123,axiom,
complete_Sup(nat) ).
tff(tcon_Nat_Onat___Complete__Lattices_OInf_124,axiom,
complete_Inf(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__modulo_125,axiom,
semidom_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide_126,axiom,
semidom_divide(nat) ).
tff(tcon_Nat_Onat___Num_Osemiring__numeral_127,axiom,
semiring_numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__add_128,axiom,
semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__less__one_129,axiom,
zero_less_one(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring_130,axiom,
comm_semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder__bot_131,axiom,
order_bot(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_132,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Countable_Ocountable_133,axiom,
countable(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__neq__one_134,axiom,
zero_neq_one(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_135,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_136,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__mult_137,axiom,
monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__add_138,axiom,
monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_139,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_140,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Orderings_Ono__top_141,axiom,
no_top(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_142,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_143,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_144,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring_145,axiom,
semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_146,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Lattices_Osup_147,axiom,
sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Oinf_148,axiom,
inf(nat) ).
tff(tcon_Nat_Onat___Power_Opower_149,axiom,
power(nat) ).
tff(tcon_Nat_Onat___Num_Onumeral_150,axiom,
numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ozero_151,axiom,
zero(nat) ).
tff(tcon_Nat_Onat___Groups_Oplus_152,axiom,
plus(nat) ).
tff(tcon_Nat_Onat___Groups_Oone_153,axiom,
one(nat) ).
tff(tcon_Nat_Onat___Rings_Odvd_154,axiom,
dvd(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_155,axiom,
heap(nat) ).
tff(tcon_Nat_Onat___HOL_Oequal_156,axiom,
cl_HOL_Oequal(nat) ).
tff(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_157,axiom,
quickc3360725361186068524ustive(num) ).
tff(tcon_Num_Onum___Quickcheck__Random_Orandom_158,axiom,
quickcheck_random(num) ).
tff(tcon_Num_Onum___Orderings_Opreorder_159,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_160,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_161,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_162,axiom,
ord(num) ).
tff(tcon_Num_Onum___Groups_Oplus_163,axiom,
plus(num) ).
tff(tcon_Num_Onum___HOL_Oequal_164,axiom,
cl_HOL_Oequal(num) ).
tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_165,axiom,
semiri1453513574482234551roduct(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_166,axiom,
ordere1937475149494474687imp_le(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167,axiom,
strict9044650504122735259up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168,axiom,
ordere580206878836729694up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169,axiom,
ordere2412721322843649153imp_le(rat) ).
tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_170,axiom,
quickc3360725361186068524ustive(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171,axiom,
strict7427464778891057005id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172,axiom,
ordere8940638589300402666id_add(rat) ).
tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_173,axiom,
linord715952674999750819strict(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_174,axiom,
linord4140545234300271783up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_175,axiom,
linord181362715937106298miring(rat) ).
tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_176,axiom,
quickc658316121487927005ustive(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_177,axiom,
semiri3467727345109120633visors(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_178,axiom,
ordere6658533253407199908up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_179,axiom,
ordere166539214618696060dd_abs(rat) ).
tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_180,axiom,
ordere6911136660526730532id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_181,axiom,
linord5086331880401160121up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_182,axiom,
cancel2418104881723323429up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_183,axiom,
ring_15535105094025558882visors(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_184,axiom,
cancel1802427076303600483id_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_185,axiom,
linord4710134922213307826strict(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_186,axiom,
comm_s4317794764714335236cancel(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_187,axiom,
linord6961819062388156250ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_188,axiom,
ordered_ab_group_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_189,axiom,
cancel_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semidom_190,axiom,
linordered_semidom(rat) ).
tff(tcon_Rat_Orat___Quickcheck__Random_Orandom_191,axiom,
quickcheck_random(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_192,axiom,
semilattice_sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_193,axiom,
semilattice_inf(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_194,axiom,
ab_semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_195,axiom,
comm_monoid_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_196,axiom,
ab_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_197,axiom,
comm_monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring_198,axiom,
linordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__idom_199,axiom,
linordered_idom(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_200,axiom,
comm_semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_201,axiom,
comm_semiring_0(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__mult_202,axiom,
semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Osemidom__divide_203,axiom,
semidom_divide(rat) ).
tff(tcon_Rat_Orat___Num_Osemiring__numeral_204,axiom,
semiring_numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__add_205,axiom,
semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__less__one_206,axiom,
zero_less_one(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring_207,axiom,
comm_semiring(rat) ).
tff(tcon_Rat_Orat___Nat_Osemiring__char__0_208,axiom,
semiring_char_0(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__group__add_209,axiom,
ab_group_add(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0(rat) ).
tff(tcon_Rat_Orat___Countable_Ocountable_210,axiom,
countable(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__neq__one_211,axiom,
zero_neq_one(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring_212,axiom,
ordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_213,axiom,
idom_abs_sgn(rat) ).
tff(tcon_Rat_Orat___Orderings_Opreorder_214,axiom,
preorder(rat) ).
tff(tcon_Rat_Orat___Orderings_Olinorder_215,axiom,
linorder(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__mult_216,axiom,
monoid_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_217,axiom,
comm_ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__add_218,axiom,
monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1_219,axiom,
semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__0_220,axiom,
semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__top_221,axiom,
no_top(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__bot_222,axiom,
no_bot(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_223,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring_224,axiom,
comm_ring(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_225,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Num_Oneg__numeral_226,axiom,
neg_numeral(rat) ).
tff(tcon_Rat_Orat___Nat_Oring__char__0_227,axiom,
ring_char_0(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring_228,axiom,
semiring(rat) ).
tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_229,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Groups_Ouminus_230,axiom,
uminus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_231,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Lattices_Osup_232,axiom,
sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Oinf_233,axiom,
inf(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
tff(tcon_Rat_Orat___Power_Opower_234,axiom,
power(rat) ).
tff(tcon_Rat_Orat___Num_Onumeral_235,axiom,
numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Ozero_236,axiom,
zero(rat) ).
tff(tcon_Rat_Orat___Groups_Oplus_237,axiom,
plus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring_238,axiom,
ring(rat) ).
tff(tcon_Rat_Orat___Groups_Oone_239,axiom,
one(rat) ).
tff(tcon_Rat_Orat___Rings_Odvd_240,axiom,
dvd(rat) ).
tff(tcon_Rat_Orat___HOL_Oequal_241,axiom,
cl_HOL_Oequal(rat) ).
tff(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_242,axiom,
! [A19: $tType] :
( quickc3360725361186068524ustive(A19)
=> quickc3360725361186068524ustive(set(A19)) ) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243,axiom,
! [A19: $tType] : comple6319245703460814977attice(set(A19)) ).
tff(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_244,axiom,
! [A19: $tType] :
( quickc658316121487927005ustive(A19)
=> quickc658316121487927005ustive(set(A19)) ) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_245,axiom,
! [A19: $tType] : boolea8198339166811842893lgebra(set(A19)) ).
tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_246,axiom,
! [A19: $tType] : comple9053668089753744459l_ccpo(set(A19)) ).
tff(tcon_Set_Oset___Quickcheck__Random_Orandom_247,axiom,
! [A19: $tType] :
( quickcheck_random(A19)
=> quickcheck_random(set(A19)) ) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_248,axiom,
! [A19: $tType] : semilattice_sup(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_249,axiom,
! [A19: $tType] : semilattice_inf(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice_250,axiom,
! [A19: $tType] : bounded_lattice(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_OSup_251,axiom,
! [A19: $tType] : complete_Sup(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_OInf_252,axiom,
! [A19: $tType] : complete_Inf(set(A19)) ).
tff(tcon_Set_Oset___Orderings_Oorder__top_253,axiom,
! [A19: $tType] : order_top(set(A19)) ).
tff(tcon_Set_Oset___Orderings_Oorder__bot_254,axiom,
! [A19: $tType] : order_bot(set(A19)) ).
tff(tcon_Set_Oset___Countable_Ocountable_255,axiom,
! [A19: $tType] :
( finite_finite(A19)
=> countable(set(A19)) ) ).
tff(tcon_Set_Oset___Orderings_Opreorder_256,axiom,
! [A19: $tType] : preorder(set(A19)) ).
tff(tcon_Set_Oset___Finite__Set_Ofinite_257,axiom,
! [A19: $tType] :
( finite_finite(A19)
=> finite_finite(set(A19)) ) ).
tff(tcon_Set_Oset___Orderings_Oorder_258,axiom,
! [A19: $tType] : order(set(A19)) ).
tff(tcon_Set_Oset___Orderings_Oord_259,axiom,
! [A19: $tType] : ord(set(A19)) ).
tff(tcon_Set_Oset___Groups_Ouminus_260,axiom,
! [A19: $tType] : uminus(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Osup_261,axiom,
! [A19: $tType] : sup(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Oinf_262,axiom,
! [A19: $tType] : inf(set(A19)) ).
tff(tcon_Set_Oset___HOL_Oequal_263,axiom,
! [A19: $tType] :
( cl_HOL_Oequal(A19)
=> cl_HOL_Oequal(set(A19)) ) ).
tff(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
quickc3360725361186068524ustive(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_265,axiom,
comple6319245703460814977attice(bool) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_266,axiom,
boolea8198339166811842893lgebra(bool) ).
tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_267,axiom,
comple9053668089753744459l_ccpo(bool) ).
tff(tcon_HOL_Obool___Quickcheck__Random_Orandom_268,axiom,
quickcheck_random(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_269,axiom,
semilattice_sup(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_270,axiom,
semilattice_inf(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice_271,axiom,
bounded_lattice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_OSup_272,axiom,
complete_Sup(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_OInf_273,axiom,
complete_Inf(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder__top_274,axiom,
order_top(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder__bot_275,axiom,
order_bot(bool) ).
tff(tcon_HOL_Obool___Countable_Ocountable_276,axiom,
countable(bool) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_277,axiom,
preorder(bool) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_278,axiom,
linorder(bool) ).
tff(tcon_HOL_Obool___Finite__Set_Ofinite_279,axiom,
finite_finite(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder_280,axiom,
order(bool) ).
tff(tcon_HOL_Obool___Orderings_Oord_281,axiom,
ord(bool) ).
tff(tcon_HOL_Obool___Groups_Ouminus_282,axiom,
uminus(bool) ).
tff(tcon_HOL_Obool___Lattices_Osup_283,axiom,
sup(bool) ).
tff(tcon_HOL_Obool___Lattices_Oinf_284,axiom,
inf(bool) ).
tff(tcon_HOL_Obool___Heap_Oheap_285,axiom,
heap(bool) ).
tff(tcon_HOL_Obool___HOL_Oequal_286,axiom,
cl_HOL_Oequal(bool) ).
tff(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_287,axiom,
! [A19: $tType] :
( typerep2(A19)
=> quickc3360725361186068524ustive(ref(A19)) ) ).
tff(tcon_Heap_Oref___Quickcheck__Random_Orandom_288,axiom,
! [A19: $tType] :
( typerep2(A19)
=> quickcheck_random(ref(A19)) ) ).
tff(tcon_Heap_Oref___Countable_Ocountable_289,axiom,
! [A19: $tType] : countable(ref(A19)) ).
tff(tcon_Heap_Oref___Heap_Oheap_290,axiom,
! [A19: $tType] : heap(ref(A19)) ).
tff(tcon_Heap_Oref___HOL_Oequal_291,axiom,
! [A19: $tType] : cl_HOL_Oequal(ref(A19)) ).
tff(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_292,axiom,
! [A19: $tType] :
( quickc3360725361186068524ustive(A19)
=> quickc3360725361186068524ustive(list(A19)) ) ).
tff(tcon_List_Olist___Quickcheck__Random_Orandom_293,axiom,
! [A19: $tType] :
( quickcheck_random(A19)
=> quickcheck_random(list(A19)) ) ).
tff(tcon_List_Olist___Countable_Ocountable_294,axiom,
! [A19: $tType] :
( countable(A19)
=> countable(list(A19)) ) ).
tff(tcon_List_Olist___Heap_Oheap_295,axiom,
! [A19: $tType] :
( heap(A19)
=> heap(list(A19)) ) ).
tff(tcon_List_Olist___HOL_Oequal_296,axiom,
! [A19: $tType] : cl_HOL_Oequal(list(A19)) ).
tff(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_297,axiom,
! [A19: $tType] :
( typerep2(A19)
=> quickc3360725361186068524ustive(array(A19)) ) ).
tff(tcon_Heap_Oarray___Quickcheck__Random_Orandom_298,axiom,
! [A19: $tType] :
( typerep2(A19)
=> quickcheck_random(array(A19)) ) ).
tff(tcon_Heap_Oarray___Countable_Ocountable_299,axiom,
! [A19: $tType] : countable(array(A19)) ).
tff(tcon_Heap_Oarray___Heap_Oheap_300,axiom,
! [A19: $tType] : heap(array(A19)) ).
tff(tcon_Heap_Oarray___HOL_Oequal_301,axiom,
! [A19: $tType] : cl_HOL_Oequal(array(A19)) ).
tff(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_302,axiom,
quickc3360725361186068524ustive(char) ).
tff(tcon_String_Ochar___Quickcheck__Random_Orandom_303,axiom,
quickcheck_random(char) ).
tff(tcon_String_Ochar___Countable_Ocountable_304,axiom,
countable(char) ).
tff(tcon_String_Ochar___Finite__Set_Ofinite_305,axiom,
finite_finite(char) ).
tff(tcon_String_Ochar___Heap_Oheap_306,axiom,
heap(char) ).
tff(tcon_String_Ochar___HOL_Oequal_307,axiom,
cl_HOL_Oequal(char) ).
tff(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_308,axiom,
! [A19: $tType,A20: $tType] :
( ( quickc3360725361186068524ustive(A19)
& quickc3360725361186068524ustive(A20) )
=> quickc3360725361186068524ustive(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_309,axiom,
! [A19: $tType,A20: $tType] :
( ( quickcheck_random(A19)
& quickcheck_random(A20) )
=> quickcheck_random(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Countable_Ocountable_310,axiom,
! [A19: $tType,A20: $tType] :
( ( countable(A19)
& countable(A20) )
=> countable(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_311,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& finite_finite(A20) )
=> finite_finite(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Heap_Oheap_312,axiom,
! [A19: $tType,A20: $tType] :
( ( heap(A19)
& heap(A20) )
=> heap(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___HOL_Oequal_313,axiom,
! [A19: $tType,A20: $tType] : cl_HOL_Oequal(sum_sum(A19,A20)) ).
tff(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_314,axiom,
! [A19: $tType] :
( quickc3360725361186068524ustive(A19)
=> quickc3360725361186068524ustive(option(A19)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_315,axiom,
! [A19: $tType] :
( comple6319245703460814977attice(A19)
=> comple6319245703460814977attice(option(A19)) ) ).
tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_316,axiom,
! [A19: $tType] :
( comple6319245703460814977attice(A19)
=> comple9053668089753744459l_ccpo(option(A19)) ) ).
tff(tcon_Option_Ooption___Quickcheck__Random_Orandom_317,axiom,
! [A19: $tType] :
( quickcheck_random(A19)
=> quickcheck_random(option(A19)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_318,axiom,
! [A19: $tType] :
( semilattice_sup(A19)
=> semilattice_sup(option(A19)) ) ).
tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_319,axiom,
! [A19: $tType] :
( semilattice_inf(A19)
=> semilattice_inf(option(A19)) ) ).
tff(tcon_Option_Ooption___Lattices_Obounded__lattice_320,axiom,
! [A19: $tType] :
( bounded_lattice_top(A19)
=> bounded_lattice(option(A19)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_OSup_321,axiom,
! [A19: $tType] :
( comple6319245703460814977attice(A19)
=> complete_Sup(option(A19)) ) ).
tff(tcon_Option_Ooption___Complete__Lattices_OInf_322,axiom,
! [A19: $tType] :
( comple6319245703460814977attice(A19)
=> complete_Inf(option(A19)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__top_323,axiom,
! [A19: $tType] :
( order_top(A19)
=> order_top(option(A19)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder__bot_324,axiom,
! [A19: $tType] :
( order(A19)
=> order_bot(option(A19)) ) ).
tff(tcon_Option_Ooption___Countable_Ocountable_325,axiom,
! [A19: $tType] :
( countable(A19)
=> countable(option(A19)) ) ).
tff(tcon_Option_Ooption___Orderings_Opreorder_326,axiom,
! [A19: $tType] :
( preorder(A19)
=> preorder(option(A19)) ) ).
tff(tcon_Option_Ooption___Orderings_Olinorder_327,axiom,
! [A19: $tType] :
( linorder(A19)
=> linorder(option(A19)) ) ).
tff(tcon_Option_Ooption___Finite__Set_Ofinite_328,axiom,
! [A19: $tType] :
( finite_finite(A19)
=> finite_finite(option(A19)) ) ).
tff(tcon_Option_Ooption___Orderings_Oorder_329,axiom,
! [A19: $tType] :
( order(A19)
=> order(option(A19)) ) ).
tff(tcon_Option_Ooption___Orderings_Oord_330,axiom,
! [A19: $tType] :
( preorder(A19)
=> ord(option(A19)) ) ).
tff(tcon_Option_Ooption___Lattices_Osup_331,axiom,
! [A19: $tType] :
( sup(A19)
=> sup(option(A19)) ) ).
tff(tcon_Option_Ooption___Lattices_Oinf_332,axiom,
! [A19: $tType] :
( inf(A19)
=> inf(option(A19)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_333,axiom,
! [A19: $tType] :
( heap(A19)
=> heap(option(A19)) ) ).
tff(tcon_Option_Ooption___HOL_Oequal_334,axiom,
! [A19: $tType] : cl_HOL_Oequal(option(A19)) ).
tff(tcon_Predicate_Oseq___HOL_Oequal_335,axiom,
! [A19: $tType] : cl_HOL_Oequal(seq(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_336,axiom,
! [A19: $tType] : comple6319245703460814977attice(pred(A19)) ).
tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_337,axiom,
! [A19: $tType] : boolea8198339166811842893lgebra(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_338,axiom,
! [A19: $tType] : comple9053668089753744459l_ccpo(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_339,axiom,
! [A19: $tType] : semilattice_sup(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_340,axiom,
! [A19: $tType] : semilattice_inf(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Obounded__lattice_341,axiom,
! [A19: $tType] : bounded_lattice(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_OSup_342,axiom,
! [A19: $tType] : complete_Sup(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_OInf_343,axiom,
! [A19: $tType] : complete_Inf(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder__top_344,axiom,
! [A19: $tType] : order_top(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder__bot_345,axiom,
! [A19: $tType] : order_bot(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Opreorder_346,axiom,
! [A19: $tType] : preorder(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Oorder_347,axiom,
! [A19: $tType] : order(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Oord_348,axiom,
! [A19: $tType] : ord(pred(A19)) ).
tff(tcon_Predicate_Opred___Groups_Ouminus_349,axiom,
! [A19: $tType] : uminus(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Osup_350,axiom,
! [A19: $tType] : sup(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Oinf_351,axiom,
! [A19: $tType] : inf(pred(A19)) ).
tff(tcon_Predicate_Opred___HOL_Oequal_352,axiom,
! [A19: $tType] : cl_HOL_Oequal(pred(A19)) ).
tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_353,axiom,
boolea8198339166811842893lgebra(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_354,axiom,
semilattice_sup(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_355,axiom,
semilattice_inf(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_356,axiom,
bounded_lattice(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_357,axiom,
ab_semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_358,axiom,
comm_monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_359,axiom,
semigroup_mult(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__top_360,axiom,
order_top(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_361,axiom,
order_bot(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Opreorder_362,axiom,
preorder(assn) ).
tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_363,axiom,
monoid_mult(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oorder_364,axiom,
order(assn) ).
tff(tcon_Assertions_Oassn___Orderings_Oord_365,axiom,
ord(assn) ).
tff(tcon_Assertions_Oassn___Groups_Ouminus_366,axiom,
uminus(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Osup_367,axiom,
sup(assn) ).
tff(tcon_Assertions_Oassn___Lattices_Oinf_368,axiom,
inf(assn) ).
tff(tcon_Assertions_Oassn___Power_Opower_369,axiom,
power(assn) ).
tff(tcon_Assertions_Oassn___Groups_Oone_370,axiom,
one(assn) ).
tff(tcon_Assertions_Oassn___Rings_Odvd_371,axiom,
dvd(assn) ).
tff(tcon_Typerep_Otyperep___Countable_Ocountable_372,axiom,
countable(typerep) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_373,axiom,
heap(typerep) ).
tff(tcon_Typerep_Otyperep___HOL_Oequal_374,axiom,
cl_HOL_Oequal(typerep) ).
tff(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_375,axiom,
! [A19: $tType] :
( quickc3360725361186068524ustive(A19)
=> quickc3360725361186068524ustive(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_376,axiom,
! [A19: $tType] :
( preorder(A19)
=> ordere6658533253407199908up_add(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_377,axiom,
! [A19: $tType] : cancel2418104881723323429up_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_378,axiom,
! [A19: $tType] : cancel1802427076303600483id_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_379,axiom,
! [A19: $tType] : cancel_semigroup_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_380,axiom,
! [A19: $tType] :
( quickcheck_random(A19)
=> quickcheck_random(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_381,axiom,
! [A19: $tType] : comm_monoid_diff(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_382,axiom,
! [A19: $tType] : ab_semigroup_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_383,axiom,
! [A19: $tType] : comm_monoid_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Complete__Lattices_OSup_384,axiom,
! [A19: $tType] : complete_Sup(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Complete__Lattices_OInf_385,axiom,
! [A19: $tType] : complete_Inf(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_386,axiom,
! [A19: $tType] : semigroup_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Orderings_Opreorder_387,axiom,
! [A19: $tType] :
( preorder(A19)
=> preorder(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_388,axiom,
! [A19: $tType] : monoid_add(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Orderings_Oorder_389,axiom,
! [A19: $tType] :
( preorder(A19)
=> order(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Orderings_Oord_390,axiom,
! [A19: $tType] :
( preorder(A19)
=> ord(multiset(A19)) ) ).
tff(tcon_Multiset_Omultiset___Groups_Ozero_391,axiom,
! [A19: $tType] : zero(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___Groups_Oplus_392,axiom,
! [A19: $tType] : plus(multiset(A19)) ).
tff(tcon_Multiset_Omultiset___HOL_Oequal_393,axiom,
! [A19: $tType] :
( cl_HOL_Oequal(A19)
=> cl_HOL_Oequal(multiset(A19)) ) ).
tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_394,axiom,
! [A19: $tType,A20: $tType] :
( ( quickc3360725361186068524ustive(A19)
& quickc3360725361186068524ustive(A20) )
=> quickc3360725361186068524ustive(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_395,axiom,
! [A19: $tType,A20: $tType] :
( ( quickc658316121487927005ustive(A19)
& quickc658316121487927005ustive(A20) )
=> quickc658316121487927005ustive(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_396,axiom,
! [A19: $tType,A20: $tType] :
( ( quickcheck_random(A19)
& quickcheck_random(A20) )
=> quickcheck_random(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Countable_Ocountable_397,axiom,
! [A19: $tType,A20: $tType] :
( ( countable(A19)
& countable(A20) )
=> countable(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_398,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& finite_finite(A20) )
=> finite_finite(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_399,axiom,
! [A19: $tType,A20: $tType] :
( ( heap(A19)
& heap(A20) )
=> heap(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___HOL_Oequal_400,axiom,
! [A19: $tType,A20: $tType] : cl_HOL_Oequal(product_prod(A19,A20)) ).
tff(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_401,axiom,
quickc3360725361186068524ustive(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_402,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_403,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_404,axiom,
comple9053668089753744459l_ccpo(product_unit) ).
tff(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_405,axiom,
quickcheck_random(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_406,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_407,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_408,axiom,
bounded_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_409,axiom,
complete_Sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_410,axiom,
complete_Inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_411,axiom,
order_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_412,axiom,
order_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Countable_Ocountable_413,axiom,
countable(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_414,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_415,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_416,axiom,
finite_finite(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_417,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_418,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ouminus_419,axiom,
uminus(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osup_420,axiom,
sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Oinf_421,axiom,
inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_422,axiom,
heap(product_unit) ).
tff(tcon_Product__Type_Ounit___HOL_Oequal_423,axiom,
cl_HOL_Oequal(product_unit) ).
tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_424,axiom,
! [A19: $tType] :
( quickcheck_random(A19)
=> quickcheck_random(heap_ext(A19)) ) ).
tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_425,axiom,
! [A19: $tType] : cl_HOL_Oequal(heap_ext(A19)) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_426,axiom,
bit_un5681908812861735899ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_427,axiom,
semiri1453513574482234551roduct(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_428,axiom,
euclid5411537665997757685th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_429,axiom,
euclid8789492081693882211th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_430,axiom,
ordere1937475149494474687imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_431,axiom,
euclid3128863361964157862miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_432,axiom,
euclid4440199948858584721cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_433,axiom,
unique1627219031080169319umeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_434,axiom,
euclid8851590272496341667cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_435,axiom,
strict9044650504122735259up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_436,axiom,
ordere580206878836729694up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_437,axiom,
ordere2412721322843649153imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_438,axiom,
bit_se359711467146920520ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_439,axiom,
quickc3360725361186068524ustive(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_440,axiom,
strict7427464778891057005id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_441,axiom,
ordere8940638589300402666id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_442,axiom,
euclid3725896446679973847miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_443,axiom,
linord715952674999750819strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_444,axiom,
linord4140545234300271783up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_445,axiom,
bit_ri3973907225187159222ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_446,axiom,
linord181362715937106298miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_447,axiom,
quickc658316121487927005ustive(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_448,axiom,
semiri3467727345109120633visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_449,axiom,
ordere6658533253407199908up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_450,axiom,
ordere166539214618696060dd_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_451,axiom,
ordere6911136660526730532id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_452,axiom,
linord5086331880401160121up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_453,axiom,
cancel2418104881723323429up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_454,axiom,
ring_15535105094025558882visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_455,axiom,
cancel1802427076303600483id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_456,axiom,
linord4710134922213307826strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_457,axiom,
comm_s4317794764714335236cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_458,axiom,
bit_semiring_bits(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_459,axiom,
linord6961819062388156250ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_460,axiom,
ordered_ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_461,axiom,
cancel_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_462,axiom,
linordered_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_463,axiom,
quickcheck_random(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_464,axiom,
ab_semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_465,axiom,
algebraic_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_466,axiom,
comm_monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_467,axiom,
ab_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_468,axiom,
semiring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_469,axiom,
comm_monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_470,axiom,
semiring_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_471,axiom,
linordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_472,axiom,
linordered_idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_473,axiom,
comm_semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_474,axiom,
comm_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_475,axiom,
semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_476,axiom,
semidom_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_477,axiom,
semidom_divide(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_478,axiom,
semiring_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_479,axiom,
semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_480,axiom,
zero_less_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_481,axiom,
comm_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_482,axiom,
semiring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_483,axiom,
ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_484,axiom,
zero_neq_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_485,axiom,
ordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_486,axiom,
idom_abs_sgn(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_487,axiom,
ring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_488,axiom,
preorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_489,axiom,
linorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_490,axiom,
monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_491,axiom,
comm_ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_492,axiom,
monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_493,axiom,
semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_494,axiom,
semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_495,axiom,
group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_496,axiom,
comm_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_497,axiom,
order(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_498,axiom,
neg_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_499,axiom,
ring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_500,axiom,
semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_501,axiom,
ord(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_502,axiom,
uminus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_503,axiom,
ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Power_Opower_504,axiom,
power(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_505,axiom,
numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_506,axiom,
zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_507,axiom,
plus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring_508,axiom,
ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oone_509,axiom,
one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_510,axiom,
dvd(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_511,axiom,
cl_HOL_Oequal(code_integer) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_512,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_513,axiom,
euclid5411537665997757685th_nat(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_514,axiom,
ordere1937475149494474687imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_515,axiom,
euclid3128863361964157862miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_516,axiom,
euclid4440199948858584721cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_517,axiom,
strict9044650504122735259up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_518,axiom,
ordere580206878836729694up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_519,axiom,
ordere2412721322843649153imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_520,axiom,
bit_se359711467146920520ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_521,axiom,
quickc3360725361186068524ustive(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_522,axiom,
strict7427464778891057005id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_523,axiom,
ordere8940638589300402666id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_524,axiom,
euclid3725896446679973847miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_525,axiom,
linord4140545234300271783up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_526,axiom,
linord181362715937106298miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_527,axiom,
quickc658316121487927005ustive(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_528,axiom,
semiri3467727345109120633visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_529,axiom,
ordere6658533253407199908up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_530,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_531,axiom,
cancel2418104881723323429up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_532,axiom,
cancel1802427076303600483id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_533,axiom,
comm_s4317794764714335236cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_534,axiom,
bit_semiring_bits(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_535,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_536,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_537,axiom,
quickcheck_random(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_538,axiom,
ab_semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_539,axiom,
algebraic_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_540,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_541,axiom,
comm_monoid_diff(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_542,axiom,
ab_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_543,axiom,
semiring_parity(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_544,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_545,axiom,
semiring_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_546,axiom,
comm_semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_547,axiom,
comm_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_548,axiom,
semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_549,axiom,
semidom_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_550,axiom,
semidom_divide(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_551,axiom,
semiring_numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_552,axiom,
semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_553,axiom,
zero_less_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_554,axiom,
comm_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_555,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_556,axiom,
zero_neq_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_557,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_558,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_559,axiom,
monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_560,axiom,
monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_561,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_562,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_563,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_564,axiom,
semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_565,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Power_Opower_566,axiom,
power(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Onumeral_567,axiom,
numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ozero_568,axiom,
zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oplus_569,axiom,
plus(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oone_570,axiom,
one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Odvd_571,axiom,
dvd(code_natural) ).
tff(tcon_Code__Numeral_Onatural___HOL_Oequal_572,axiom,
cl_HOL_Oequal(code_natural) ).
tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_573,axiom,
! [A19: $tType] :
( quickcheck_random(A19)
=> quickcheck_random(heap_Time_Heap(A19)) ) ).
tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_574,axiom,
! [A19: $tType] : cl_HOL_Oequal(heap_Time_Heap(A19)) ).
% Helper facts (16)
tff(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] : if(A,fFalse,X2,Y) = Y ).
tff(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] : if(A,fTrue,X2,Y) = X2 ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fconj(P,Q))
| pp(Q) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fconj(P,Q))
| pp(P) ) ).
tff(help_fconj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(fconj(P,Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(fdisj(P,Q))
| pp(P)
| pp(Q) ) ).
tff(help_fdisj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(Q)
| pp(fdisj(P,Q)) ) ).
tff(help_fdisj_1_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(P)
| pp(fdisj(P,Q)) ) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( X2 != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),Y)) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),Y))
| ( X2 = Y ) ) ).
tff(help_fChoice_1_1_T,axiom,
! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).
% Free types (1)
tff(tfree_0,hypothesis,
heap(a) ).
% Conjectures (1)
tff(conj_0,conjecture,
aa(heap_ext(product_unit),option(product_prod(array(a),product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(array(a)),fun(heap_ext(product_unit),option(product_prod(array(a),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(a)),array_of_list(a,xs)),h) != none(product_prod(array(a),product_prod(heap_ext(product_unit),nat))) ).
%------------------------------------------------------------------------------