TPTP Problem File: ITP208_2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP208_2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Array_Time 00247_008755
% 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 : 0019_Array_Time_00247_008755 [Des22]
% Status : Theorem
% Rating : 0.50 v8.1.0
% Syntax : Number of formulae : 9605 (2354 unt;1225 typ; 0 def)
% Number of atoms : 22119 (7002 equ)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 15375 (1636 ~; 291 |;1333 &)
% (1643 <=>;10472 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 5 avg)
% Maximal term depth : 18 ( 2 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 1076 ( 773 >; 303 *; 0 +; 0 <<)
% Number of predicates : 241 ( 238 usr; 3 prp; 0-7 aty)
% Number of functors : 975 ( 975 usr; 80 con; 0-8 aty)
% Number of variables : 25063 (22983 !; 395 ?;25063 :)
% (1685 !>; 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 14:11:03.523
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
tff(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: ( $tType * $tType * $tType ) > $tType ).
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_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_Nitpick_Ofun__box,type,
fun_box: ( $tType * $tType ) > $tType ).
tff(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
tff(ty_t_String_Oliteral,type,
literal: $tType ).
tff(ty_t_Sum__Type_Osum,type,
sum_sum: ( $tType * $tType ) > $tType ).
tff(ty_t_Option_Ooption,type,
option: $tType > $tType ).
tff(ty_t_Nitpick_Oword,type,
word: $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_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 (1199)
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osize,type,
size:
!>[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_Oidom,type,
idom:
!>[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_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[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_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom,type,
semidom:
!>[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_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[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_Lattices_Olattice,type,
lattice:
!>[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_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_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[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_Orderings_Owellorder,type,
wellorder:
!>[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_Fields_Ofield__abs__sgn,type,
field_abs_sgn:
!>[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_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[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_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[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_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Odistrib__lattice,type,
distrib_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_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
tff(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[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_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[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_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[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_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
tff(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[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_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[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_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[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_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[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_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[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_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[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_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[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] : fun(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] : ( 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] : fun(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] : ( 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] : fun(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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bh____,type,
aTP_Lamp_bh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bi____,type,
aTP_Lamp_bi:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : ( 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] : ( 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] : fun(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] : ( 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] : ( 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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : ( 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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dr____,type,
aTP_Lamp_dr:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__eg____,type,
aTP_Lamp_eg:
!>[A: $tType,B: $tType] : fun(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] : ( 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] : ( 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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fk____,type,
aTP_Lamp_fk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fl____,type,
aTP_Lamp_fl:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : ( 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] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : ( 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] : ( 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] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ib____,type,
aTP_Lamp_ib:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__iv____,type,
aTP_Lamp_iv:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jf____,type,
aTP_Lamp_jf:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jp____,type,
aTP_Lamp_jp:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jz____,type,
aTP_Lamp_jz:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kj____,type,
aTP_Lamp_kj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kk____,type,
aTP_Lamp_kk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kl____,type,
aTP_Lamp_kl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__km____,type,
aTP_Lamp_km:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : ( 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] : ( 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] : ( 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] : ( 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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ln____,type,
aTP_Lamp_ln:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ls____,type,
aTP_Lamp_ls:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lt____,type,
aTP_Lamp_lt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lu____,type,
aTP_Lamp_lu:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mh____,type,
aTP_Lamp_mh:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ml____,type,
aTP_Lamp_ml:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ms____,type,
aTP_Lamp_ms:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mt____,type,
aTP_Lamp_mt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mu____,type,
aTP_Lamp_mu:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mx____,type,
aTP_Lamp_mx:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nm____,type,
aTP_Lamp_nm:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nt____,type,
aTP_Lamp_nt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nu____,type,
aTP_Lamp_nu:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nv____,type,
aTP_Lamp_nv:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oj____,type,
aTP_Lamp_oj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ok____,type,
aTP_Lamp_ok:
!>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : fun(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] : ( 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] : ( 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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pu____,type,
aTP_Lamp_pu:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qj____,type,
aTP_Lamp_qj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qk____,type,
aTP_Lamp_qk:
!>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qs____,type,
aTP_Lamp_qs:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qt____,type,
aTP_Lamp_qt:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rn____,type,
aTP_Lamp_rn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ro____,type,
aTP_Lamp_ro:
!>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rr____,type,
aTP_Lamp_rr:
!>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sb____,type,
aTP_Lamp_sb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sc____,type,
aTP_Lamp_sc:
!>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sf____,type,
aTP_Lamp_sf:
!>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(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] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__th____,type,
aTP_Lamp_th:
!>[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_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_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] : ( ( nat * 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) * nat ) > 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) * 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_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).
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_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__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_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_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_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_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_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_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_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_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 > fun(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 ) > bool ) ).
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_Oxor__num,type,
bit_un2480387367778600638or_num: ( num * num ) > option(num) ).
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__axioms,type,
boolea6902313364301356556axioms:
!>[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_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * fun(A,fun(A,A)) ) > $o ) ).
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] : fun(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_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( fun(A,A) > set(A) ) ).
tff(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( fun(A,A) > fun(A,bool) ) ).
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_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : fun(set(A),bool) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : fun(set(A),bool) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
condit622319405099724424ng_bdd:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
condit16957441358409770ng_bdd:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),bool) ) ).
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_Divides_Oadjust__div,type,
adjust_div: product_prod(int,int) > int ).
tff(sy_c_Divides_Oadjust__mod,type,
adjust_mod: ( 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_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] : fun(set(B),nat) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : fun(set(A),bool) ).
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__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).
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(B,C) > fun(fun(A,B),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__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( fun(A,nat) > $o ) ).
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__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( itself(A) > nat ) ).
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__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__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_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_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > array(A) ) ).
tff(sy_c_Heap_Oarray_Osize__array,type,
size_array:
!>[A: $tType] : ( ( fun(A,nat) * array(A) ) > nat ) ).
tff(sy_c_Heap_Oempty,type,
empty: heap_ext(product_unit) ).
tff(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( ( heap_ext(Z) * typerep * nat ) > list(nat) ) ).
tff(sy_c_Heap_Oheap_Oheap__ext,type,
heap_ext2:
!>[Z: $tType] : ( ( fun(typerep,fun(nat,list(nat))) * fun(typerep,fun(nat,nat)) * nat * 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__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( 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(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B) * 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(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C) * 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] : ( 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_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( fun(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: fun(num,int) ).
tff(sy_c_Int_ORep__Integ,type,
rep_Integ: fun(int,product_prod(nat,nat)) ).
tff(sy_c_Int_Ocr__int,type,
cr_int: fun(product_prod(nat,nat),fun(int,bool)) ).
tff(sy_c_Int_Odup,type,
dup: int > int ).
tff(sy_c_Int_Oint_OAbs__int,type,
abs_int: fun(set(product_prod(nat,nat)),int) ).
tff(sy_c_Int_Oint_ORep__int,type,
rep_int: fun(int,set(product_prod(nat,nat))) ).
tff(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > set(product_prod(int,int)) ).
tff(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > set(product_prod(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] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices_Osemilattice__axioms,type,
semilattice_axioms:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices_Osemilattice__order__axioms,type,
semila6385135966242565138axioms:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).
tff(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
lattic1883929316492267755max_on:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).
tff(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > fun(B,bool) ) ).
tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > fun(B,bool) ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
lattic5652469242046573047tr_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(set(A),A) ) ).
tff(sy_c_Lattices__Big_Osemilattice__order__neutr__set,type,
lattic3600114342068043075tr_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
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)) > fun(set(A),A) ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : fun(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] : ( ( list(A) * 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_Ocoset,type,
coset:
!>[A: $tType] : ( list(A) > set(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_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) * B ) > B ) ).
tff(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * set(B) * fun(B,A) ) > $o ) ).
tff(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).
tff(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > fun(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_Olexord,type,
lexord:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * 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] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
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) * list(A) ) > list(Aa) ) ).
tff(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( 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__ex,type,
list_ex:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
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_Omeasures,type,
measures:
!>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).
tff(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).
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] : fun(list(A),fun(list(A),bool)) ).
tff(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).
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] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_OremoveAll,type,
removeAll:
!>[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_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) ) > $o ) ).
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_Otranspose,type,
transpose:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
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__aux,type,
upto_aux: ( int * int * list(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_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list(product_prod(A,B)) * 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_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).
tff(sy_c_Nat_OSuc,type,
suc: fun(nat,nat) ).
tff(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : fun(nat,fun(A,A)) ).
tff(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).
tff(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).
tff(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).
tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,bool) ) ).
tff(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : set(A) ).
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_Oint__decode,type,
nat_int_decode: fun(nat,int) ).
tff(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: fun(int,nat) ).
tff(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: fun(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: fun(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: fun(nat,product_prod(nat,nat)) ).
tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > fun(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: fun(set(nat),nat) ).
tff(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: fun(nat,sum_sum(nat,nat)) ).
tff(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: fun(sum_sum(nat,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: fun(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: 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] : fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))) ).
tff(sy_c_Num_Onum_Orec__num,type,
rec_num:
!>[A: $tType] : fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(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_Opow,type,
pow: ( num * num ) > num ).
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,type,
old_Push:
!>[B: $tType] : ( ( sum_sum(B,nat) * fun(nat,sum_sum(B,nat)) * nat ) > sum_sum(B,nat) ) ).
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_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_Old__Datatype_Onode_ORep__Node,type,
old_Rep_Node:
!>[A: $tType,B: $tType] : ( old_node(A,B) > product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) ) ).
tff(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( ( nat * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).
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] : ( ( B * fun(A,B) * option(A) ) > B ) ).
tff(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).
tff(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( option(A) > A ) ).
tff(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).
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_OLeast,type,
least:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(fun(A,bool),A) ) ).
tff(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).
tff(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).
tff(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( fun(A,bool) > 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] : fun(A,fun(A,A)) ).
tff(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : fun(fun(A,B),bool) ).
tff(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A ) > $o ) ).
tff(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).
tff(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
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_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_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( 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_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( 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_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
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_Quotient_Oquot__type_Oabs,type,
quot_abs:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(set(A),B) ) > fun(A,B) ) ).
tff(sy_c_Quotient_Oquot__type_Orep,type,
quot_rep:
!>[B: $tType,A: $tType] : ( fun(B,set(A)) > fun(B,A) ) ).
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_Opick,type,
pick:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(code_natural,A) ) ).
tff(sy_c_Random_Orange,type,
range: ( code_natural * product_prod(code_natural,code_natural) ) > product_prod(code_natural,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_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_ORats,type,
field_char_0_Rats:
!>[A: $tType] : set(A) ).
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: fun(rat,bool) ).
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_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
tuple_1188178415141063261rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,product_prod(B,C)) * fun(product_prod(B,C),A) ) > tuple_isomorphism(A,B,C) ) ).
tff(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
tuple_9181185373184732606rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,nat) * fun(B,nat) * fun(C,nat) * tuple_isomorphism(A,B,C) ) > nat ) ).
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_OPowp,type,
powp:
!>[A: $tType] : ( ( fun(A,bool) * set(A) ) > $o ) ).
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_Oirreflp,type,
irreflp:
!>[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_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] : fun(A,fun(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] : fun(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(A,bool) > set(A) ) ).
tff(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set(A) * set(A) ) > $o ) ).
tff(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > set(B) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * 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) * 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] : ( 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] : ( A > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( 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] : ( A > set(A) ) ).
tff(sy_c_String_OLiteral,type,
literal2: ( bool * bool * bool * bool * bool * bool * bool * literal ) > literal ).
tff(sy_c_String_Ochar_OChar,type,
char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).
tff(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : fun(char,A) ).
tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : fun(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_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) * sum_sum(A,B) ) > C ) ).
tff(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $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_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_OTyperep,type,
typerep3: ( literal * list(typerep) ) > typerep ).
tff(sy_c_Typerep_Otyperep_Osize__typerep,type,
size_typerep: fun(typerep,nat) ).
tff(sy_c_Typerep_Otyperep__class_Otyperep,type,
typerep_typerep:
!>[A: $tType] : ( itself(A) > typerep ) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).
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_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_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( set(set(A)) > $o ) ).
tff(sy_c_Zorn_Ochains,type,
chains:
!>[A: $tType] : ( set(set(A)) > set(set(set(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_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > $o ) ).
tff(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > set(A) ) ).
tff(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
pred_s596693808085603175closed:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) ) > set(set(A)) ) ).
tff(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
pred_s7749564232668923593losedp:
!>[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: 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_a,type,
a2: array(a) ).
tff(sy_v_h,type,
h: heap_ext(product_unit) ).
tff(sy_v_h_H,type,
h2: heap_ext(product_unit) ).
tff(sy_v_n,type,
n: nat ).
tff(sy_v_r,type,
r: nat ).
tff(sy_v_thesis,type,
thesis: $o ).
% Relevant facts (7701)
tff(fact_0_assms,axiom,
heap_Time_effect(nat,array_len(a,a2),h,h2,r,n) ).
% assms
tff(fact_1_that,axiom,
( ( r = aa(array(a),nat,array_length(a,h2),a2) )
=> ( ( h2 = h )
=> ( ( n = one_one(nat) )
=> thesis ) ) ) ).
% that
tff(fact_2_effect__lengthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),H2: heap_ext(product_unit),R: nat,A3: array(A),N: nat] :
( ( H = H2 )
=> ( ( R = aa(array(A),nat,array_length(A,H2),A3) )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(nat,array_len(A,A3),H2,H,R,N) ) ) ) ) ).
% effect_lengthI
tff(fact_3_one__natural_Orsp,axiom,
one_one(nat) = one_one(nat) ).
% one_natural.rsp
tff(fact_4_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
% one_reorient
tff(fact_5_length__update,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [B2: array(B),I: nat,V: B,H2: heap_ext(product_unit)] : array_length(A,array_update(B,B2,I,V,H2)) = array_length(A,H2) ) ).
% length_update
tff(fact_6_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_7_fact__1,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).
% fact_1
tff(fact_8_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_9_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_10_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_11_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_12_round__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).
% round_1
tff(fact_13_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M: nat,N: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
<=> ( M = N ) ) ) ).
% of_nat_eq_iff
tff(fact_14_of__nat__fact,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),semiring_char_0_fact(nat,N)) = semiring_char_0_fact(A,N) ) ).
% of_nat_fact
tff(fact_15_euclidean__size__of__nat,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: nat] : aa(A,nat,euclid6346220572633701492n_size(A),aa(nat,A,semiring_1_of_nat(A),N)) = N ) ).
% euclidean_size_of_nat
tff(fact_16_round__of__nat,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [N: nat] : archimedean_round(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).
% round_of_nat
tff(fact_17_one__integer_Orsp,axiom,
one_one(int) = one_one(int) ).
% one_integer.rsp
tff(fact_18_update__swap__neqIndex,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,I2: nat,A3: array(A),V: A,V2: A,H2: heap_ext(product_unit)] :
( ( I != I2 )
=> ( array_update(A,A3,I,V,array_update(A,A3,I2,V2,H2)) = array_update(A,A3,I2,V2,array_update(A,A3,I,V,H2)) ) ) ) ).
% update_swap_neqIndex
tff(fact_19_int__ops_I2_J,axiom,
aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).
% int_ops(2)
tff(fact_20_effect__deterministic_I3_J,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),A3: A,N: nat,H3: heap_ext(product_unit),B2: A,N2: nat] :
( heap_Time_effect(A,F,H2,H,A3,N)
=> ( heap_Time_effect(A,F,H2,H3,B2,N2)
=> ( N = N2 ) ) ) ).
% effect_deterministic(3)
tff(fact_21_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),A3: A,N: nat,H3: heap_ext(product_unit),B2: A,N2: nat] :
( heap_Time_effect(A,F,H2,H,A3,N)
=> ( heap_Time_effect(A,F,H2,H3,B2,N2)
=> ( H = H3 ) ) ) ).
% effect_deterministic(2)
tff(fact_22_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),A3: A,N: nat,H3: heap_ext(product_unit),B2: A,N2: nat] :
( heap_Time_effect(A,F,H2,H,A3,N)
=> ( heap_Time_effect(A,F,H2,H3,B2,N2)
=> ( A3 = B2 ) ) ) ).
% effect_deterministic(1)
tff(fact_23_effect__ifE,axiom,
! [A: $tType,C2: bool,T2: heap_Time_Heap(A),E: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,if(heap_Time_Heap(A),C2,T2,E),H2,H,R,N)
=> ( ( pp(C2)
=> ~ heap_Time_effect(A,T2,H2,H,R,N) )
=> ~ ( ~ pp(C2)
=> ~ heap_Time_effect(A,E,H2,H,R,N) ) ) ) ).
% effect_ifE
tff(fact_24_effect__ifI,axiom,
! [A: $tType,C2: bool,T2: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat,E: heap_Time_Heap(A)] :
( ( pp(C2)
=> heap_Time_effect(A,T2,H2,H,R,N) )
=> ( ( ~ pp(C2)
=> heap_Time_effect(A,E,H2,H,R,N) )
=> heap_Time_effect(A,if(heap_Time_Heap(A),C2,T2,E),H2,H,R,N) ) ) ).
% effect_ifI
tff(fact_25_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_26_present__update,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [B2: array(B),I: nat,V: B,H2: heap_ext(product_unit)] : array_present(A,array_update(B,B2,I,V,H2)) = array_present(A,H2) ) ).
% present_update
tff(fact_27_fact__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).
% fact_0
tff(fact_28_effect__tapI,axiom,
! [A: $tType,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,F: fun(heap_ext(product_unit),A)] :
( ( H = H2 )
=> ( ( R = aa(heap_ext(product_unit),A,F,H2) )
=> heap_Time_effect(A,heap_Time_tap(A,F),H2,H,R,one_one(nat)) ) ) ).
% effect_tapI
tff(fact_29_effect__tapE,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_tap(A,F),H2,H,R,N)
=> ~ ( ( H = H2 )
=> ( ( R = aa(heap_ext(product_unit),A,F,H2) )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_tapE
tff(fact_30_update__swap__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [A3: array(A),A4: array(B),I: nat,V: A,I2: nat,V2: B,H2: heap_ext(product_unit)] :
( array_noteq(A,B,A3,A4)
=> ( array_update(A,A3,I,V,array_update(B,A4,I2,V2,H2)) = array_update(B,A4,I2,V2,array_update(A,A3,I,V,H2)) ) ) ) ).
% update_swap_neq
tff(fact_31_unequal,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),A4: array(A)] :
( ( A3 != A4 )
<=> array_noteq(A,A,A3,A4) ) ) ).
% unequal
tff(fact_32_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_33_round__0,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).
% round_0
tff(fact_34_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
<=> ( M = zero_zero(nat) ) ) ) ).
% of_nat_eq_0_iff
tff(fact_35_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] :
( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
<=> ( zero_zero(nat) = N ) ) ) ).
% of_nat_0_eq_iff
tff(fact_36_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).
% of_nat_0
tff(fact_37_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A] :
( ( aa(A,nat,euclid6346220572633701492n_size(A),B2) = zero_zero(nat) )
<=> ( B2 = zero_zero(A) ) ) ) ).
% euclidean_size_eq_0_iff
tff(fact_38_size__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ( aa(A,nat,euclid6346220572633701492n_size(A),zero_zero(A)) = zero_zero(nat) ) ) ).
% size_0
tff(fact_39_nat__int__comparison_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( A3 = B2 )
<=> ( aa(nat,int,semiring_1_of_nat(int),A3) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).
% nat_int_comparison(1)
tff(fact_40_zero__natural_Orsp,axiom,
zero_zero(nat) = zero_zero(nat) ).
% zero_natural.rsp
tff(fact_41_division__segment__not__0,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A] : aa(A,A,euclid7384307370059645450egment(A),A3) != zero_zero(A) ) ).
% division_segment_not_0
tff(fact_42_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: fun(A,bool)] :
( pp(member(A,A3,collect(A,P)))
<=> pp(aa(A,bool,P,A3)) ) ).
% mem_Collect_eq
tff(fact_43_Collect__mem__eq,axiom,
! [A: $tType,A5: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,bool),A5)) = A5 ).
% Collect_mem_eq
tff(fact_44_Collect__cong,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) )
=> ( collect(A,P) = collect(A,Q) ) ) ).
% Collect_cong
tff(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
% ext
tff(fact_46_int__if,axiom,
! [P: bool,A3: nat,B2: nat] :
( ( pp(P)
=> ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A3,B2)) = aa(nat,int,semiring_1_of_nat(int),A3) ) )
& ( ~ pp(P)
=> ( aa(nat,int,semiring_1_of_nat(int),if(nat,P,A3,B2)) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ) ).
% int_if
tff(fact_47_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
% zero_reorient
tff(fact_48_noteq__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) ) ) ).
% noteq_sym
tff(fact_49_noteq__irrefl,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A)] : ~ array_noteq(A,A,R,R) ) ).
% noteq_irrefl
tff(fact_50_fact__nonzero,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semiri3467727345109120633visors(A) )
=> ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).
% fact_nonzero
tff(fact_51_division__segment__nat__def,axiom,
! [N: nat] : aa(nat,nat,euclid7384307370059645450egment(nat),N) = one_one(nat) ).
% division_segment_nat_def
tff(fact_52_division__segment__eq__iff,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,euclid7384307370059645450egment(A),A3) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),B2) )
=> ( A3 = B2 ) ) ) ) ).
% division_segment_eq_iff
tff(fact_53_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
% zero_neq_one
tff(fact_54_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_55_effect__ureturnI,axiom,
! [A: $tType,H2: heap_ext(product_unit),H: heap_ext(product_unit),X: A] :
( ( H2 = H )
=> heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H2,H,X,zero_zero(nat)) ) ).
% effect_ureturnI
tff(fact_56_effect__ureturnE,axiom,
! [A: $tType,X: A,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H2,H,R,N)
=> ~ ( ( R = X )
=> ( ( H = H2 )
=> ( N != zero_zero(nat) ) ) ) ) ).
% effect_ureturnE
tff(fact_57_effect__assertE,axiom,
! [A: $tType,P: fun(A,bool),X: A,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_assert(A,P,X),H2,H,R,N)
=> ~ ( pp(aa(A,bool,P,X))
=> ( ( R = X )
=> ( ( H = H2 )
=> ( N != one_one(nat) ) ) ) ) ) ).
% effect_assertE
tff(fact_58_effect__assertI,axiom,
! [A: $tType,P: fun(A,bool),X: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
( pp(aa(A,bool,P,X))
=> ( ( H = H2 )
=> ( ( R = X )
=> ( ( N = one_one(nat) )
=> heap_Time_effect(A,heap_Time_assert(A,P,X),H2,H,R,N) ) ) ) ) ).
% effect_assertI
tff(fact_59_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_60_division__segment__euclidean__size,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,euclid7384307370059645450egment(A),A3)),aa(nat,A,semiring_1_of_nat(A),aa(A,nat,euclid6346220572633701492n_size(A),A3))) = A3 ) ).
% division_segment_euclidean_size
tff(fact_61_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
<=> ( B2 != zero_zero(A) ) ) ) ).
% euclidean_size_greater_0_iff
tff(fact_62_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_63_of__nat__le__0__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)))
<=> ( M = zero_zero(nat) ) ) ) ).
% of_nat_le_0_iff
tff(fact_64_effect__returnE,axiom,
! [A: $tType,X: A,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H2,H,R,N)
=> ~ ( ( R = X )
=> ( ( H = H2 )
=> ( N != one_one(nat) ) ) ) ) ).
% effect_returnE
tff(fact_65_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( group_add(B)
=> ! [B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,uminus_uminus(B),B2)) = B2 ) ).
% verit_minus_simplify(4)
tff(fact_66_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_67_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_68_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).
% Suc_le_mono
tff(fact_69_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( aa(nat,nat,suc,Nat) = aa(nat,nat,suc,Nat2) )
<=> ( Nat = Nat2 ) ) ).
% old.nat.inject
tff(fact_70_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y2) )
<=> ( X22 = Y2 ) ) ).
% nat.inject
tff(fact_71_mult__cancel2,axiom,
! [M: nat,K: nat,N: 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) )
<=> ( ( M = N )
| ( K = zero_zero(nat) ) ) ) ).
% mult_cancel2
tff(fact_72_mult__cancel1,axiom,
! [K: nat,M: nat,N: 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) )
<=> ( ( M = N )
| ( K = zero_zero(nat) ) ) ) ).
% mult_cancel1
tff(fact_73_mult__0__right,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).
% mult_0_right
tff(fact_74_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
| ( N = zero_zero(nat) ) ) ) ).
% mult_is_0
tff(fact_75_le0,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).
% le0
tff(fact_76_bot__nat__0_Oextremum,axiom,
! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A3)) ).
% bot_nat_0.extremum
tff(fact_77_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_78_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_79_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_80_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_81_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% mult_zero_left
tff(fact_82_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% mult_zero_right
tff(fact_83_mult__eq__0__iff,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
<=> ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
% mult_eq_0_iff
tff(fact_84_mult__cancel__left,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A3: A,B2: 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) )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
tff(fact_85_mult__cancel__right,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [A3: A,C2: A,B2: 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) )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
tff(fact_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_mult__minus__left,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% mult_minus_left
tff(fact_96_minus__mult__minus,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) ) ).
% minus_mult_minus
tff(fact_97_mult__minus__right,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% mult_minus_right
tff(fact_98_of__nat__le__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
% of_nat_le_iff
tff(fact_99_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).
% one_le_mult_iff
tff(fact_100_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
<=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% one_eq_mult_iff
tff(fact_101_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% mult_eq_1_iff
tff(fact_102_of__nat__mult,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),times_times(nat),M),N)) = 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)) ) ).
% of_nat_mult
tff(fact_103_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% Suc_less_eq
tff(fact_104_Suc__mono,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).
% Suc_mono
tff(fact_105_lessI,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).
% lessI
tff(fact_106_mult__le__cancel2,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),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
<=> ( 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),M),N)) ) ) ).
% mult_le_cancel2
tff(fact_107_nat__0__less__mult__iff,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),times_times(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)) ) ) ).
% nat_0_less_mult_iff
tff(fact_108_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(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)))
<=> ( 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(nat),M),N)) ) ) ).
% mult_less_cancel2
tff(fact_109_less__nat__zero__code,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
% less_nat_zero_code
tff(fact_110_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% neq0_conv
tff(fact_111_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A3)) ) ).
% bot_nat_0.not_eq_extremum
tff(fact_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_of__nat__less__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
% of_nat_less_iff
tff(fact_127_zero__less__Suc,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).
% zero_less_Suc
tff(fact_128_less__Suc0,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
<=> ( N = zero_zero(nat) ) ) ).
% less_Suc0
tff(fact_129_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_130_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_131_fact__Suc,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N))),semiring_char_0_fact(A,N)) ) ).
% fact_Suc
tff(fact_132_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
% of_nat_0_less_iff
tff(fact_133_zero__integer_Orsp,axiom,
zero_zero(int) = zero_zero(int) ).
% zero_integer.rsp
tff(fact_134_one__less__mult,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).
% one_less_mult
tff(fact_135_n__less__m__mult__n,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),aa(nat,nat,suc,zero_zero(nat))),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).
% n_less_m_mult_n
tff(fact_136_n__less__n__mult__m,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),aa(nat,nat,suc,zero_zero(nat))),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).
% n_less_n_mult_m
tff(fact_137_ex__least__nat__less,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,N))
=> ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
=> ? [K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),N))
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),K2))
=> ~ pp(aa(nat,bool,P,I3)) )
& pp(aa(nat,bool,P,aa(nat,nat,suc,K2))) ) ) ) ).
% ex_least_nat_less
tff(fact_138_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).
% mult_le_cancel_iff1
tff(fact_139_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).
% mult_le_cancel_iff2
tff(fact_140_division__segment__int__def,axiom,
! [K: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( aa(int,int,euclid7384307370059645450egment(int),K) = one_one(int) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( aa(int,int,euclid7384307370059645450egment(int),K) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% division_segment_int_def
tff(fact_141_verit__la__disequality,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: 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),B2),A3)) ) ) ).
% verit_la_disequality
tff(fact_142_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord2810124833399127020strict(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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_143_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),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(A),A3),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(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_right_disj
tff(fact_144_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_strict_right_mono
tff(fact_145_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),C2),zero_zero(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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_strict_right_mono_neg
tff(fact_146_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),C2),A3)),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(A),A3),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(A),B2),A3)) ) ) ) ) ).
% mult_less_cancel_left_disj
tff(fact_147_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_strict_left_mono
tff(fact_148_mult__less__le__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( 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)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).
% mult_less_le_imp_less
tff(fact_149_mult__le__less__imp__less,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( 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)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).
% mult_le_less_imp_less
tff(fact_150_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),C2),zero_zero(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)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_strict_left_mono_neg
tff(fact_151_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( 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(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(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% mult_less_cancel_left_pos
tff(fact_152_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: 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(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(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ).
% mult_less_cancel_left_neg
tff(fact_153_mult__right__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),times_times(A),A3),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),A3),B2)) ) ) ) ).
% mult_right_le_imp_le
tff(fact_154_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [B2: 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),times_times(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),zero_zero(A)),B2)) ) ) ) ).
% zero_less_mult_pos2
tff(fact_155_mult__left__le__imp__le,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),times_times(A),C2),A3)),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),A3),B2)) ) ) ) ).
% mult_left_le_imp_le
tff(fact_156_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),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(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% mult_le_cancel_left_pos
tff(fact_157_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [C2: A,A3: A,B2: 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),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(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% mult_le_cancel_left_neg
tff(fact_158_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),times_times(A),A3),B2)))
=> ( 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)) ) ) ) ).
% zero_less_mult_pos
tff(fact_159_mult__less__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),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),A3),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),A3)) ) ) ) ) ).
% mult_less_cancel_right
tff(fact_160_mult__strict__mono_H,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( 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)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).
% mult_strict_mono'
tff(fact_161_mult__right__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(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),times_times(A),A3),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),A3),B2)) ) ) ) ).
% mult_right_less_imp_less
tff(fact_162_mult__less__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),C2),A3)),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),A3),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),A3)) ) ) ) ) ).
% mult_less_cancel_left
tff(fact_163_mult__strict__mono,axiom,
! [A: $tType] :
( linord8928482502909563296strict(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( 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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).
% mult_strict_mono
tff(fact_164_mult__left__less__imp__less,axiom,
! [A: $tType] :
( linordered_semiring(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),times_times(A),C2),A3)),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),A3),B2)) ) ) ) ).
% mult_left_less_imp_less
tff(fact_165_mult__le__cancel__right,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),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),A3),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),A3)) ) ) ) ) ).
% mult_le_cancel_right
tff(fact_166_mult__le__cancel__left,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),C2),A3)),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),A3),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),A3)) ) ) ) ) ).
% mult_le_cancel_left
tff(fact_167_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ordere2520102378445227354miring(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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_168_zero__less__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),B2)))
<=> ( ( 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),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).
% zero_less_mult_iff
tff(fact_169_mult__pos__neg2,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).
% mult_pos_neg2
tff(fact_170_mult__pos__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),times_times(A),A3),B2))) ) ) ) ).
% mult_pos_pos
tff(fact_171_mult__pos__neg,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),B2),zero_zero(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),B2)),zero_zero(A))) ) ) ) ).
% mult_pos_neg
tff(fact_172_mult__neg__pos,axiom,
! [A: $tType] :
( linord8928482502909563296strict(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),zero_zero(A)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).
% mult_neg_pos
tff(fact_173_zero__le__mult__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),B2)))
<=> ( ( 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),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).
% zero_le_mult_iff
tff(fact_174_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ordered_semiring_0(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),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),times_times(A),B2),A3)),zero_zero(A))) ) ) ) ).
% mult_nonneg_nonpos2
tff(fact_175_mult__less__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),B2)),zero_zero(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),zero_zero(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),zero_zero(A)),B2)) ) ) ) ) ).
% mult_less_0_iff
tff(fact_176_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(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),zero_zero(A)),B2))
=> 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)),zero_zero(A))) ) ) ) ).
% mult_nonpos_nonneg
tff(fact_177_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ordered_semiring_0(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),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),times_times(A),A3),B2)),zero_zero(A))) ) ) ) ).
% mult_nonneg_nonpos
tff(fact_178_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ordered_semiring_0(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),times_times(A),A3),B2))) ) ) ) ).
% mult_nonneg_nonneg
tff(fact_179_split__mult__neg__le,axiom,
! [A: $tType] :
( ordered_semiring_0(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),B2),zero_zero(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),zero_zero(A)),B2)) ) )
=> 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)),zero_zero(A))) ) ) ).
% split_mult_neg_le
tff(fact_180_mult__le__0__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),times_times(A),A3),B2)),zero_zero(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),zero_zero(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),zero_zero(A)),B2)) ) ) ) ) ).
% mult_le_0_iff
tff(fact_181_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(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),A3),A3)),zero_zero(A))) ) ).
% not_square_less_zero
tff(fact_182_mult__neg__neg,axiom,
! [A: $tType] :
( linord4710134922213307826strict(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),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).
% mult_neg_neg
tff(fact_183_mult__right__mono,axiom,
! [A: $tType] :
( ordered_semiring(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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_right_mono
tff(fact_184_mult__right__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(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),C2),zero_zero(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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ) ).
% mult_right_mono_neg
tff(fact_185_mult__left__mono,axiom,
! [A: $tType] :
( ordered_semiring(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),zero_zero(A)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_left_mono
tff(fact_186_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ordered_ring(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),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ) ).
% mult_nonpos_nonpos
tff(fact_187_mult__left__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(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),C2),zero_zero(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)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ).
% mult_left_mono_neg
tff(fact_188_split__mult__pos__le,axiom,
! [A: $tType] :
( ordered_ring(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),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),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ) ).
% split_mult_pos_le
tff(fact_189_zero__le__square,axiom,
! [A: $tType] :
( linordered_ring(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),times_times(A),A3),A3))) ) ).
% zero_le_square
tff(fact_190_mult__mono_H,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( 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)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).
% mult_mono'
tff(fact_191_mult__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( 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)),C2))
=> 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)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).
% mult_mono
tff(fact_192_verit__negate__coefficient_I2_J,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,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3))) ) ) ).
% verit_negate_coefficient(2)
tff(fact_193_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).
% verit_negate_coefficient(3)
tff(fact_194_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).
% verit_comp_simplify1(1)
tff(fact_195_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).
% verit_comp_simplify1(2)
tff(fact_196_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( linorder(B)
=> ! [B3: B,A4: B] :
( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B3),A4))
<=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A4),B3)) ) ) ).
% verit_comp_simplify1(3)
tff(fact_197_square__eq__iff,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
<=> ( ( A3 = B2 )
| ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).
% square_eq_iff
tff(fact_198_minus__mult__commute,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% minus_mult_commute
tff(fact_199_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_neqE_linordered_idom
tff(fact_200_infinite__descent__measure,axiom,
! [A: $tType,P: fun(A,bool),V3: fun(A,nat),X: A] :
( ! [X2: A] :
( ~ pp(aa(A,bool,P,X2))
=> ? [Y3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V3,Y3)),aa(A,nat,V3,X2)))
& ~ pp(aa(A,bool,P,Y3)) ) )
=> pp(aa(A,bool,P,X)) ) ).
% infinite_descent_measure
tff(fact_201_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: fun(nat,fun(nat,bool))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( ! [X2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X2),X2))
=> ( ! [X2: nat,Y4: nat,Z3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X2),Y4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,Y4),Z3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,X2),Z3)) ) )
=> ( ! [N3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,N3),aa(nat,nat,suc,N3)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),R2,M),N)) ) ) ) ) ).
% transitive_stepwise_le
tff(fact_202_less__mono__imp__le__mono,axiom,
! [F: fun(nat,nat),I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F,I4)),aa(nat,nat,F,J2))) )
=> ( 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,F,I)),aa(nat,nat,F,J))) ) ) ).
% less_mono_imp_le_mono
tff(fact_203_Suc__mult__less__cancel1,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),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% Suc_mult_less_cancel1
tff(fact_204_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
<=> ( N = M ) ) ) ).
% not_less_less_Suc_eq
tff(fact_205_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(nat,bool,P,M))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),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_at_least
tff(fact_206_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( wellorder(B)
=> ! [F: fun(A,B),P: fun(A,bool),A3: A] :
( ! [X2: A] :
( ! [Y3: A] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Y3)),aa(A,B,F,X2)))
=> pp(aa(A,bool,P,Y3)) )
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% measure_induct_rule
tff(fact_207_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( ( M != N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
% le_neq_implies_less
tff(fact_208_Suc__mult__le__cancel1,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),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% Suc_mult_le_cancel1
tff(fact_209_strict__inc__induct,axiom,
! [I: nat,J: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( ! [I4: nat] :
( ( J = aa(nat,nat,suc,I4) )
=> pp(aa(nat,bool,P,I4)) )
=> ( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J))
=> ( pp(aa(nat,bool,P,aa(nat,nat,suc,I4)))
=> pp(aa(nat,bool,P,I4)) ) )
=> pp(aa(nat,bool,P,I)) ) ) ) ).
% strict_inc_induct
tff(fact_210_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
% linorder_neqE_nat
tff(fact_211_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| ( M = N ) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% less_or_eq_imp_le
tff(fact_212_le__eq__less__or__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),M),N))
| ( M = N ) ) ) ).
% le_eq_less_or_eq
tff(fact_213_infinite__descent,axiom,
! [P: fun(nat,bool),N: nat] :
( ! [N3: nat] :
( ~ pp(aa(nat,bool,P,N3))
=> ? [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N3))
& ~ pp(aa(nat,bool,P,M2)) ) )
=> pp(aa(nat,bool,P,N)) ) ).
% infinite_descent
tff(fact_214_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
<=> ( M = N ) ) ).
% Suc_mult_cancel1
tff(fact_215_nat__less__induct,axiom,
! [P: fun(nat,bool),N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N3))
=> pp(aa(nat,bool,P,M2)) )
=> pp(aa(nat,bool,P,N3)) )
=> pp(aa(nat,bool,P,N)) ) ).
% nat_less_induct
tff(fact_216_less__irrefl__nat,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
% less_irrefl_nat
tff(fact_217_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% less_imp_le_nat
tff(fact_218_less__Suc__induct,axiom,
! [I: nat,J: nat,P: fun(nat,fun(nat,bool))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( ! [I4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I4),aa(nat,nat,suc,I4)))
=> ( ! [I4: nat,J2: nat,K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I4),J2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I4),K2)) ) ) ) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I),J)) ) ) ) ).
% less_Suc_induct
tff(fact_219_le__imp__less__Suc,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),M),aa(nat,nat,suc,N))) ) ).
% le_imp_less_Suc
tff(fact_220_full__nat__induct,axiom,
! [P: fun(nat,bool),N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N3))
=> pp(aa(nat,bool,P,M2)) )
=> pp(aa(nat,bool,P,N3)) )
=> pp(aa(nat,bool,P,N)) ) ).
% full_nat_induct
tff(fact_221_not__less__eq__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_eq(nat),aa(nat,nat,suc,N)),M)) ) ).
% not_less_eq_eq
tff(fact_222_measure__induct,axiom,
! [B: $tType,A: $tType] :
( wellorder(B)
=> ! [F: fun(A,B),P: fun(A,bool),A3: A] :
( ! [X2: A] :
( ! [Y3: A] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Y3)),aa(A,B,F,X2)))
=> pp(aa(A,bool,P,Y3)) )
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% measure_induct
tff(fact_223_less__trans__Suc,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),J),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K)) ) ) ).
% less_trans_Suc
tff(fact_224_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
=> ( S != T2 ) ) ).
% less_not_refl3
tff(fact_225_less__not__refl2,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( M != N ) ) ).
% less_not_refl2
tff(fact_226_less__eq__Suc__le,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).
% less_eq_Suc_le
tff(fact_227_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% less_Suc_eq_le
tff(fact_228_le__less__Suc__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),N),aa(nat,nat,suc,M)))
<=> ( N = M ) ) ) ).
% le_less_Suc_eq
tff(fact_229_Suc__n__not__le__n,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).
% Suc_n_not_le_n
tff(fact_230_less__not__refl,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
% less_not_refl
tff(fact_231_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% Suc_less_SucD
tff(fact_232_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
=> ( M = N ) ) ) ).
% less_antisym
tff(fact_233_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
<=> ? [M3: nat] :
( ( M = aa(nat,nat,suc,M3) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M3)) ) ) ).
% Suc_less_eq2
tff(fact_234_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% Suc_le_lessD
tff(fact_235_All__less__Suc,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(nat,nat,suc,N)))
=> pp(aa(nat,bool,P,I5)) )
<=> ( pp(aa(nat,bool,P,N))
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
=> pp(aa(nat,bool,P,I5)) ) ) ) ).
% All_less_Suc
tff(fact_236_not__less__eq,axiom,
! [M: nat,N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M))) ) ).
% not_less_eq
tff(fact_237_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).
% nat_neq_iff
tff(fact_238_nat__less__le,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& ( M != N ) ) ) ).
% nat_less_le
tff(fact_239_n__not__Suc__n,axiom,
! [N: nat] : N != aa(nat,nat,suc,N) ).
% n_not_Suc_n
tff(fact_240_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| ( M = N ) ) ) ).
% less_Suc_eq
tff(fact_241_Ex__less__Suc,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(nat,nat,suc,N)))
& pp(aa(nat,bool,P,I5)) )
<=> ( pp(aa(nat,bool,P,N))
| ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
& pp(aa(nat,bool,P,I5)) ) ) ) ).
% Ex_less_Suc
tff(fact_242_inc__induct,axiom,
! [I: nat,J: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,P,J))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),J))
=> ( pp(aa(nat,bool,P,aa(nat,nat,suc,N3)))
=> pp(aa(nat,bool,P,N3)) ) ) )
=> pp(aa(nat,bool,P,I)) ) ) ) ).
% inc_induct
tff(fact_243_dec__induct,axiom,
! [I: nat,J: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( pp(aa(nat,bool,P,I))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),J))
=> ( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) ) )
=> pp(aa(nat,bool,P,J)) ) ) ) ).
% dec_induct
tff(fact_244_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
=> ( X = Y ) ) ).
% Suc_inject
tff(fact_245_less__SucI,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).
% less_SucI
tff(fact_246_less__SucE,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( M = N ) ) ) ).
% less_SucE
tff(fact_247_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
| ( M = aa(nat,nat,suc,N) ) ) ) ).
% le_Suc_eq
tff(fact_248_Suc__lessI,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( ( aa(nat,nat,suc,M) != N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N)) ) ) ).
% Suc_lessI
tff(fact_249_Suc__lessE,axiom,
! [I: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K))
=> ~ ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
=> ( K != aa(nat,nat,suc,J2) ) ) ) ).
% Suc_lessE
tff(fact_250_Suc__lessD,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% Suc_lessD
tff(fact_251_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% Suc_le_eq
tff(fact_252_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M4))
=> ? [M5: nat] : M4 = aa(nat,nat,suc,M5) ) ).
% Suc_le_D
tff(fact_253_le__SucI,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),M),aa(nat,nat,suc,N))) ) ).
% le_SucI
tff(fact_254_le__SucE,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( M = aa(nat,nat,suc,N) ) ) ) ).
% le_SucE
tff(fact_255_Suc__leI,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N)) ) ).
% Suc_leI
tff(fact_256_Suc__leD,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% Suc_leD
tff(fact_257_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K))
=> ( ( K != aa(nat,nat,suc,I) )
=> ~ ! [J2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
=> ( K != aa(nat,nat,suc,J2) ) ) ) ) ).
% Nat.lessE
tff(fact_258_square__eq__1__iff,axiom,
! [A: $tType] :
( ring_15535105094025558882visors(A)
=> ! [X: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
<=> ( ( X = one_one(A) )
| ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).
% square_eq_1_iff
tff(fact_259_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_260_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_261_mult_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)) ) ).
% mult.left_commute
tff(fact_262_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_263_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_264_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_265_mult_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) ) ).
% mult.commute
tff(fact_266_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_267_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_268_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_269_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A),N: nat,M: nat] :
( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N3)),aa(nat,A,F,aa(nat,nat,suc,N3))))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N)),aa(nat,A,F,M)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).
% lift_Suc_mono_less_iff
tff(fact_270_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_271_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A),N: nat,N2: nat] :
( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,aa(nat,nat,suc,N3))),aa(nat,A,F,N3)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N2)),aa(nat,A,F,N))) ) ) ) ).
% lift_Suc_antimono_le
tff(fact_272_lift__Suc__mono__less,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A),N: nat,N2: nat] :
( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N3)),aa(nat,A,F,aa(nat,nat,suc,N3))))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N)),aa(nat,A,F,N2))) ) ) ) ).
% lift_Suc_mono_less
tff(fact_273_lift__Suc__mono__le,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A),N: nat,N2: nat] :
( ! [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N3)),aa(nat,A,F,aa(nat,nat,suc,N3))))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N)),aa(nat,A,F,N2))) ) ) ) ).
% lift_Suc_mono_le
tff(fact_274_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_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)) ) ).
% ab_semigroup_mult_class.mult_ac(1)
tff(fact_275_fold__if__return,axiom,
! [A: $tType,B2: bool,C2: A,D2: 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,D2)) ) )
& ( ~ pp(B2)
=> ( aa(A,heap_Time_Heap(A),heap_Time_return(A),D2) = aa(A,heap_Time_Heap(A),heap_Time_return(A),if(A,B2,C2,D2)) ) ) ) ).
% fold_if_return
tff(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_mult__less__iff1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).
% mult_less_iff1
tff(fact_285_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_286_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_287_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_288_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_289_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
<=> ( ( M = zero_zero(nat) )
| ? [J3: nat] :
( ( M = aa(nat,nat,suc,J3) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N)) ) ) ) ).
% less_Suc_eq_0_disj
tff(fact_290_gr0__implies__Suc,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).
% gr0_implies_Suc
tff(fact_291_All__less__Suc2,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(nat,nat,suc,N)))
=> pp(aa(nat,bool,P,I5)) )
<=> ( pp(aa(nat,bool,P,zero_zero(nat)))
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
=> pp(aa(nat,bool,P,aa(nat,nat,suc,I5))) ) ) ) ).
% All_less_Suc2
tff(fact_292_gr0__conv__Suc,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
<=> ? [M6: nat] : N = aa(nat,nat,suc,M6) ) ).
% gr0_conv_Suc
tff(fact_293_Ex__less__Suc2,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(nat,nat,suc,N)))
& pp(aa(nat,bool,P,I5)) )
<=> ( pp(aa(nat,bool,P,zero_zero(nat)))
| ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
& pp(aa(nat,bool,P,aa(nat,nat,suc,I5))) ) ) ) ).
% Ex_less_Suc2
tff(fact_294_mult__less__mono2,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),zero_zero(nat)),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ) ).
% mult_less_mono2
tff(fact_295_mult__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),zero_zero(nat)),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ) ).
% mult_less_mono1
tff(fact_296_ex__least__nat__le,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,N))
=> ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
=> ? [K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K2))
=> ~ pp(aa(nat,bool,P,I3)) )
& pp(aa(nat,bool,P,K2)) ) ) ) ).
% ex_least_nat_le
tff(fact_297_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).
% less_imp_of_nat_less
tff(fact_298_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
% of_nat_less_imp_less
tff(fact_299_of__nat__mono,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [I: nat,J: nat] :
( 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(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J))) ) ) ).
% of_nat_mono
tff(fact_300_int__ops_I7_J,axiom,
! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% int_ops(7)
tff(fact_301_nat__int__comparison_I3_J,axiom,
! [A3: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).
% nat_int_comparison(3)
tff(fact_302_mult__not__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) )
=> ( ( A3 != zero_zero(A) )
& ( B2 != zero_zero(A) ) ) ) ) ).
% mult_not_zero
tff(fact_303_divisors__zero,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = zero_zero(A) )
=> ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
% divisors_zero
tff(fact_304_no__zero__divisors,axiom,
! [A: $tType] :
( semiri3467727345109120633visors(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) != zero_zero(A) ) ) ) ) ).
% no_zero_divisors
tff(fact_305_mult__left__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(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) )
<=> ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
tff(fact_306_mult__right__cancel,axiom,
! [A: $tType] :
( semiri6575147826004484403cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(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) )
<=> ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
tff(fact_307_fact__mono,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ).
% fact_mono
tff(fact_308_euclidean__size__mult,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A] : aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)) ) ).
% euclidean_size_mult
tff(fact_309_round__mono,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).
% round_mono
tff(fact_310_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_311_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_312_mult__right__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( 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),X),Y)),X)) ) ) ) ) ).
% mult_right_le_one_le
tff(fact_313_mult__left__le__one__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( 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),X)),X)) ) ) ) ) ).
% mult_left_le_one_le
tff(fact_314_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_315_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_316_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_317_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_318_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N3)),X))) ) ) ).
% ex_less_of_nat_mult
tff(fact_319_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_320_size__mult__mono_H,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).
% size_mult_mono'
tff(fact_321_size__mult__mono,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).
% size_mult_mono
tff(fact_322_fact__less__mono,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [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),M),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).
% fact_less_mono
tff(fact_323_fact__ge__Suc__0__nat,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).
% fact_ge_Suc_0_nat
tff(fact_324_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_325_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_326_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_327_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_328_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_329_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).
% less_numeral_extra(3)
tff(fact_330_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_331_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_332_zero__le,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).
% zero_le
tff(fact_333_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).
% le_numeral_extra(3)
tff(fact_334_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_335_exists__least__lemma,axiom,
! [P: fun(nat,bool)] :
( ~ pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ? [X_1: nat] : pp(aa(nat,bool,P,X_1))
=> ? [N3: nat] :
( ~ pp(aa(nat,bool,P,N3))
& pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) ) ) ).
% exists_least_lemma
tff(fact_336_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).
% not0_implies_Suc
tff(fact_337_Zero__not__Suc,axiom,
! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).
% Zero_not_Suc
tff(fact_338_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).
% Zero_neq_Suc
tff(fact_339_Suc__neq__Zero,axiom,
! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).
% Suc_neq_Zero
tff(fact_340_zero__induct,axiom,
! [P: fun(nat,bool),K: nat] :
( pp(aa(nat,bool,P,K))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,suc,N3)))
=> pp(aa(nat,bool,P,N3)) )
=> pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).
% zero_induct
tff(fact_341_diff__induct,axiom,
! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
( ! [X2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X2),zero_zero(nat)))
=> ( ! [Y4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y4)))
=> ( ! [X2: nat,Y4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X2),Y4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y4))) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ) ).
% diff_induct
tff(fact_342_nat__induct,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
% nat_induct
tff(fact_343_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).
% old.nat.exhaust
tff(fact_344_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat = aa(nat,nat,suc,X22) )
=> ( Nat != zero_zero(nat) ) ) ).
% nat.discI
tff(fact_345_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).
% old.nat.distinct(1)
tff(fact_346_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).
% old.nat.distinct(2)
tff(fact_347_nat_Odistinct_I1_J,axiom,
! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).
% nat.distinct(1)
tff(fact_348_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_349_mult__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).
% mult_0
tff(fact_350_mult__of__nat__commute,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).
% mult_of_nat_commute
tff(fact_351_infinite__descent0__measure,axiom,
! [A: $tType,V3: fun(A,nat),P: fun(A,bool),X: A] :
( ! [X2: A] :
( ( aa(A,nat,V3,X2) = zero_zero(nat) )
=> pp(aa(A,bool,P,X2)) )
=> ( ! [X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V3,X2)))
=> ( ~ pp(aa(A,bool,P,X2))
=> ? [Y3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V3,Y3)),aa(A,nat,V3,X2)))
& ~ pp(aa(A,bool,P,Y3)) ) ) )
=> pp(aa(A,bool,P,X)) ) ) ).
% infinite_descent0_measure
tff(fact_352_infinite__descent0,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(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))
=> ? [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N3))
& ~ pp(aa(nat,bool,P,M2)) ) ) )
=> pp(aa(nat,bool,P,N)) ) ) ).
% infinite_descent0
tff(fact_353_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( N != zero_zero(nat) ) ) ).
% gr_implies_not0
tff(fact_354_less__zeroE,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
% less_zeroE
tff(fact_355_not__less0,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
% not_less0
tff(fact_356_not__gr0,axiom,
! [N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
<=> ( N = zero_zero(nat) ) ) ).
% not_gr0
tff(fact_357_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% gr0I
tff(fact_358_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),zero_zero(nat))) ).
% bot_nat_0.extremum_strict
tff(fact_359_reals__Archimedean2,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N3))) ) ).
% reals_Archimedean2
tff(fact_360_le__0__eq,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
<=> ( N = zero_zero(nat) ) ) ).
% le_0_eq
tff(fact_361_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
=> ( A3 = zero_zero(nat) ) ) ).
% bot_nat_0.extremum_uniqueI
tff(fact_362_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),zero_zero(nat)))
<=> ( A3 = zero_zero(nat) ) ) ).
% bot_nat_0.extremum_unique
tff(fact_363_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).
% less_eq_nat.simps(1)
tff(fact_364_real__arch__simple,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N3))) ) ).
% real_arch_simple
tff(fact_365_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_366_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_367_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_368_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_369_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_370_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_371_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_372_fact__mono__nat,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),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ).
% fact_mono_nat
tff(fact_373_fact__ge__self,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),semiring_char_0_fact(nat,N))) ).
% fact_ge_self
tff(fact_374_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_375_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_376_of__nat__neq__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).
% of_nat_neq_0
tff(fact_377_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).
% of_nat_less_0_iff
tff(fact_378_of__nat__0__le__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).
% of_nat_0_le_iff
tff(fact_379_One__nat__def,axiom,
one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).
% One_nat_def
tff(fact_380_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_381_fact__gt__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).
% fact_gt_zero
tff(fact_382_fact__not__neg,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).
% fact_not_neg
tff(fact_383_fact__ge__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).
% fact_ge_zero
tff(fact_384_division__segment__mult,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,euclid7384307370059645450egment(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),aa(A,A,euclid7384307370059645450egment(A),A3)),aa(A,A,euclid7384307370059645450egment(A),B2)) ) ) ) ) ).
% division_segment_mult
tff(fact_385_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_386_int__ops_I1_J,axiom,
aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).
% int_ops(1)
tff(fact_387_fact__less__mono__nat,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),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).
% fact_less_mono_nat
tff(fact_388_effect__returnI,axiom,
! [A: $tType,H2: heap_ext(product_unit),H: heap_ext(product_unit),X: A] :
( ( H2 = H )
=> heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H2,H,X,one_one(nat)) ) ).
% effect_returnI
tff(fact_389_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
<=> ( ( N = zero_zero(nat) )
& ( M = zero_zero(nat) ) ) ) ).
% negative_eq_positive
tff(fact_390_nat__mult__less__cancel__disj,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),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
<=> ( 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(nat),M),N)) ) ) ).
% nat_mult_less_cancel_disj
tff(fact_391_nat__mult__le__cancel__disj,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),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
<=> ( 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),M),N)) ) ) ).
% nat_mult_le_cancel_disj
tff(fact_392_compl__less__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% compl_less_compl_iff
tff(fact_393_compl__le__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% compl_le_compl_iff
tff(fact_394_field__le__mult__one__interval,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: 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),X)),Y)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% field_le_mult_one_interval
tff(fact_395_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero(int) )
=> ( ! [N3: nat] :
( ( K = aa(nat,int,semiring_1_of_nat(int),N3) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) )
=> ~ ! [N3: nat] :
( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) ) ) ) ).
% int_cases3
tff(fact_396_not__zle__0__negative,axiom,
! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))) ).
% not_zle_0_negative
tff(fact_397_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] : M != aa(nat,int,semiring_1_of_nat(int),N3)
=> ~ ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ) ) ).
% int_cases4
tff(fact_398_negative__zle,axiom,
! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M))) ).
% negative_zle
tff(fact_399_lift__collapse,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X: B] : aa(B,heap_Time_Heap(A),heap_Time_lift(B,A,F),X) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,F,X)) ).
% lift_collapse
tff(fact_400_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),X)) = X ) ).
% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_401_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,uminus_uminus(A),X) = aa(A,A,uminus_uminus(A),Y) )
<=> ( X = Y ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_402_negative__zless,axiom,
! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M))) ).
% negative_zless
tff(fact_403_verit__la__generic,axiom,
! [A3: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),X))
| ( A3 = X )
| pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),A3)) ) ).
% verit_la_generic
tff(fact_404_Nat_Oex__has__greatest__nat,axiom,
! [P: fun(nat,bool),K: nat,B2: nat] :
( pp(aa(nat,bool,P,K))
=> ( ! [Y4: nat] :
( pp(aa(nat,bool,P,Y4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
=> ? [X2: nat] :
( pp(aa(nat,bool,P,X2))
& ! [Y3: nat] :
( pp(aa(nat,bool,P,Y3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),X2)) ) ) ) ) ).
% Nat.ex_has_greatest_nat
tff(fact_405_nat__le__linear,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),N),M)) ) ).
% nat_le_linear
tff(fact_406_mult__le__mono2,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),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ).
% mult_le_mono2
tff(fact_407_mult__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),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ).
% mult_le_mono1
tff(fact_408_mult__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),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L))) ) ) ).
% mult_le_mono
tff(fact_409_le__antisym,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),N),M))
=> ( M = N ) ) ) ).
% le_antisym
tff(fact_410_le__square,axiom,
! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).
% le_square
tff(fact_411_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% eq_imp_le
tff(fact_412_le__trans,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))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K)) ) ) ).
% le_trans
tff(fact_413_le__refl,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).
% le_refl
tff(fact_414_le__cube,axiom,
! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)))) ).
% le_cube
tff(fact_415_less__int__code_I1_J,axiom,
~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),zero_zero(int))) ).
% less_int_code(1)
tff(fact_416_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J))
=> ( 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(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J))) ) ) ).
% zmult_zless_mono2
tff(fact_417_not__int__zless__negative,axiom,
! [N: nat,M: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).
% not_int_zless_negative
tff(fact_418_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ).
% int_one_le_iff_zero_less
tff(fact_419_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
<=> ( ( M = one_one(int) )
& ( N = one_one(int) ) ) ) ) ).
% pos_zmult_eq_1_iff
tff(fact_420_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J))) ) ) ).
% zmult_zless_mono2_lemma
tff(fact_421_negative__zless__0,axiom,
! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).
% negative_zless_0
tff(fact_422_negD,axiom,
! [X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
=> ? [N3: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).
% negD
tff(fact_423_nat__int__comparison_I2_J,axiom,
! [A3: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) ) ).
% nat_int_comparison(2)
tff(fact_424_pos__int__cases,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ~ ! [N3: nat] :
( ( K = aa(nat,int,semiring_1_of_nat(int),N3) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) ) ) ).
% pos_int_cases
tff(fact_425_zero__less__imp__eq__int,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
& ( K = aa(nat,int,semiring_1_of_nat(int),N3) ) ) ) ).
% zero_less_imp_eq_int
tff(fact_426_neg__int__cases,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ~ ! [N3: nat] :
( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3)) ) ) ).
% neg_int_cases
tff(fact_427_linordered__field__no__ub,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X3: A] :
? [X_12: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X_12)) ) ).
% linordered_field_no_ub
tff(fact_428_linordered__field__no__lb,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X3: A] :
? [Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X3)) ) ).
% linordered_field_no_lb
tff(fact_429_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: 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) )
<=> ( ( K = zero_zero(nat) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
tff(fact_430_times__int__code_I2_J,axiom,
! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).
% times_int_code(2)
tff(fact_431_times__int__code_I1_J,axiom,
! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).
% times_int_code(1)
tff(fact_432_uminus__int__code_I1_J,axiom,
aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).
% uminus_int_code(1)
tff(fact_433_less__eq__int__code_I1_J,axiom,
pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).
% less_eq_int_code(1)
tff(fact_434_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
<=> ( ( ( M = one_one(int) )
& ( N = one_one(int) ) )
| ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
& ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).
% zmult_eq_1_iff
tff(fact_435_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
=> ( ( M = one_one(int) )
| ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
tff(fact_436_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(nat,int,semiring_1_of_nat(int),N) )
<=> ( M = N ) ) ).
% int_int_eq
tff(fact_437_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] : Z2 != aa(nat,int,semiring_1_of_nat(int),N3)
=> ~ ! [N3: nat] : Z2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ).
% int_cases2
tff(fact_438_zle__int,axiom,
! [M: nat,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% zle_int
tff(fact_439_compl__le__swap2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).
% compl_le_swap2
tff(fact_440_compl__le__swap1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).
% compl_le_swap1
tff(fact_441_compl__mono,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).
% compl_mono
tff(fact_442_compl__less__swap2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).
% compl_less_swap2
tff(fact_443_compl__less__swap1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,uminus_uminus(A),X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).
% compl_less_swap1
tff(fact_444_nat__mult__eq__cancel1,axiom,
! [K: nat,M: 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),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
<=> ( M = N ) ) ) ).
% nat_mult_eq_cancel1
tff(fact_445_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: 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),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)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
% nat_mult_le_cancel1
tff(fact_446_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: 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(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)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).
% nat_mult_less_cancel1
tff(fact_447_int__of__nat__induct,axiom,
! [P: fun(int,bool),Z2: int] :
( ! [N3: nat] : pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),N3)))
=> ( ! [N3: nat] : pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3)))))
=> pp(aa(int,bool,P,Z2)) ) ) ).
% int_of_nat_induct
tff(fact_448_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] : Z2 != aa(nat,int,semiring_1_of_nat(int),N3)
=> ~ ! [N3: nat] : Z2 != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).
% int_cases
tff(fact_449_int__zle__neg,axiom,
! [N: nat,M: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
<=> ( ( N = zero_zero(nat) )
& ( M = zero_zero(nat) ) ) ) ).
% int_zle_neg
tff(fact_450_zero__le__imp__eq__int,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ? [N3: nat] : K = aa(nat,int,semiring_1_of_nat(int),N3) ) ).
% zero_le_imp_eq_int
tff(fact_451_nonpos__int__cases,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
=> ~ ! [N3: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ).
% nonpos_int_cases
tff(fact_452_nonneg__int__cases,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ~ ! [N3: nat] : K != aa(nat,int,semiring_1_of_nat(int),N3) ) ).
% nonneg_int_cases
tff(fact_453_negative__zle__0,axiom,
! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int))) ).
% negative_zle_0
tff(fact_454_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( uminus(B)
=> ! [A5: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A5),X) = aa(B,B,uminus_uminus(B),aa(A,B,A5,X)) ) ).
% uminus_apply
tff(fact_455_dual__order_Orefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),A3)) ) ).
% dual_order.refl
tff(fact_456_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).
% order_refl
tff(fact_457_imp__le__cong,axiom,
! [X: int,X4: int,P: bool,P2: bool] :
( ( X = X4 )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
=> ( pp(P)
<=> pp(P2) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> pp(P) )
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
=> pp(P2) ) ) ) ) ).
% imp_le_cong
tff(fact_458_conj__le__cong,axiom,
! [X: int,X4: int,P: bool,P2: bool] :
( ( X = X4 )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
=> ( pp(P)
<=> pp(P2) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
& pp(P) )
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
& pp(P2) ) ) ) ) ).
% conj_le_cong
tff(fact_459_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: fun(A,bool),K: A,F: fun(A,nat),B2: nat] :
( pp(aa(A,bool,P,K))
=> ( ! [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)),B2)) )
=> ? [X2: A] :
( pp(aa(A,bool,P,X2))
& ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,Y3)),aa(A,nat,F,X2))) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
tff(fact_460_nat__descend__induct,axiom,
! [N: nat,P: fun(nat,bool),M: nat] :
( ! [K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
=> pp(aa(nat,bool,P,K2)) )
=> ( ! [K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
=> ( ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),I3))
=> pp(aa(nat,bool,P,I3)) )
=> pp(aa(nat,bool,P,K2)) ) )
=> pp(aa(nat,bool,P,M)) ) ) ).
% nat_descend_induct
tff(fact_461_dependent__nat__choice,axiom,
! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
( ? [X_1: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_1))
=> ( ! [X2: A,N3: nat] :
( pp(aa(A,bool,aa(nat,fun(A,bool),P,N3),X2))
=> ? [Y3: A] :
( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N3)),Y3))
& pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N3),X2),Y3)) ) )
=> ? [F2: fun(nat,A)] :
! [N4: nat] :
( pp(aa(A,bool,aa(nat,fun(A,bool),P,N4),aa(nat,A,F2,N4)))
& pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N4),aa(nat,A,F2,N4)),aa(nat,A,F2,aa(nat,nat,suc,N4)))) ) ) ) ).
% dependent_nat_choice
tff(fact_462_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero(nat) )
=> ~ ! [N3: nat] : X != aa(nat,nat,suc,N3) ) ).
% list_decode.cases
tff(fact_463_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
tff(fact_464_nle__le,axiom,
! [A: $tType] :
( linorder(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),B2),A3))
& ( B2 != A3 ) ) ) ) ).
% nle_le
tff(fact_465_le__cases3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X)) )
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).
% le_cases3
tff(fact_466_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ( X = Y )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).
% order_class.order_eq_iff
tff(fact_467_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( 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),A3),C2)) ) ) ) ).
% ord_eq_le_trans
tff(fact_468_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( B2 = C2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).
% ord_le_eq_trans
tff(fact_469_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( X = Y ) ) ) ) ).
% order_antisym
tff(fact_470_order_Otrans,axiom,
! [A: $tType] :
( preorder(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),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2)) ) ) ) ).
% order.trans
tff(fact_471_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).
% order_trans
tff(fact_472_linorder__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B4))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B4)) )
=> ( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ).
% linorder_wlog
tff(fact_473_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
<=> ( 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),A3),B2)) ) ) ) ).
% dual_order.eq_iff
tff(fact_474_dual__order_Oantisym,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: 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),A3),B2))
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
tff(fact_475_dual__order_Otrans,axiom,
! [A: $tType] :
( preorder(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),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A3)) ) ) ) ).
% dual_order.trans
tff(fact_476_antisym,axiom,
! [A: $tType] :
( order(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),B2),A3))
=> ( A3 = B2 ) ) ) ) ).
% antisym
tff(fact_477_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))) ) ) ).
% le_funD
tff(fact_478_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X))) ) ) ).
% le_funE
tff(fact_479_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)))
=> pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G)) ) ) ).
% le_funI
tff(fact_480_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
<=> ! [X5: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X5)),aa(A,B,G,X5))) ) ) ).
% le_fun_def
tff(fact_481_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: 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),B2),A3)) ) ) ) ).
% Orderings.order_eq_iff
tff(fact_482_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
=> ( ! [X2: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,C2))) ) ) ) ) ).
% order_subst1
tff(fact_483_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C2: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,B2)),C2))
=> ( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,X2)),aa(A,C,F,Y4))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,A3)),C2)) ) ) ) ) ).
% order_subst2
tff(fact_484_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( ( X = Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% order_eq_refl
tff(fact_485_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% linorder_linear
tff(fact_486_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( ( A3 = aa(B,A,F,B2) )
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
=> ( ! [X2: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,C2))) ) ) ) ) ).
% ord_eq_le_subst
tff(fact_487_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( aa(A,B,F,B2) = C2 )
=> ( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y4))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,A3)),C2)) ) ) ) ) ).
% ord_le_eq_subst
tff(fact_488_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% linorder_le_cases
tff(fact_489_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% order_antisym_conv
tff(fact_490_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ord(C)
=> ! [F3: D] :
? [Z3: C] :
! [X3: C] :
( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),X3),Z3))
=> ( F3 = F3 ) ) ) ).
% minf(11)
tff(fact_491_minf_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X3)) ) ) ).
% minf(7)
tff(fact_492_minf_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),T2)) ) ) ).
% minf(5)
tff(fact_493_minf_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> ( X3 != T2 ) ) ) ).
% minf(4)
tff(fact_494_minf_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> ( X3 != T2 ) ) ) ).
% minf(3)
tff(fact_495_minf_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,bool),P2: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P2,X2)) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q2,X2)) ) )
=> ? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> ( ( pp(aa(A,bool,P,X3))
| pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P2,X3))
| pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).
% minf(2)
tff(fact_496_minf_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,bool),P2: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P2,X2)) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z4))
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q2,X2)) ) )
=> ? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> ( ( pp(aa(A,bool,P,X3))
& pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P2,X3))
& pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).
% minf(1)
tff(fact_497_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ord(C)
=> ! [F3: D] :
? [Z3: C] :
! [X3: C] :
( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),Z3),X3))
=> ( F3 = F3 ) ) ) ).
% pinf(11)
tff(fact_498_pinf_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X3)) ) ) ).
% pinf(7)
tff(fact_499_pinf_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),T2)) ) ) ).
% pinf(5)
tff(fact_500_pinf_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> ( X3 != T2 ) ) ) ).
% pinf(4)
tff(fact_501_pinf_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> ( X3 != T2 ) ) ) ).
% pinf(3)
tff(fact_502_pinf_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,bool),P2: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P2,X2)) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q2,X2)) ) )
=> ? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> ( ( pp(aa(A,bool,P,X3))
| pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P2,X3))
| pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).
% pinf(2)
tff(fact_503_pinf_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,bool),P2: fun(A,bool),Q: fun(A,bool),Q2: fun(A,bool)] :
( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P2,X2)) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),X2))
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q2,X2)) ) )
=> ? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> ( ( pp(aa(A,bool,P,X3))
& pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P2,X3))
& pp(aa(A,bool,Q2,X3)) ) ) ) ) ) ) ).
% pinf(1)
tff(fact_504_lt__ex,axiom,
! [A: $tType] :
( no_bot(A)
=> ! [X: A] :
? [Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X)) ) ).
% lt_ex
tff(fact_505_gt__ex,axiom,
! [A: $tType] :
( no_top(A)
=> ! [X: A] :
? [X_12: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_12)) ) ).
% gt_ex
tff(fact_506_dense,axiom,
! [A: $tType] :
( dense_order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ? [Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),Y)) ) ) ) ).
% dense
tff(fact_507_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
% less_imp_neq
tff(fact_508_order_Oasym,axiom,
! [A: $tType] :
( preorder(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),B2),A3)) ) ) ).
% order.asym
tff(fact_509_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( 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),A3),C2)) ) ) ) ).
% ord_eq_less_trans
tff(fact_510_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( ( B2 = C2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).
% ord_less_eq_trans
tff(fact_511_less__induct,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),A3: A] :
( ! [X2: A] :
( ! [Y3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X2))
=> pp(aa(A,bool,P,Y3)) )
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,A3)) ) ) ).
% less_induct
tff(fact_512_antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% antisym_conv3
tff(fact_513_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ( X != Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_cases
tff(fact_514_dual__order_Oasym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: 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),A3),B2)) ) ) ).
% dual_order.asym
tff(fact_515_dual__order_Oirrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),A3)) ) ).
% dual_order.irrefl
tff(fact_516_exists__least__iff,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool)] :
( ? [X_13: A] : pp(aa(A,bool,P,X_13))
<=> ? [N5: A] :
( pp(aa(A,bool,P,N5))
& ! [M6: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M6),N5))
=> ~ pp(aa(A,bool,P,M6)) ) ) ) ) ).
% exists_least_iff
tff(fact_517_linorder__less__wlog,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,fun(A,bool)),A3: A,B2: A] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),B4))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B4)) )
=> ( ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A6),A6))
=> ( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,B4),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A6),B4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P,A3),B2)) ) ) ) ) ).
% linorder_less_wlog
tff(fact_518_order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(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),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).
% order.strict_trans
tff(fact_519_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
tff(fact_520_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( preorder(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),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).
% dual_order.strict_trans
tff(fact_521_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
tff(fact_522_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
tff(fact_523_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_neqE
tff(fact_524_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% order_less_asym
tff(fact_525_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ( X != Y )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
% linorder_neq_iff
tff(fact_526_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(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),B2),A3)) ) ) ).
% order_less_asym'
tff(fact_527_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).
% order_less_trans
tff(fact_528_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( ( A3 = aa(B,A,F,B2) )
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
=> ( ! [X2: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C2))) ) ) ) ) ).
% ord_eq_less_subst
tff(fact_529_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [A3: A,B2: A,F: fun(A,B),C2: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( ( aa(A,B,F,B2) = C2 )
=> ( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y4))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,A3)),C2)) ) ) ) ) ).
% ord_less_eq_subst
tff(fact_530_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).
% order_less_irrefl
tff(fact_531_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
=> ( ! [X2: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C2))) ) ) ) ) ).
% order_less_subst1
tff(fact_532_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C2: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,B2)),C2))
=> ( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,X2)),aa(A,C,F,Y4))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,A3)),C2)) ) ) ) ) ).
% order_less_subst2
tff(fact_533_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% order_less_not_sym
tff(fact_534_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,P: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> pp(P) ) ) ) ).
% order_less_imp_triv
tff(fact_535_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% linorder_less_linear
tff(fact_536_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
tff(fact_537_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
tff(fact_538_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% order_less_imp_not_less
tff(fact_539_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( uminus(B)
=> ! [A5: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A5),X3) = aa(B,B,uminus_uminus(B),aa(A,B,A5,X3)) ) ).
% fun_Compl_def
tff(fact_540_ex__has__least__nat,axiom,
! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
( pp(aa(A,bool,P,K))
=> ? [X2: A] :
( pp(aa(A,bool,P,X2))
& ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,X2)),aa(A,nat,M,Y3))) ) ) ) ).
% ex_has_least_nat
tff(fact_541_minf_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X3)) ) ) ).
% minf(8)
tff(fact_542_minf_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),T2)) ) ) ).
% minf(6)
tff(fact_543_pinf_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X3)) ) ) ).
% pinf(8)
tff(fact_544_pinf_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),T2)) ) ) ).
% pinf(6)
tff(fact_545_leD,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
% leD
tff(fact_546_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% leI
tff(fact_547_nless__le,axiom,
! [A: $tType] :
( order(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),A3),B2))
| ( A3 = B2 ) ) ) ) ).
% nless_le
tff(fact_548_antisym__conv1,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% antisym_conv1
tff(fact_549_antisym__conv2,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( X = Y ) ) ) ) ).
% antisym_conv2
tff(fact_550_dense__ge,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,Y: A] :
( ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).
% dense_ge
tff(fact_551_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,Z2: A] :
( ! [X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ).
% dense_le
tff(fact_552_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).
% less_le_not_le
tff(fact_553_not__le__imp__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
% not_le_imp_less
tff(fact_554_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(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(A),A3),B2))
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
tff(fact_555_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(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),A3),B2))
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
tff(fact_556_order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(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(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).
% order.strict_trans1
tff(fact_557_order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(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_eq(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2)) ) ) ) ).
% order.strict_trans2
tff(fact_558_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(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),A3),B2))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ).
% order.strict_iff_not
tff(fact_559_dense__ge__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
=> ( ! [W: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),W))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% dense_ge_bounded
tff(fact_560_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ! [W: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z2)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% dense_le_bounded
tff(fact_561_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: 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(A),B2),A3))
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
tff(fact_562_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: 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_eq(A),B2),A3))
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
tff(fact_563_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( preorder(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(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).
% dual_order.strict_trans1
tff(fact_564_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( preorder(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_eq(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A3)) ) ) ) ).
% dual_order.strict_trans2
tff(fact_565_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: 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_eq(A),B2),A3))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).
% dual_order.strict_iff_not
tff(fact_566_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(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),A3),B2)) ) ) ).
% order.strict_implies_order
tff(fact_567_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: 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_eq(A),B2),A3)) ) ) ).
% dual_order.strict_implies_order
tff(fact_568_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y ) ) ) ) ).
% order_le_less
tff(fact_569_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& ( X != Y ) ) ) ) ).
% order_less_le
tff(fact_570_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% linorder_not_le
tff(fact_571_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% linorder_not_less
tff(fact_572_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% order_less_imp_le
tff(fact_573_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( ( A3 != B2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).
% order_le_neq_trans
tff(fact_574_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [A3: A,B2: 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(A),A3),B2)) ) ) ) ).
% order_neq_le_trans
tff(fact_575_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).
% order_le_less_trans
tff(fact_576_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2)) ) ) ) ).
% order_less_le_trans
tff(fact_577_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
=> ( ! [X2: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C2))) ) ) ) ) ).
% order_le_less_subst1
tff(fact_578_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C2: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,B2)),C2))
=> ( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,X2)),aa(A,C,F,Y4))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,A3)),C2)) ) ) ) ) ).
% order_le_less_subst2
tff(fact_579_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [A3: A,F: fun(B,A),B2: B,C2: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,B2)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
=> ( ! [X2: B,Y4: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X2),Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,C2))) ) ) ) ) ).
% order_less_le_subst1
tff(fact_580_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( order(C)
& order(A) )
=> ! [A3: A,B2: A,F: fun(A,C),C2: C] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F,B2)),C2))
=> ( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,X2)),aa(A,C,F,Y4))) )
=> pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F,A3)),C2)) ) ) ) ) ).
% order_less_le_subst2
tff(fact_581_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
% linorder_le_less_linear
tff(fact_582_complete__interval,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [A3: A,B2: A,P: fun(A,bool)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,P,A3))
=> ( ~ pp(aa(A,bool,P,B2))
=> ? [C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),B2))
& ! [X3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),C3)) )
=> pp(aa(A,bool,P,X3)) )
& ! [D3: A] :
( ! [X2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),D3)) )
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D3),C3)) ) ) ) ) ) ) ).
% complete_interval
tff(fact_583_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_584_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_585_zero__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).
% zero_le_ceiling
tff(fact_586_ceiling__less__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).
% ceiling_less_zero
tff(fact_587_one__less__nat__eq,axiom,
! [Z2: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z2)) ) ).
% one_less_nat_eq
tff(fact_588_split__div_H,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))) )
| ? [Q3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)),M))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q3))))
& pp(aa(nat,bool,P,Q3)) ) ) ) ).
% split_div'
tff(fact_589_frac__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( archimedean_frac(A,X) = X )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).
% frac_eq
tff(fact_590_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),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_591_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_592_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_593_minus__apply,axiom,
! [B: $tType,A: $tType] :
( minus(B)
=> ! [A5: fun(A,B),B5: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A5),B5),X) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A5,X)),aa(A,B,B5,X)) ) ).
% minus_apply
tff(fact_594_mod__mod__trivial,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),modulo_modulo(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% mod_mod_trivial
tff(fact_595_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),one_one(nat)))
<=> ( M = one_one(nat) ) ) ).
% nat_dvd_1_iff_1
tff(fact_596_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ).
% int_dvd_int_iff
tff(fact_597_sgn__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).
% sgn_sgn
tff(fact_598_frac__frac,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_frac(A,archimedean_frac(A,X)) = archimedean_frac(A,X) ) ).
% frac_frac
tff(fact_599_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_600_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_601_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_602_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_603_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_604_dvd__0__left__iff,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
<=> ( A3 = zero_zero(A) ) ) ) ).
% dvd_0_left_iff
tff(fact_605_dvd__0__right,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),zero_zero(A))) ) ).
% dvd_0_right
tff(fact_606_division__ring__divide__zero,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% division_ring_divide_zero
tff(fact_607_divide__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,C2: A,B2: 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) )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_right
tff(fact_608_divide__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
<=> ( ( C2 = zero_zero(A) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_left
tff(fact_609_divide__eq__0__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
<=> ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
% divide_eq_0_iff
tff(fact_610_div__by__0,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% div_by_0
tff(fact_611_div__0,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% div_0
tff(fact_612_times__divide__eq__right,axiom,
! [A: $tType] :
( division_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),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) ) ).
% times_divide_eq_right
tff(fact_613_divide__divide__eq__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(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),C2)),B2) ) ).
% divide_divide_eq_right
tff(fact_614_divide__divide__eq__left,axiom,
! [A: $tType] :
( field(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),divide_divide(A),A3),B2)),C2) = 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)) ) ).
% divide_divide_eq_left
tff(fact_615_times__divide__eq__left,axiom,
! [A: $tType] :
( field(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ).
% times_divide_eq_left
tff(fact_616_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_617_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_618_mod__self,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),A3) = zero_zero(A) ) ).
% mod_self
tff(fact_619_mod__by__0,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),zero_zero(A)) = A3 ) ).
% mod_by_0
tff(fact_620_mod__0,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% mod_0
tff(fact_621_minus__dvd__iff,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,uminus_uminus(A),X)),Y))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y)) ) ) ).
% minus_dvd_iff
tff(fact_622_dvd__minus__iff,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,uminus_uminus(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y)) ) ) ).
% dvd_minus_iff
tff(fact_623_div__minus__minus,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ).
% div_minus_minus
tff(fact_624_dvd__1__left,axiom,
! [K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K)) ).
% dvd_1_left
tff(fact_625_dvd__1__iff__1,axiom,
! [M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,suc,zero_zero(nat))))
<=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).
% dvd_1_iff_1
tff(fact_626_div__dvd__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ) ).
% div_dvd_div
tff(fact_627_minus__mod__self2,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ).
% minus_mod_self2
tff(fact_628_mod__minus__minus,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) ) ).
% mod_minus_minus
tff(fact_629_diff__Suc__Suc,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).
% diff_Suc_Suc
tff(fact_630_Suc__diff__diff,axiom,
! [M: nat,N: 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),aa(nat,nat,suc,M)),N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),K) ).
% Suc_diff_diff
tff(fact_631_diff__0__eq__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).
% diff_0_eq_0
tff(fact_632_diff__self__eq__0,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).
% diff_self_eq_0
tff(fact_633_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(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)))
<=> ( ( K = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
% nat_mult_dvd_cancel_disj
tff(fact_634_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = I ) ) ).
% diff_diff_cancel
tff(fact_635_sgn__0,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).
% sgn_0
tff(fact_636_sgn__mult__dvd__iff,axiom,
! [R: int,L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
& ( ( R = zero_zero(int) )
=> ( K = zero_zero(int) ) ) ) ) ).
% sgn_mult_dvd_iff
tff(fact_637_mult__sgn__dvd__iff,axiom,
! [L: int,R: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
& ( ( R = zero_zero(int) )
=> ( K = zero_zero(int) ) ) ) ) ).
% mult_sgn_dvd_iff
tff(fact_638_dvd__sgn__mult__iff,axiom,
! [L: int,R: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
| ( R = zero_zero(int) ) ) ) ).
% dvd_sgn_mult_iff
tff(fact_639_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R))))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
| ( R = zero_zero(int) ) ) ) ).
% dvd_mult_sgn_iff
tff(fact_640_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_641_sgn__minus,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A3)) ) ).
% sgn_minus
tff(fact_642_sgn__divide,axiom,
! [A: $tType] :
( field_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).
% sgn_divide
tff(fact_643_mod__less,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),modulo_modulo(nat),M),N) = M ) ) ).
% mod_less
tff(fact_644_nat__int,axiom,
! [N: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),N)) = N ).
% nat_int
tff(fact_645_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_646_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_647_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_648_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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)))
<=> ( ( C2 = zero_zero(A) )
| pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).
% dvd_mult_cancel_left
tff(fact_649_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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)))
<=> ( ( C2 = zero_zero(A) )
| pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).
% dvd_mult_cancel_right
tff(fact_650_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).
% dvd_times_left_cancel_iff
tff(fact_651_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).
% dvd_times_right_cancel_iff
tff(fact_652_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = zero_zero(A) ) )
& ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
tff(fact_653_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
tff(fact_654_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
tff(fact_655_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
tff(fact_656_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( field(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
tff(fact_657_div__mult__mult1,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% div_mult_mult1
tff(fact_658_div__mult__mult2,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% div_mult_mult2
tff(fact_659_div__mult__mult1__if,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2 = zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = zero_zero(A) ) )
& ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).
% div_mult_mult1_if
tff(fact_660_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(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)),A3) = B2 ) ) ) ).
% nonzero_mult_div_cancel_left
tff(fact_661_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( semidom_divide(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),times_times(A),A3),B2)),B2) = A3 ) ) ) ).
% nonzero_mult_div_cancel_right
tff(fact_662_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_663_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( group_add(B)
=> ! [B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).
% verit_minus_simplify(3)
tff(fact_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_unit__prod,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A))) ) ) ) ).
% unit_prod
tff(fact_674_mod__mult__self1__is__0,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),times_times(A),B2),A3)),B2) = zero_zero(A) ) ).
% mod_mult_self1_is_0
tff(fact_675_mod__mult__self2__is__0,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),times_times(A),A3),B2)),B2) = zero_zero(A) ) ).
% mod_mult_self2_is_0
tff(fact_676_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_677_divide__minus1,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).
% divide_minus1
tff(fact_678_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_679_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)) = B2 ) ) ) ).
% dvd_mult_div_cancel
tff(fact_680_dvd__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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),B2),A3)),A3) = B2 ) ) ) ).
% dvd_div_mult_self
tff(fact_681_unit__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A))) ) ) ) ).
% unit_div
tff(fact_682_unit__div__1__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),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_683_unit__div__1__div__1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_684_div__diff,axiom,
! [A: $tType] :
( idom_modulo(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).
% div_diff
tff(fact_685_dvd__imp__mod__0,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A3) = zero_zero(A) ) ) ) ).
% dvd_imp_mod_0
tff(fact_686_mod__div__trivial,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),B2) = zero_zero(A) ) ).
% mod_div_trivial
tff(fact_687_minus__mod__self1,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).
% minus_mod_self1
tff(fact_688_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,sgn_sgn(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% sgn_less
tff(fact_689_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% sgn_greater
tff(fact_690_zero__less__diff,axiom,
! [N: nat,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),minus_minus(nat),N),M)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).
% zero_less_diff
tff(fact_691_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% diff_is_0_eq
tff(fact_692_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) ) ) ).
% diff_is_0_eq'
tff(fact_693_divide__sgn,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B2)) ) ).
% divide_sgn
tff(fact_694_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_695_mod__by__Suc__0,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).
% mod_by_Suc_0
tff(fact_696_div__by__Suc__0,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).
% div_by_Suc_0
tff(fact_697_div__less,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),divide_divide(nat),M),N) = zero_zero(nat) ) ) ).
% div_less
tff(fact_698_ceiling__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).
% ceiling_zero
tff(fact_699_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(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)) = zero_zero(nat) ) )
& ( ( K != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(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)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ) ).
% nat_mult_div_cancel_disj
tff(fact_700_ceiling__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
% ceiling_one
tff(fact_701_ceiling__of__nat,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [N: nat] : archimedean_ceiling(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).
% ceiling_of_nat
tff(fact_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_unit__mult__div__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_714_unit__div__mult__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_715_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_716_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_717_Suc__pred,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),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).
% Suc_pred
tff(fact_718_nat__1,axiom,
aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).
% nat_1
tff(fact_719_nat__le__0,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int)))
=> ( aa(int,nat,nat2,Z2) = zero_zero(nat) ) ) ).
% nat_le_0
tff(fact_720_nat__0__iff,axiom,
! [I: int] :
( ( aa(int,nat,nat2,I) = zero_zero(nat) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) ) ).
% nat_0_iff
tff(fact_721_zless__nat__conj,axiom,
! [W2: int,Z2: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).
% zless_nat_conj
tff(fact_722_div__mult__self1__is__m,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),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),N) = M ) ) ).
% div_mult_self1_is_m
tff(fact_723_div__mult__self__is__m,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),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),N) = M ) ) ).
% div_mult_self_is_m
tff(fact_724_ceiling__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).
% ceiling_diff_one
tff(fact_725_int__nat__eq,axiom,
! [Z2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = Z2 ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = zero_zero(int) ) ) ) ).
% int_nat_eq
tff(fact_726_nat__zminus__int,axiom,
! [N: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).
% nat_zminus_int
tff(fact_727_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_728_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_729_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_730_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_731_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_732_ceiling__le__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).
% ceiling_le_zero
tff(fact_733_zero__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).
% zero_less_ceiling
tff(fact_734_zero__less__nat__eq,axiom,
! [Z2: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ).
% zero_less_nat_eq
tff(fact_735_ceiling__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).
% ceiling_less_one
tff(fact_736_one__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).
% one_le_ceiling
tff(fact_737_ceiling__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).
% ceiling_le_one
tff(fact_738_one__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).
% one_less_ceiling
tff(fact_739_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,B),G: fun(A,B)] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F),G))
<=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F),G))
& ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G),F)) ) ) ) ).
% less_fun_def
tff(fact_740_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( minus(B)
=> ! [A5: fun(A,B),B5: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A5),B5),X3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A5,X3)),aa(A,B,B5,X3)) ) ).
% fun_diff_def
tff(fact_741_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = zero_zero(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).
% mod_eq_0_iff_dvd
tff(fact_742_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A3) = zero_zero(A) ) ) ) ).
% dvd_eq_mod_eq_0
tff(fact_743_mod__0__imp__dvd,axiom,
! [A: $tType] :
( semiring_modulo(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).
% mod_0_imp_dvd
tff(fact_744_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = A3 )
<=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ).
% mod_eq_self_iff_div_eq_0
tff(fact_745_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ) ).
% dvd_div_eq_0_iff
tff(fact_746_dvd__diff,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2))) ) ) ) ).
% dvd_diff
tff(fact_747_dvd__refl,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),A3)) ) ).
% dvd_refl
tff(fact_748_dvd__trans,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% dvd_trans
tff(fact_749_dvd__mod__iff,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3)) ) ) ) ).
% dvd_mod_iff
tff(fact_750_dvd__minus__mod,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)))) ) ).
% dvd_minus_mod
tff(fact_751_dvd__div__eq__iff,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( ( 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) )
<=> ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
tff(fact_752_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( semidom_modulo(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3)) ) ) ) ).
% dvd_mod_imp_dvd
tff(fact_753_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A,C2: A,B2: 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) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
tff(fact_754_div__div__div__same,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [D2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).
% div_div_div_same
tff(fact_755_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2) )
=> ( ( A3 = B2 )
<=> ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
tff(fact_756_diff__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) ) ).
% diff_right_commute
tff(fact_757_dvd__diff__commute,axiom,
! [A: $tType] :
( euclid5891614535332579305n_ring(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2))) ) ) ).
% dvd_diff_commute
tff(fact_758_mod__diff__eq,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(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),minus_minus(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),minus_minus(A),A3),B2)),C2) ) ).
% mod_diff_eq
tff(fact_759_dvd__mod,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [K: A,M: A,N: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),K),M))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),K),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),K),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),M),N))) ) ) ) ).
% dvd_mod
tff(fact_760_mod__diff__cong,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,C2: A,A4: A,B2: A,B3: 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),A4),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),B3),C2) )
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A4),B3)),C2) ) ) ) ) ).
% mod_diff_cong
tff(fact_761_mod__eq__dvd__iff,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) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2))) ) ) ).
% mod_eq_dvd_iff
tff(fact_762_mod__diff__left__eq,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(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),minus_minus(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),minus_minus(A),A3),B2)),C2) ) ).
% mod_diff_left_eq
tff(fact_763_mod__diff__right__eq,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),minus_minus(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),minus_minus(A),A3),B2)),C2) ) ).
% mod_diff_right_eq
tff(fact_764_mod__mod__cancel,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) ) ) ) ).
% mod_mod_cancel
tff(fact_765_diff__commute,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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).
% diff_commute
tff(fact_766_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).
% dvd_diff_nat
tff(fact_767_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(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,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% field_char_0_class.of_nat_div
tff(fact_768_diff__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),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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).
% diff_divide_distrib
tff(fact_769_modulo__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),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)) ).
% modulo_nat_def
tff(fact_770_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(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),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) ) ).
% minus_mult_div_eq_mod
tff(fact_771_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( semiring_modulo(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),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)) ) ).
% minus_mod_eq_mult_div
tff(fact_772_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( semiring_modulo(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),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) ) ).
% minus_mod_eq_div_mult
tff(fact_773_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( semiring_modulo(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),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) ) ).
% minus_div_mult_eq_mod
tff(fact_774_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,D2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),C2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(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),D2)) ) ) ) ) ).
% div_mult_div_if_dvd
tff(fact_775_dvd__mult__imp__div,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ).
% dvd_mult_imp_div
tff(fact_776_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),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) ) ) ) ).
% dvd_div_mult2_eq
tff(fact_777_div__div__eq__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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),divide_divide(A),B2),C2)) = 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) ) ) ) ) ).
% div_div_eq_right
tff(fact_778_div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( 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) ) ) ) ).
% div_mult_swap
tff(fact_779_dvd__div__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) ) ) ) ).
% dvd_div_mult
tff(fact_780_unit__div__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_781_div__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% div_unit_dvd_iff
tff(fact_782_dvd__div__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% dvd_div_unit_iff
tff(fact_783_of__nat__mod,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_mod
tff(fact_784_dvd__neg__div,axiom,
! [A: $tType] :
( idom_divide(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).
% dvd_neg_div
tff(fact_785_dvd__div__neg,axiom,
! [A: $tType] :
( idom_divide(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).
% dvd_div_neg
tff(fact_786_mod__if,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),modulo_modulo(nat),M),N) = M ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) ) ) ) ).
% mod_if
tff(fact_787_le__mod__geq,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),modulo_modulo(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) ) ) ).
% le_mod_geq
tff(fact_788_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
% dvd_minus_self
tff(fact_789_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_790_of__nat__dvd__iff,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
% of_nat_dvd_iff
tff(fact_791_less__eq__dvd__minus,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),dvd_dvd(nat),M),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).
% less_eq_dvd_minus
tff(fact_792_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ) ) ).
% dvd_diffD1
tff(fact_793_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M)) ) ) ) ).
% dvd_diffD
tff(fact_794_mod__nat__eqI,axiom,
! [R: nat,N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),R)))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = R ) ) ) ) ).
% mod_nat_eqI
tff(fact_795_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_796_of__nat__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [R: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R))))) ) ).
% of_nat_ceiling
tff(fact_797_add__divide__eq__if__simps_I4_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),minus_minus(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),minus_minus(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),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z2)),B2)),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(4)
tff(fact_798_diff__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% diff_frac_eq
tff(fact_799_diff__divide__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),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),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),Y)),Z2) ) ) ) ).
% diff_divide_eq_iff
tff(fact_800_divide__diff__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).
% divide_diff_eq_iff
tff(fact_801_dvd__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = C2 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).
% dvd_div_eq_mult
tff(fact_802_div__dvd__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).
% div_dvd_iff_mult
tff(fact_803_dvd__div__iff__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2)) ) ) ) ) ).
% dvd_div_iff_mult
tff(fact_804_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( A3 != zero_zero(A) )
=> ( ( C2 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D2))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
tff(fact_805_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_806_unit__eq__div1,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_807_unit__eq__div2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_808_div__mult__unit2,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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_809_unit__div__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_810_unit__div__mult__swap,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_811_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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_812_div__less__mono,axiom,
! [A5: nat,B5: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A5),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A5),N) = zero_zero(nat) )
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B5),N) = zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),N))) ) ) ) ) ).
% div_less_mono
tff(fact_813_mod__greater__zero__iff__not__dvd,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),modulo_modulo(nat),M),N)))
<=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ).
% mod_greater_zero_iff_not_dvd
tff(fact_814_of__nat__diff,axiom,
! [A: $tType] :
( semiring_1_cancel(A)
=> ! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).
% of_nat_diff
tff(fact_815_division__segment__mod,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,euclid7384307370059645450egment(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) = aa(A,A,euclid7384307370059645450egment(A),B2) ) ) ) ) ).
% division_segment_mod
tff(fact_816_mod__mult__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),times_times(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),times_times(A),A3),B2)),C2) ) ).
% mod_mult_eq
tff(fact_817_mod__mult__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,A4: A,B2: A,B3: 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),A4),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),B3),C2) )
=> ( 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) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A4),B3)),C2) ) ) ) ) ).
% mod_mult_cong
tff(fact_818_mod__mult__mult2,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),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),C2) ) ).
% mod_mult_mult2
tff(fact_819_mult__mod__right,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(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)) ) ).
% mult_mod_right
tff(fact_820_mod__mult__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),times_times(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),times_times(A),A3),B2)),C2) ) ).
% mod_mult_left_eq
tff(fact_821_mod__mult__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),times_times(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),times_times(A),A3),B2)),C2) ) ).
% mod_mult_right_eq
tff(fact_822_diff__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D2: 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),D2),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),D2))) ) ) ) ).
% diff_mono
tff(fact_823_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_824_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_825_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2) )
=> ( 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),D2)) ) ) ) ).
% diff_eq_diff_less_eq
tff(fact_826_mod__minus__eq,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).
% mod_minus_eq
tff(fact_827_mod__minus__cong,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A,A4: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A4),B2) )
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A4)),B2) ) ) ) ).
% mod_minus_cong
tff(fact_828_mod__minus__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B2)) ) ).
% mod_minus_right
tff(fact_829_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_830_diff__strict__mono,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,D2: 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),D2),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),D2))) ) ) ) ).
% diff_strict_mono
tff(fact_831_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2) )
=> ( 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),D2)) ) ) ) ).
% diff_eq_diff_less
tff(fact_832_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_833_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_834_right__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(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),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(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)) ) ).
% right_diff_distrib'
tff(fact_835_left__diff__distrib_H,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(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)) ) ).
% left_diff_distrib'
tff(fact_836_right__diff__distrib,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),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(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)) ) ).
% right_diff_distrib
tff(fact_837_left__diff__distrib,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),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),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).
% left_diff_distrib
tff(fact_838_inf__period_I1_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
( ! [X2: A,K2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
=> ( ! [X2: A,K2: A] :
( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
=> ! [X3: A,K3: A] :
( ( pp(aa(A,bool,P,X3))
& pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))))
& pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))) ) ) ) ) ) ).
% inf_period(1)
tff(fact_839_inf__period_I2_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [P: fun(A,bool),D4: A,Q: fun(A,bool)] :
( ! [X2: A,K2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
=> ( ! [X2: A,K2: A] :
( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K2),D4)))) )
=> ! [X3: A,K3: A] :
( ( pp(aa(A,bool,P,X3))
| pp(aa(A,bool,Q,X3)) )
<=> ( pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))))
| pp(aa(A,bool,Q,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4)))) ) ) ) ) ) ).
% inf_period(2)
tff(fact_840_dvd__field__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
<=> ( ( A3 = zero_zero(A) )
=> ( B2 = zero_zero(A) ) ) ) ) ).
% dvd_field_iff
tff(fact_841_dvd__0__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
=> ( A3 = zero_zero(A) ) ) ) ).
% dvd_0_left
tff(fact_842_mod__Suc__eq,axiom,
! [M: 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),modulo_modulo(nat),M),N))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).
% mod_Suc_eq
tff(fact_843_mod__Suc__Suc__eq,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,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),N) ).
% mod_Suc_Suc_eq
tff(fact_844_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_845_dvd__triv__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3))) ) ).
% dvd_triv_right
tff(fact_846_dvd__mult__right,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ).
% dvd_mult_right
tff(fact_847_mult__dvd__mono,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A,D2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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),D2))) ) ) ) ).
% mult_dvd_mono
tff(fact_848_dvd__triv__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))) ) ).
% dvd_triv_left
tff(fact_849_dvd__mult__left,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ).
% dvd_mult_left
tff(fact_850_dvd__mult2,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).
% dvd_mult2
tff(fact_851_dvd__mult,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).
% dvd_mult
tff(fact_852_dvd__def,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
<=> ? [K4: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K4) ) ) ).
% dvd_def
tff(fact_853_dvdI,axiom,
! [A: $tType] :
( dvd(A)
=> ! [A3: A,B2: A,K: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).
% dvdI
tff(fact_854_dvdE,axiom,
! [A: $tType] :
( dvd(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).
% dvdE
tff(fact_855_one__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),one_one(A)),A3)) ) ).
% one_dvd
tff(fact_856_unit__imp__dvd,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).
% unit_imp_dvd
tff(fact_857_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ) ).
% dvd_unit_imp_unit
tff(fact_858_divide__divide__eq__left_H,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),divide_divide(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).
% divide_divide_eq_left'
tff(fact_859_divide__divide__times__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z2),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),W2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ).
% divide_divide_times_eq
tff(fact_860_times__divide__times__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,Z2: A,W2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z2),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W2)) ) ).
% times_divide_times_eq
tff(fact_861_mod__less__eq__dividend,axiom,
! [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),modulo_modulo(nat),M),N)),M)) ).
% mod_less_eq_dividend
tff(fact_862_minus__divide__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).
% minus_divide_left
tff(fact_863_minus__divide__divide,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ).
% minus_divide_divide
tff(fact_864_minus__divide__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).
% minus_divide_right
tff(fact_865_div__minus__right,axiom,
! [A: $tType] :
( euclid8851590272496341667cancel(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).
% div_minus_right
tff(fact_866_zero__induct__lemma,axiom,
! [P: fun(nat,bool),K: nat,I: nat] :
( pp(aa(nat,bool,P,K))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,suc,N3)))
=> pp(aa(nat,bool,P,N3)) )
=> pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I))) ) ) ).
% zero_induct_lemma
tff(fact_867_minus__nat_Odiff__0,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),zero_zero(nat)) = M ).
% minus_nat.diff_0
tff(fact_868_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) = zero_zero(nat) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
tff(fact_869_sgn__eq__0__iff,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% sgn_eq_0_iff
tff(fact_870_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% sgn_0_0
tff(fact_871_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ) ).
% diff_less_mono2
tff(fact_872_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N)),K)) ) ).
% less_imp_diff_less
tff(fact_873_div__le__dividend,axiom,
! [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),divide_divide(nat),M),N)),M)) ).
% div_le_dividend
tff(fact_874_div__le__mono,axiom,
! [M: nat,N: nat,K: 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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K))) ) ).
% div_le_mono
tff(fact_875_diff__le__mono2,axiom,
! [M: nat,N: nat,L: 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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ).
% diff_le_mono2
tff(fact_876_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),C2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),A3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C2),B2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A3)) ) ) ) ).
% le_diff_iff'
tff(fact_877_diff__le__self,axiom,
! [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),minus_minus(nat),M),N)),M)) ).
% diff_le_self
tff(fact_878_diff__le__mono,axiom,
! [M: nat,N: nat,L: 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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).
% diff_le_mono
tff(fact_879_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( 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_eq(nat),K),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ) ).
% Nat.diff_diff_eq
tff(fact_880_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( 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_eq(nat),K),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
% le_diff_iff
tff(fact_881_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( 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_eq(nat),K),N))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K) )
<=> ( M = N ) ) ) ) ).
% eq_diff_iff
tff(fact_882_sgn__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,A,sgn_sgn(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),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B2)) ) ).
% sgn_mult
tff(fact_883_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),Q4) ).
% div_mult2_eq
tff(fact_884_diff__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),minus_minus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(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)) ).
% diff_mult_distrib2
tff(fact_885_diff__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),minus_minus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(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)) ).
% diff_mult_distrib
tff(fact_886_uminus__dvd__conv_I2_J,axiom,
! [D2: int,T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),T2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,uminus_uminus(int),T2))) ) ).
% uminus_dvd_conv(2)
tff(fact_887_uminus__dvd__conv_I1_J,axiom,
! [D2: int,T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),T2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,uminus_uminus(int),D2)),T2)) ) ).
% uminus_dvd_conv(1)
tff(fact_888_nat__dvd__iff,axiom,
! [Z2: int,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z2)),M))
<=> ( ( 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),dvd_dvd(int),Z2),aa(nat,int,semiring_1_of_nat(int),M))) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( M = zero_zero(nat) ) ) ) ) ).
% nat_dvd_iff
tff(fact_889_frac__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != 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),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A))) ) ) ) ) ).
% frac_le_eq
tff(fact_890_frac__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != 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),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2)))
<=> 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),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),zero_zero(A))) ) ) ) ) ).
% frac_less_eq
tff(fact_891_add__divide__eq__if__simps_I6_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),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(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),minus_minus(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(6)
tff(fact_892_add__divide__eq__if__simps_I5_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),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z2)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(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),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z2))),Z2) ) ) ) ) ).
% add_divide_eq_if_simps(5)
tff(fact_893_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).
% minus_divide_diff_eq_iff
tff(fact_894_is__unitE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ~ ( ( A3 != zero_zero(A) )
=> ! [B4: A] :
( ( B4 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B4),one_one(A)))
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B4 )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B4) = A3 )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B4) = 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),B4) ) ) ) ) ) ) ) ) ) ).
% is_unitE
tff(fact_895_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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_896_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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_897_div__if,axiom,
! [M: nat,N: nat] :
( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| ( N = zero_zero(nat) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
& ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| ( N = zero_zero(nat) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).
% div_if
tff(fact_898_le__div__geq,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_eq(nat),N),M))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).
% le_div_geq
tff(fact_899_nat__zero__as__int,axiom,
zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).
% nat_zero_as_int
tff(fact_900_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_901_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_902_int__sgnE,axiom,
! [K: int] :
~ ! [N3: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N3)) ).
% int_sgnE
tff(fact_903_ex__nat,axiom,
! [P: fun(nat,bool)] :
( ? [X_13: nat] : pp(aa(nat,bool,P,X_13))
<=> ? [X5: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
& pp(aa(nat,bool,P,aa(int,nat,nat2,X5))) ) ) ).
% ex_nat
tff(fact_904_all__nat,axiom,
! [P: fun(nat,bool)] :
( ! [X_13: nat] : pp(aa(nat,bool,P,X_13))
<=> ! [X5: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
=> pp(aa(nat,bool,P,aa(int,nat,nat2,X5))) ) ) ).
% all_nat
tff(fact_905_eq__nat__nat__iff,axiom,
! [Z2: int,Z5: 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)),Z5))
=> ( ( aa(int,nat,nat2,Z2) = aa(int,nat,nat2,Z5) )
<=> ( Z2 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
tff(fact_906_divide__le__0__iff,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),B2)),zero_zero(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),zero_zero(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),zero_zero(A)),B2)) ) ) ) ) ).
% divide_le_0_iff
tff(fact_907_divide__right__mono,axiom,
! [A: $tType] :
( linordered_field(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),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(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))) ) ) ) ).
% divide_right_mono
tff(fact_908_zero__le__divide__iff,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),B2)))
<=> ( ( 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),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).
% zero_le_divide_iff
tff(fact_909_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( 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),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).
% divide_nonneg_nonneg
tff(fact_910_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),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),X),Y)),zero_zero(A))) ) ) ) ).
% divide_nonneg_nonpos
tff(fact_911_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
=> ( 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),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).
% divide_nonpos_nonneg
tff(fact_912_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(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),X),Y))) ) ) ) ).
% divide_nonpos_nonpos
tff(fact_913_divide__right__mono__neg,axiom,
! [A: $tType] :
( linordered_field(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),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),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2))) ) ) ) ).
% divide_right_mono_neg
tff(fact_914_nat__mono,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y))) ) ).
% nat_mono
tff(fact_915_divide__neg__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(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),X),Y))) ) ) ) ).
% divide_neg_neg
tff(fact_916_divide__neg__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).
% divide_neg_pos
tff(fact_917_divide__pos__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),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),X),Y)),zero_zero(A))) ) ) ) ).
% divide_pos_neg
tff(fact_918_divide__pos__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> 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),X),Y))) ) ) ) ).
% divide_pos_pos
tff(fact_919_divide__less__0__iff,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),B2)),zero_zero(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),zero_zero(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),zero_zero(A)),B2)) ) ) ) ) ).
% divide_less_0_iff
tff(fact_920_divide__less__cancel,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(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(A),A3),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(A),B2),A3)) )
& ( C2 != zero_zero(A) ) ) ) ) ).
% divide_less_cancel
tff(fact_921_zero__less__divide__iff,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)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)))
<=> ( ( 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),A3),zero_zero(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).
% zero_less_divide_iff
tff(fact_922_divide__strict__right__mono,axiom,
! [A: $tType] :
( linordered_field(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),zero_zero(A)),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(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))) ) ) ) ).
% divide_strict_right_mono
tff(fact_923_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( linordered_field(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),C2),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),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))) ) ) ) ).
% divide_strict_right_mono_neg
tff(fact_924_not__is__unit__0,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),one_one(A))) ) ).
% not_is_unit_0
tff(fact_925_frac__eq__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W2: A] :
( ( Y != zero_zero(A) )
=> ( ( Z2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).
% frac_eq_eq
tff(fact_926_divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
<=> ( ( ( C2 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq
tff(fact_927_eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq
tff(fact_928_divide__eq__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 ) ) ) ) ).
% divide_eq_imp
tff(fact_929_eq__divide__imp,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 )
=> ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).
% eq_divide_imp
tff(fact_930_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,B2: A,A3: A] :
( ( C2 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A3 )
<=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).
% nonzero_divide_eq_eq
tff(fact_931_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C2: A,A3: A,B2: A] :
( ( C2 != zero_zero(A) )
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B2 ) ) ) ) ).
% nonzero_eq_divide_eq
tff(fact_932_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_933_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) = N )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) = zero_zero(nat) ) )
& ( ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) != N )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) ) ) ) ).
% mod_Suc
tff(fact_934_is__unit__mult__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).
% is_unit_mult_iff
tff(fact_935_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% dvd_mult_unit_iff
tff(fact_936_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% mult_unit_dvd_iff
tff(fact_937_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% dvd_mult_unit_iff'
tff(fact_938_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ).
% mult_unit_dvd_iff'
tff(fact_939_unit__mult__left__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_940_unit__mult__right__cancel,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_941_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).
% nonzero_minus_divide_right
tff(fact_942_nonzero__minus__divide__divide,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),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).
% nonzero_minus_divide_divide
tff(fact_943_mod__induct,axiom,
! [P: fun(nat,bool),N: nat,P3: nat,M: nat] :
( pp(aa(nat,bool,P,N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),P3))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),P3))
=> ( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N3)),P3))) ) )
=> pp(aa(nat,bool,P,M)) ) ) ) ) ).
% mod_induct
tff(fact_944_nat__one__as__int,axiom,
one_one(nat) = aa(int,nat,nat2,one_one(int)) ).
% nat_one_as_int
tff(fact_945_mod__less__divisor,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),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)),N)) ) ).
% mod_less_divisor
tff(fact_946_mod__Suc__le__divisor,axiom,
! [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),modulo_modulo(nat),M),aa(nat,nat,suc,N))),N)) ).
% mod_Suc_le_divisor
tff(fact_947_ceiling__mono,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).
% ceiling_mono
tff(fact_948_ceiling__less__cancel,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
% ceiling_less_cancel
tff(fact_949_div__mult2__eq_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(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),divide_divide(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)) ) ).
% div_mult2_eq'
tff(fact_950_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
| ( N = zero_zero(nat) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
tff(fact_951_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( aa(nat,nat,suc,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),M),N) ) ) ).
% Suc_diff_Suc
tff(fact_952_diff__less__Suc,axiom,
! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,suc,M))) ).
% diff_less_Suc
tff(fact_953_nat__dvd__not__less,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),M),N))
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).
% nat_dvd_not_less
tff(fact_954_Suc__div__le__mono,axiom,
! [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),divide_divide(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),N))) ).
% Suc_div_le_mono
tff(fact_955_diff__less,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),zero_zero(nat)),M))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ) ) ).
% diff_less
tff(fact_956_Suc__diff__le,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),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ) ).
% Suc_diff_le
tff(fact_957_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( 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_eq(nat),K),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),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),M),N)) ) ) ) ).
% less_diff_iff
tff(fact_958_diff__less__mono,axiom,
! [A3: nat,B2: nat,C2: 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),C2),A3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),C2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C2))) ) ) ).
% diff_less_mono
tff(fact_959_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
=> ( 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),dvd_dvd(int),M),N))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M))
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
tff(fact_960_sgn__not__eq__imp,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),A3) != zero_zero(A) )
=> ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
=> ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).
% sgn_not_eq_imp
tff(fact_961_zdvd__not__zless,axiom,
! [M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N))
=> ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).
% zdvd_not_zless
tff(fact_962_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),N)))
=> 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)),I)) ) ).
% less_mult_imp_div_less
tff(fact_963_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),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))),M)) ).
% times_div_less_eq_dividend
tff(fact_964_div__times__less__eq__dividend,axiom,
! [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),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)),M)) ).
% div_times_less_eq_dividend
tff(fact_965_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)))
=> ( ( K != zero_zero(int) )
=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),N)) ) ) ).
% zdvd_mult_cancel
tff(fact_966_zdvd__mono,axiom,
! [K: int,M: int,T2: int] :
( ( K != zero_zero(int) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),T2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),T2))) ) ) ).
% zdvd_mono
tff(fact_967_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_968_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_969_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),B2) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
tff(fact_970_euclidean__size__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_971_fact__dvd,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M))) ) ) ).
% fact_dvd
tff(fact_972_is__unit__division__segment,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,euclid7384307370059645450egment(A),A3)),one_one(A))) ) ).
% is_unit_division_segment
tff(fact_973_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K != zero_zero(int) )
=> ( aa(int,int,euclid7384307370059645450egment(int),K) = aa(int,int,sgn_sgn(int),K) ) ) ).
% division_segment_eq_sgn
tff(fact_974_ceiling__ge__round,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X))) ) ).
% ceiling_ge_round
tff(fact_975_frac__ge__0,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).
% frac_ge_0
tff(fact_976_frac__lt__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).
% frac_lt_1
tff(fact_977_nat__mono__iff,axiom,
! [Z2: int,W2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).
% nat_mono_iff
tff(fact_978_frac__le,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,W2: A,Z2: A] :
( 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),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).
% frac_le
tff(fact_979_frac__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A,W2: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).
% frac_less
tff(fact_980_frac__less2,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A,W2: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2))) ) ) ) ) ) ).
% frac_less2
tff(fact_981_divide__le__cancel,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(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),A3),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),A3)) ) ) ) ) ).
% divide_le_cancel
tff(fact_982_divide__nonneg__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),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),X),Y)),zero_zero(A))) ) ) ) ).
% divide_nonneg_neg
tff(fact_983_divide__nonneg__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> 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),X),Y))) ) ) ) ).
% divide_nonneg_pos
tff(fact_984_divide__nonpos__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(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),X),Y))) ) ) ) ).
% divide_nonpos_neg
tff(fact_985_divide__nonpos__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),zero_zero(A))) ) ) ) ).
% divide_nonpos_pos
tff(fact_986_divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: 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),divide_divide(A),B2),C2)),A3))
<=> ( ( 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(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),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(A),C2),zero_zero(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)),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(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% divide_less_eq
tff(fact_987_less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(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(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
& ( ~ 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(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq
tff(fact_988_neg__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: 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(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
<=> 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)),B2)) ) ) ) ).
% neg_divide_less_eq
tff(fact_989_neg__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: 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(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% neg_less_divide_eq
tff(fact_990_pos__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( 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(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% pos_divide_less_eq
tff(fact_991_pos__less__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( 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(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
<=> 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)),B2)) ) ) ) ).
% pos_less_divide_eq
tff(fact_992_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2)) ) ) ) ).
% mult_imp_div_pos_less
tff(fact_993_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).
% mult_imp_less_div_pos
tff(fact_994_divide__strict__left__mono,axiom,
! [A: $tType] :
( linordered_field(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),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(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),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).
% divide_strict_left_mono
tff(fact_995_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(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),C2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(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),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).
% divide_strict_left_mono_neg
tff(fact_996_unit__dvdE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_997_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_998_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_999_zless__nat__eq__int__zless,axiom,
! [M: nat,Z2: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(int,nat,nat2,Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z2)) ) ).
% zless_nat_eq_int_zless
tff(fact_1000_int__eq__iff,axiom,
! [M: nat,Z2: int] :
( ( aa(nat,int,semiring_1_of_nat(int),M) = Z2 )
<=> ( ( M = aa(int,nat,nat2,Z2) )
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) ) ) ).
% int_eq_iff
tff(fact_1001_nat__0__le,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z2)) = Z2 ) ) ).
% nat_0_le
tff(fact_1002_nat__le__iff,axiom,
! [X: int,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),N))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).
% nat_le_iff
tff(fact_1003_eq__minus__divide__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B2) ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_minus_divide_eq
tff(fact_1004_minus__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A3 )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% minus_divide_eq_eq
tff(fact_1005_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = C2 )
<=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
tff(fact_1006_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) )
<=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
tff(fact_1007_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_1008_mod__le__divisor,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_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)),N)) ) ).
% mod_le_divisor
tff(fact_1009_fact__mod,axiom,
! [A: $tType] :
( ( linordered_semidom(A)
& semidom_modulo(A) )
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).
% fact_mod
tff(fact_1010_mod__size__less,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2))),aa(A,nat,euclid6346220572633701492n_size(A),B2))) ) ) ).
% mod_size_less
tff(fact_1011_diff__Suc__less,axiom,
! [N: nat,I: 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),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I))),N)) ) ).
% diff_Suc_less
tff(fact_1012_div__greater__zero__iff,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),divide_divide(nat),M),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).
% div_greater_zero_iff
tff(fact_1013_div__le__mono2,axiom,
! [M: nat,N: nat,K: 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_eq(nat),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M))) ) ) ).
% div_le_mono2
tff(fact_1014_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_1015_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
=> ( 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),K),N)) ) ) ).
% dvd_imp_le
tff(fact_1016_nat__mult__div__cancel1,axiom,
! [K: nat,M: 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),divide_divide(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)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ).
% nat_mult_div_cancel1
tff(fact_1017_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q4)),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4))) ) ) ).
% div_less_iff_less_mult
tff(fact_1018_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: 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),dvd_dvd(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)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).
% nat_mult_dvd_cancel1
tff(fact_1019_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(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)))
=> ( 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),dvd_dvd(nat),M),N)) ) ) ).
% dvd_mult_cancel
tff(fact_1020_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_1021_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_1022_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_1023_euclidean__size__times__unit,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_1024_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z2),N))
=> ( 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),Z2),N)) ) ) ).
% zdvd_imp_le
tff(fact_1025_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2))) ) ) ) ) ).
% dvd_proper_imp_size_less
tff(fact_1026_dvd__imp__size__le,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( ( B2 != zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2))) ) ) ) ).
% dvd_imp_size_le
tff(fact_1027_ex__gt__or__lt,axiom,
! [A: $tType] :
( condit5016429287641298734tinuum(A)
=> ! [A3: A] :
? [B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B4))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),A3)) ) ) ).
% ex_gt_or_lt
tff(fact_1028_of__nat__euclidean__size,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] : aa(nat,A,semiring_1_of_nat(A),aa(A,nat,euclid6346220572633701492n_size(A),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,euclid7384307370059645450egment(A),A3)) ) ).
% of_nat_euclidean_size
tff(fact_1029_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(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),semiring_char_0_fact(nat,N))) ) ) ).
% dvd_fact
tff(fact_1030_zsgn__def,axiom,
! [I: int] :
( ( ( I = zero_zero(int) )
=> ( aa(int,int,sgn_sgn(int),I) = zero_zero(int) ) )
& ( ( I != zero_zero(int) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
=> ( aa(int,int,sgn_sgn(int),I) = one_one(int) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
=> ( aa(int,int,sgn_sgn(int),I) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).
% zsgn_def
tff(fact_1031_nat__less__eq__zless,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).
% nat_less_eq_zless
tff(fact_1032_nat__le__eq__zle,axiom,
! [W2: int,Z2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W2))
| pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ) ).
% nat_le_eq_zle
tff(fact_1033_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( aa(int,nat,nat2,W2) = M )
<=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
=> ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
=> ( M = zero_zero(nat) ) ) ) ) ).
% nat_eq_iff
tff(fact_1034_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M = aa(int,nat,nat2,W2) )
<=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
=> ( W2 = aa(nat,int,semiring_1_of_nat(int),M) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
=> ( M = zero_zero(nat) ) ) ) ) ).
% nat_eq_iff2
tff(fact_1035_divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: 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),divide_divide(A),B2),C2)),A3))
<=> ( ( 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),B2),aa(A,A,aa(A,fun(A,A),times_times(A),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(A),C2),zero_zero(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)),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),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% divide_le_eq
tff(fact_1036_le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(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),aa(A,A,aa(A,fun(A,A),times_times(A),A3),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
& ( ~ 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),A3),zero_zero(A))) ) ) ) ) ) ) ).
% le_divide_eq
tff(fact_1037_divide__left__mono,axiom,
! [A: $tType] :
( linordered_field(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),zero_zero(A)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).
% divide_left_mono
tff(fact_1038_neg__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: 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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),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)),B2)) ) ) ) ).
% neg_divide_le_eq
tff(fact_1039_neg__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: 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),A3),aa(A,A,aa(A,fun(A,A),divide_divide(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),times_times(A),A3),C2))) ) ) ) ).
% neg_le_divide_eq
tff(fact_1040_pos__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( 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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% pos_divide_le_eq
tff(fact_1041_pos__le__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
<=> 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)),B2)) ) ) ) ).
% pos_le_divide_eq
tff(fact_1042_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)),Z2)) ) ) ) ).
% mult_imp_div_pos_le
tff(fact_1043_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,Z2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y))) ) ) ) ).
% mult_imp_le_div_pos
tff(fact_1044_divide__left__mono__neg,axiom,
! [A: $tType] :
( linordered_field(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),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))) ) ) ) ) ).
% divide_left_mono_neg
tff(fact_1045_split__nat,axiom,
! [P: fun(nat,bool),I: int] :
( pp(aa(nat,bool,P,aa(int,nat,nat2,I)))
<=> ( ! [N5: nat] :
( ( I = aa(nat,int,semiring_1_of_nat(int),N5) )
=> pp(aa(nat,bool,P,N5)) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
=> pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).
% split_nat
tff(fact_1046_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_1047_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_1048_le__nat__iff,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(int,nat,nat2,K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ) ).
% le_nat_iff
tff(fact_1049_nat__mult__distrib,axiom,
! [Z2: int,Z5: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) ) ) ).
% nat_mult_distrib
tff(fact_1050_less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(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(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
& ( ~ 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(A),A3),zero_zero(A))) ) ) ) ) ) ) ).
% less_minus_divide_eq
tff(fact_1051_minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
<=> ( ( 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(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),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(A),C2),zero_zero(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)),aa(A,A,uminus_uminus(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(A),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% minus_divide_less_eq
tff(fact_1052_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: 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(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% neg_less_minus_divide_eq
tff(fact_1053_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: 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(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
<=> 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)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% neg_minus_divide_less_eq
tff(fact_1054_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( 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(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
<=> 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)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% pos_less_minus_divide_eq
tff(fact_1055_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( 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(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% pos_minus_divide_less_eq
tff(fact_1056_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)),M))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q4))))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = Q4 ) ) ) ).
% div_nat_eqI
tff(fact_1057_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q4))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q4)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),N)) ) ) ).
% less_eq_div_iff_mult_less_eq
tff(fact_1058_sgn__if,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( ( ( X = zero_zero(A) )
=> ( aa(A,A,sgn_sgn(A),X) = zero_zero(A) ) )
& ( ( X != zero_zero(A) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( aa(A,A,sgn_sgn(A),X) = one_one(A) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ( aa(A,A,sgn_sgn(A),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).
% sgn_if
tff(fact_1059_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_1060_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_1061_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_1062_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(aa(nat,bool,aa(nat,fun(nat,bool),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_1063_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(aa(nat,bool,aa(nat,fun(nat,bool),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_1064_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,euclid7384307370059645450egment(A),A3) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
| ( B2 = zero_zero(A) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_1065_nat__less__iff,axiom,
! [W2: int,M: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W2)),M))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).
% nat_less_iff
tff(fact_1066_le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(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),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) )
& ( ~ 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),A3),zero_zero(A))) ) ) ) ) ) ) ).
% le_minus_divide_eq
tff(fact_1067_minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: 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,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
<=> ( ( 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),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),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(A),C2),zero_zero(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)),aa(A,A,uminus_uminus(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),zero_zero(A)),A3)) ) ) ) ) ) ) ).
% minus_divide_le_eq
tff(fact_1068_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: 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),A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% neg_le_minus_divide_eq
tff(fact_1069_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: 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),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),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)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% neg_minus_divide_le_eq
tff(fact_1070_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))))
<=> 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)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).
% pos_le_minus_divide_eq
tff(fact_1071_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,B2: A,A3: A] :
( 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),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))) ) ) ) ).
% pos_minus_divide_le_eq
tff(fact_1072_nat__mult__distrib__neg,axiom,
! [Z2: int,Z5: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int)))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z2))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z5))) ) ) ).
% nat_mult_distrib_neg
tff(fact_1073_nat__approx__posE,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [E: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
=> ~ ! [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)))),E)) ) ) ).
% nat_approx_posE
tff(fact_1074_mult__ceiling__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(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(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B2)))) ) ) ) ).
% mult_ceiling_le
tff(fact_1075_bits__mod__div__trivial,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),B2) = zero_zero(A) ) ).
% bits_mod_div_trivial
tff(fact_1076_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_1077_choose__dvd,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),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,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),semiring_char_0_fact(A,N))) ) ) ).
% choose_dvd
tff(fact_1078_div__geq,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),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N)) ) ) ) ).
% div_geq
tff(fact_1079_bits__mod__0,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% bits_mod_0
tff(fact_1080_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_1081_bits__div__by__0,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% bits_div_by_0
tff(fact_1082_bits__div__0,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% bits_div_0
tff(fact_1083_unique__euclidean__semiring__numeral__class_Omod__less,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),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_1084_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( 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),zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2))) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_1085_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).
% mod_neg_neg_trivial
tff(fact_1086_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).
% mod_pos_pos_trivial
tff(fact_1087_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).
% div_neg_neg_trivial
tff(fact_1088_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).
% div_pos_pos_trivial
tff(fact_1089_zle__diff1__eq,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),one_one(int))))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ).
% zle_diff1_eq
tff(fact_1090_dvd__antisym,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M))
=> ( M = N ) ) ) ).
% dvd_antisym
tff(fact_1091_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),M)) ) ) ).
% zdvd_zdiffD
tff(fact_1092_zmod__zminus1__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A3)),B2) = zero_zero(int) ) )
& ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A3)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2)) ) ) ) ).
% zmod_zminus1_eq_if
tff(fact_1093_zmod__zminus2__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,uminus_uminus(int),B2)) = zero_zero(int) ) )
& ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2)),B2) ) ) ) ).
% zmod_zminus2_eq_if
tff(fact_1094_zdiv__zminus1__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2 != zero_zero(int) )
=> ( ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B2) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) )
& ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),one_one(int)) ) ) ) ) ).
% zdiv_zminus1_eq_if
tff(fact_1095_zdiv__zminus2__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2 != zero_zero(int) )
=> ( ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) )
& ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2) != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))),one_one(int)) ) ) ) ) ).
% zdiv_zminus2_eq_if
tff(fact_1096_zdiv__mono__strict,axiom,
! [A5: int,B5: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A5),B5))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
=> ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A5),N) = zero_zero(int) )
=> ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B5),N) = zero_zero(int) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N))) ) ) ) ) ).
% zdiv_mono_strict
tff(fact_1097_mod__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),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L) ) ) ) ).
% mod_pos_geq
tff(fact_1098_sgn__mod,axiom,
! [L: int,K: int] :
( ( L != zero_zero(int) )
=> ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
=> ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).
% sgn_mod
tff(fact_1099_minus__int__code_I1_J,axiom,
! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).
% minus_int_code(1)
tff(fact_1100_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W2)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W2)) ).
% int_distrib(3)
tff(fact_1101_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z22)) ).
% int_distrib(4)
tff(fact_1102_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M5: nat,N3: nat] : Z2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M5)),aa(nat,int,semiring_1_of_nat(int),N3)) ).
% int_diff_cases
tff(fact_1103_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),K)),M)) ) ).
% zmod_le_nonneg_dividend
tff(fact_1104_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
<=> ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).
% zmod_eq_0_iff
tff(fact_1105_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
=> ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q5) ) ).
% zmod_eq_0D
tff(fact_1106_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).
% zmod_zminus2_not_zero
tff(fact_1107_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).
% zmod_zminus1_not_zero
tff(fact_1108_div__neg__pos__less0,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).
% div_neg_pos_less0
tff(fact_1109_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).
% neg_imp_zdiv_neg_iff
tff(fact_1110_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).
% pos_imp_zdiv_neg_iff
tff(fact_1111_zmod__int,axiom,
! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% zmod_int
tff(fact_1112_zdiv__int,axiom,
! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% zdiv_int
tff(fact_1113_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))),L)) ) ) ).
% minus_mod_int_eq
tff(fact_1114_zmod__minus1,axiom,
! [B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).
% zmod_minus1
tff(fact_1115_neg__mod__conj,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2)),zero_zero(int)))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2))) ) ) ).
% neg_mod_conj
tff(fact_1116_pos__mod__conj,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2)))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B2)),B2)) ) ) ).
% pos_mod_conj
tff(fact_1117_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),K) = I )
<=> ( ( K = zero_zero(int) )
| ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K)) )
| ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).
% zmod_trivial_iff
tff(fact_1118_mod__int__pos__iff,axiom,
! [K: int,L: 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),modulo_modulo(int),K),L)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
| ( ( L = zero_zero(int) )
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) )
| pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).
% mod_int_pos_iff
tff(fact_1119_neg__mod__bound,axiom,
! [L: int,K: int] :
( 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),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))) ) ).
% neg_mod_bound
tff(fact_1120_Euclidean__Division_Opos__mod__bound,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(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),L)) ) ).
% Euclidean_Division.pos_mod_bound
tff(fact_1121_zdiv__mono1,axiom,
! [A3: int,A4: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2))) ) ) ).
% zdiv_mono1
tff(fact_1122_zdiv__mono2,axiom,
! [A3: int,B3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))) ) ) ) ).
% zdiv_mono2
tff(fact_1123_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
<=> ( ( K = zero_zero(int) )
| ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K)) )
| ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int)))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I)) ) ) ) ).
% zdiv_eq_0_iff
tff(fact_1124_zdiv__mono1__neg,axiom,
! [A3: int,A4: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),A4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))) ) ) ).
% zdiv_mono1_neg
tff(fact_1125_zdiv__mono2__neg,axiom,
! [A3: int,B3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2))) ) ) ) ).
% zdiv_mono2_neg
tff(fact_1126_div__int__pos__iff,axiom,
! [K: int,L: 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),divide_divide(int),K),L)))
<=> ( ( K = zero_zero(int) )
| ( L = zero_zero(int) )
| ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
| ( 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(int),L),zero_zero(int))) ) ) ) ).
% div_int_pos_iff
tff(fact_1127_div__positive__int,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
=> ( 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),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))) ) ) ).
% div_positive_int
tff(fact_1128_div__nonneg__neg__le0,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).
% div_nonneg_neg_le0
tff(fact_1129_div__nonpos__pos__le0,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),zero_zero(int))) ) ) ).
% div_nonpos_pos_le0
tff(fact_1130_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: 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(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I)) ) ) ).
% pos_imp_zdiv_pos_iff
tff(fact_1131_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(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),divide_divide(int),A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).
% neg_imp_zdiv_nonneg_iff
tff(fact_1132_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).
% pos_imp_zdiv_nonneg_iff
tff(fact_1133_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
tff(fact_1134_int__div__less__self,axiom,
! [X: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),K)),X)) ) ) ).
% int_div_less_self
tff(fact_1135_minus__int__code_I2_J,axiom,
! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).
% minus_int_code(2)
tff(fact_1136_int__le__induct,axiom,
! [I: int,K: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),K))
=> ( pp(aa(int,bool,P,K))
=> ( ! [I4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I4),K))
=> ( pp(aa(int,bool,P,I4))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I4),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_le_induct
tff(fact_1137_int__less__induct,axiom,
! [I: int,K: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K))
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))))
=> ( ! [I4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I4),K))
=> ( pp(aa(int,bool,P,I4))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I4),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_less_induct
tff(fact_1138_div__eq__minus1,axiom,
! [B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).
% div_eq_minus1
tff(fact_1139_neg__mod__sign,axiom,
! [L: int,K: int] :
( 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_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),zero_zero(int))) ) ).
% neg_mod_sign
tff(fact_1140_Euclidean__Division_Opos__mod__sign,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),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))) ) ).
% Euclidean_Division.pos_mod_sign
tff(fact_1141_plusinfinity,axiom,
! [D2: int,P2: fun(int,bool),P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P2,X2))
<=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
=> ( ? [Z4: int] :
! [X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X2))
=> ( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P2,X2)) ) )
=> ( ? [X_1: int] : pp(aa(int,bool,P2,X_1))
=> ? [X_12: int] : pp(aa(int,bool,P,X_12)) ) ) ) ) ).
% plusinfinity
tff(fact_1142_minusinfinity,axiom,
! [D2: int,P1: fun(int,bool),P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P1,X2))
<=> pp(aa(int,bool,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
=> ( ? [Z4: int] :
! [X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Z4))
=> ( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P1,X2)) ) )
=> ( ? [X_1: int] : pp(aa(int,bool,P1,X_1))
=> ? [X_12: int] : pp(aa(int,bool,P,X_12)) ) ) ) ) ).
% minusinfinity
tff(fact_1143_zdiv__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),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),C2) ) ) ).
% zdiv_zmult2_eq
tff(fact_1144_int__minus,axiom,
! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)))) ).
% int_minus
tff(fact_1145_nat__mod__distrib,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).
% nat_mod_distrib
tff(fact_1146_decr__mult__lemma,axiom,
! [D2: int,P: fun(int,bool),K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ! [X2: int] :
( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D2))) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [X3: int] :
( pp(aa(int,bool,P,X3))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).
% decr_mult_lemma
tff(fact_1147_int__ops_I6_J,axiom,
! [A3: nat,B2: nat] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
=> ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = zero_zero(int) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)))
=> ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ) ) ).
% int_ops(6)
tff(fact_1148_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z5))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z5),Z2))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) ) ) ) ).
% nat_diff_distrib
tff(fact_1149_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ) ).
% nat_diff_distrib'
tff(fact_1150_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A5),B5))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N)))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N))) ) ) ).
% verit_less_mono_div_int2
tff(fact_1151_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).
% nat_div_distrib'
tff(fact_1152_nat__div__distrib,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y)) ) ) ).
% nat_div_distrib
tff(fact_1153_zdiff__int__split,axiom,
! [P: fun(int,bool),X: nat,Y: nat] :
( pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Y))))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
& ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).
% zdiff_int_split
tff(fact_1154_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,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_eq(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),A3)) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_1155_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( 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),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),B2)) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_1156_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D2) = zero_zero(nat) )
=> ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q5) ) ).
% mod_eq_0D
tff(fact_1157_mod__geq,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),modulo_modulo(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) ) ) ).
% mod_geq
tff(fact_1158_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q4) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ) ).
% mod_eq_dvd_iff_nat
tff(fact_1159_div__positive,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( 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),A3))
=> 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),A3),B2))) ) ) ) ).
% div_positive
tff(fact_1160_unique__euclidean__semiring__numeral__class_Odiv__less,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),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1161_unique__euclidean__semiring__numeral__class_Odiv__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),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) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1162_bezout1__nat,axiom,
! [A3: nat,B2: nat] :
? [D5: nat,X2: nat,Y4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),A3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),B2))
& ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = D5 )
| ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)) = D5 ) ) ) ).
% bezout1_nat
tff(fact_1163_gcd__nat__induct,axiom,
! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
( ! [M5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),zero_zero(nat)))
=> ( ! [M5: nat,N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N3),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M5),N3)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),N3)) ) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ).
% gcd_nat_induct
tff(fact_1164_dvd__pos__nat,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),dvd_dvd(nat),M),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).
% dvd_pos_nat
tff(fact_1165_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_1166_modulo__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),modulo_modulo(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,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,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),modulo_modulo(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,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(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),modulo_modulo(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,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)))) ) ) ) ) ) ).
% modulo_int_unfold
tff(fact_1167_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) )
=> ! [Q5: A,R3: A] :
( ( aa(A,A,euclid7384307370059645450egment(A),R3) = aa(A,A,euclid7384307370059645450egment(A),B2) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
=> ( ( R3 != zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q5 )
=> ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = R3 )
=> ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B2)),R3) ) ) ) ) ) ) )
=> ( B2 = zero_zero(A) ) ) ) ) ).
% divmod_cases
tff(fact_1168_gcd__nat_Oextremum,axiom,
! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat))) ).
% gcd_nat.extremum
tff(fact_1169_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_1170_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_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_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_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
tff(fact_1187_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A,Y: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
tff(fact_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% dvd_add_triv_left_iff
tff(fact_1201_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% dvd_add_triv_right_iff
tff(fact_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [P: bool,Q: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
<=> ( pp(P)
=> pp(Q) ) ) ) ).
% of_bool_less_eq_iff
tff(fact_1211_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P: bool] :
( ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) )
<=> ~ pp(P) ) ) ).
% of_bool_eq_0_iff
tff(fact_1212_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).
% of_bool_eq(1)
tff(fact_1213_of__bool__less__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [P: bool,Q: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
<=> ( ~ pp(P)
& pp(Q) ) ) ) ).
% of_bool_less_iff
tff(fact_1214_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_1215_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_1216_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_1217_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_1218_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_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% dvd_add_times_triv_right_iff
tff(fact_1235_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% dvd_add_times_triv_left_iff
tff(fact_1236_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_1237_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_1238_div__add,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),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_1239_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_1240_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_1241_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_1242_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_1243_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P)))
<=> pp(P) ) ) ).
% zero_less_of_bool_iff
tff(fact_1244_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_1245_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_1246_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_1247_of__bool__not__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),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_1248_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( semiring_char_0(B)
& semidom_divide(B) )
=> ! [K: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K)) = zero_zero(B) ) ).
% gbinomial_0(2)
tff(fact_1249_zle__add1__eq__le,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),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),W2),Z2)) ) ).
% zle_add1_eq_le
tff(fact_1250_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_1251_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_1252_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_1253_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_1254_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_1255_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).
% gbinomial_Suc0
tff(fact_1256_of__nat__of__bool,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [P: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).
% of_nat_of_bool
tff(fact_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_sgn__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).
% sgn_mult_self_eq
tff(fact_1265_ceiling__add__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).
% ceiling_add_one
tff(fact_1266_Suc__0__mod__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(bool,nat,zero_neq_one_of_bool(nat),fNot(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),aa(nat,nat,suc,zero_zero(nat))))) ).
% Suc_0_mod_eq
tff(fact_1267_sgn__of__nat,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),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)) ) ).
% sgn_of_nat
tff(fact_1268_of__nat__gbinomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,gbinomial(nat,N),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) ) ).
% of_nat_gbinomial
tff(fact_1269_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_1270_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_1271_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_1272_Euclid__induct,axiom,
! [P: fun(nat,fun(nat,bool)),A3: nat,B2: nat] :
( ! [A6: nat,B4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),B4))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B4),A6)) )
=> ( ! [A6: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),zero_zero(nat)))
=> ( ! [A6: nat,B4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),B4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A6),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),B4))) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A3),B2)) ) ) ) ).
% Euclid_induct
tff(fact_1273_of__bool__eq__iff,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ! [P3: bool,Q4: bool] :
( ( aa(bool,A,zero_neq_one_of_bool(A),P3) = aa(bool,A,zero_neq_one_of_bool(A),Q4) )
<=> ( pp(P3)
<=> pp(Q4) ) ) ) ).
% of_bool_eq_iff
tff(fact_1274_ab__semigroup__add__class_Oadd__ac_I1_J,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)) ) ).
% ab_semigroup_add_class.add_ac(1)
tff(fact_1275_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_1276_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_1277_group__cancel_Oadd1,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),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_1278_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_1279_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_1280_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_1281_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_1282_add_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) ) ).
% add.commute
tff(fact_1283_add_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)) ) ).
% add.left_commute
tff(fact_1284_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_1285_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_1286_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_1287_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_1288_ceiling__add__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))) ) ).
% ceiling_add_le
tff(fact_1289_of__bool__conj,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).
% of_bool_conj
tff(fact_1290_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_1291_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_1292_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_1293_add__mono,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> 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),D2))) ) ) ) ).
% add_mono
tff(fact_1294_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_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_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_1306_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_1307_add__strict__mono,axiom,
! [A: $tType] :
( strict9044650504122735259up_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> 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),D2))) ) ) ) ).
% add_strict_mono
tff(fact_1308_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_1309_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_1310_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_1311_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_1312_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [A3: A,E: 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),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),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)),E)),C2) ) ).
% combine_common_factor
tff(fact_1313_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_1314_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_1315_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_1316_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_1317_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_1318_group__cancel_Osub1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A5),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_1319_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_1320_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_1321_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_1322_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_1323_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_1324_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_1325_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_1326_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_1327_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_1328_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A5: A,K: A,A3: A] :
( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
=> ( aa(A,A,uminus_uminus(A),A5) = 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_1329_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_1330_dvd__add,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ) ).
% dvd_add
tff(fact_1331_dvd__add__left__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).
% dvd_add_left_iff
tff(fact_1332_dvd__add__right__iff,axiom,
! [A: $tType] :
( comm_s4317794764714335236cancel(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).
% dvd_add_right_iff
tff(fact_1333_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_1334_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A3: nat] :
( ( A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A3) )
=> ( aa(nat,nat,suc,A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A3)) ) ) ).
% nat_arith.suc1
tff(fact_1335_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_1336_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_1337_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_1338_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_1339_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_1340_mod__add__cong,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,C2: A,A4: A,B2: A,B3: 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),A4),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),B3),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),A4),B3)),C2) ) ) ) ) ).
% mod_add_cong
tff(fact_1341_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_1342_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_1343_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_1344_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_1345_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_1346_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_1347_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_1348_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_1349_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_1350_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_1351_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_1352_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_1353_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_1354_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_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_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_1362_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_1363_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_1364_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_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W2),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),W2),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z22)) ).
% int_distrib(2)
tff(fact_1372_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: 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)),W2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W2)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W2)) ).
% int_distrib(1)
tff(fact_1373_div__mod__decomp__int,axiom,
! [A5: int,N: int] : A5 = 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),A5),N)),N)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A5),N)) ).
% div_mod_decomp_int
tff(fact_1374_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_1375_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_1376_bezout__lemma__nat,axiom,
! [D2: nat,A3: nat,B2: nat,X: nat,Y: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
=> ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y)),D2) ) )
=> ? [X2: nat,Y4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),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),X2) = 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)),D2) )
| ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),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),Y4)),D2) ) ) ) ) ) ) ).
% bezout_lemma_nat
tff(fact_1377_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D5: nat,X2: nat,Y4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),A3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),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),Y4)),D5) )
| ( 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),Y4)),D5) ) ) ) ).
% bezout_add_nat
tff(fact_1378_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_1379_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_1380_nat__add__distrib,axiom,
! [Z2: int,Z5: 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)),Z5))
=> ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) ) ) ) ).
% nat_add_distrib
tff(fact_1381_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_1382_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P))) ) ).
% zero_less_eq_of_bool
tff(fact_1383_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_1384_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_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( 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),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
tff(fact_1394_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),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),X),Y) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
tff(fact_1395_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_1396_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_1397_add__le__less__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> 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),D2))) ) ) ) ).
% add_le_less_mono
tff(fact_1398_add__less__le__mono,axiom,
! [A: $tType] :
( ordere580206878836729694up_add(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> 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),D2))) ) ) ) ).
% add_less_le_mono
tff(fact_1399_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_1400_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_1401_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_1402_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_1403_add__less__zeroD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),Y)),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).
% add_less_zeroD
tff(fact_1404_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_1405_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_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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_1412_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_1413_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_1414_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_1415_diff__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 ) ) ) ).
% diff_add
tff(fact_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_square__diff__square__factored,axiom,
! [A: $tType] :
( comm_ring(A)
=> ! [X: 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),X),X)),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),X),Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ).
% square_diff_square_factored
tff(fact_1429_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2) )
<=> ( 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)),E)),D2) ) ) ) ).
% eq_add_iff2
tff(fact_1430_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2) )
<=> ( 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)),E)),C2) = D2 ) ) ) ).
% eq_add_iff1
tff(fact_1431_pinf_I9_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X3))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).
% pinf(9)
tff(fact_1432_pinf_I10_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z3),X3))
=> ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
<=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).
% pinf(10)
tff(fact_1433_minf_I9_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Z3))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
<=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).
% minf(9)
tff(fact_1434_minf_I10_J,axiom,
! [B: $tType] :
( ( plus(B)
& linorder(B)
& dvd(B) )
=> ! [D2: B,S: B] :
? [Z3: B] :
! [X3: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Z3))
=> ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S)))
<=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D2),aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),S))) ) ) ) ).
% minf(10)
tff(fact_1435_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_1436_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_1437_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_1438_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_1439_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_1440_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) )
=> ~ ! [D5: 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),D5)) ) ) ).
% mod_eqE
tff(fact_1441_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_1442_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_1443_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_1444_less__natE,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ~ ! [Q5: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ) ).
% less_natE
tff(fact_1445_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_1446_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_1447_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_1448_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).
% less_imp_Suc_add
tff(fact_1449_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ? [K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
& ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).
% less_imp_add_positive
tff(fact_1450_mono__nat__linear__lb,axiom,
! [F: fun(nat,nat),M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F,M5)),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_1451_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))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ? [Y3: A] :
( pp(aa(A,bool,P,Y3))
& ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,Y3)),aa(A,nat,F,X2))) ) )
=> ? [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_1452_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero(nat) )
=> ? [D5: nat,X2: nat,Y4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),A3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D5),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),Y4)),D5) ) ) ) ).
% bezout_add_strong_nat
tff(fact_1453_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_1454_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_1455_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_1456_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_1457_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_1458_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_1459_plus__1__eq__Suc,axiom,
aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).
% plus_1_eq_Suc
tff(fact_1460_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_1461_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_1462_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_1463_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_1464_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_1465_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_1466_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))
=> ( ! [I4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I4))
=> ( pp(aa(int,bool,P,I4))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_ge_induct
tff(fact_1467_zless__add1__eq,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),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),W2),Z2))
| ( W2 = Z2 ) ) ) ).
% zless_add1_eq
tff(fact_1468_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))))
=> ( ! [I4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I4))
=> ( pp(aa(int,bool,P,I4))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_gr_induct
tff(fact_1469_zle__iff__zadd,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2))
<=> ? [N5: nat] : Z2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N5)) ) ).
% zle_iff_zadd
tff(fact_1470_zdvd__period,axiom,
! [A3: int,D2: int,X: int,T2: int,C2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),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),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),T2))) ) ) ).
% zdvd_period
tff(fact_1471_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),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(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N)) ) ).
% zdvd_reduce
tff(fact_1472_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),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),X),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_1473_frac__1__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).
% frac_1_eq
tff(fact_1474_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_1475_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_1476_dbl__inc__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).
% dbl_inc_def
tff(fact_1477_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_1478_field__le__epsilon,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A] :
( ! [E2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% field_le_epsilon
tff(fact_1479_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_1480_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_1481_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_1482_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_1483_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_1484_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_1485_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: 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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).
% sum_squares_ge_zero
tff(fact_1486_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_1487_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: 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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).
% not_sum_squares_lt_zero
tff(fact_1488_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_1489_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E: A,C2: A,B2: A,D2: 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),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2)))
<=> 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)),E)),D2))) ) ) ).
% le_add_iff2
tff(fact_1490_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E: A,C2: A,B2: A,D2: 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),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2)))
<=> 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)),E)),C2)),D2)) ) ) ).
% le_add_iff1
tff(fact_1491_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E: A,C2: A,B2: A,D2: 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),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2)))
<=> 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)),E)),D2))) ) ) ).
% less_add_iff2
tff(fact_1492_less__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [A3: A,E: A,C2: A,B2: A,D2: 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),E)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E)),D2)))
<=> 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)),E)),C2)),D2)) ) ) ).
% less_add_iff1
tff(fact_1493_unity__coeff__ex,axiom,
! [A: $tType] :
( ( dvd(A)
& semiring_0(A) )
=> ! [P: fun(A,bool),L: A] :
( ? [X5: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X5)))
<=> ? [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),zero_zero(A))))
& pp(aa(A,bool,P,X5)) ) ) ) ).
% unity_coeff_ex
tff(fact_1494_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_1495_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_1496_add__frac__eq,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: A,W2: 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),X),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),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),X),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2)) ) ) ) ) ).
% add_frac_eq
tff(fact_1497_add__frac__num,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,X: 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),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).
% add_frac_num
tff(fact_1498_add__num__frac,axiom,
! [A: $tType] :
( field(A)
=> ! [Y: A,Z2: A,X: 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),X),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Y))),Y) ) ) ) ).
% add_num_frac
tff(fact_1499_add__divide__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: A,Y: A] :
( ( Z2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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),X),Z2)),Y)),Z2) ) ) ) ).
% add_divide_eq_iff
tff(fact_1500_divide__add__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: 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),X),Z2)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).
% divide_add_eq_iff
tff(fact_1501_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_1502_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_1503_square__diff__one__factored,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),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),X),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ).
% square_diff_one_factored
tff(fact_1504_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_1505_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_1506_inf__period_I4_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D2: A,D4: A,T2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),D4))
=> ! [X3: A,K3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T2)))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))),T2))) ) ) ) ).
% inf_period(4)
tff(fact_1507_inf__period_I3_J,axiom,
! [A: $tType] :
( ( comm_ring(A)
& dvd(A) )
=> ! [D2: A,D4: A,T2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),D4))
=> ! [X3: A,K3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),T2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D4))),T2))) ) ) ) ).
% inf_period(3)
tff(fact_1508_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).
% gbinomial_of_nat_symmetric
tff(fact_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_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))) )
| ? [D6: nat] :
( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
& ~ pp(aa(nat,bool,P,D6)) ) ) ) ).
% nat_diff_split_asm
tff(fact_1518_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))) )
& ! [D6: nat] :
( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
=> pp(aa(nat,bool,P,D6)) ) ) ) ).
% nat_diff_split
tff(fact_1519_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_1520_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_1521_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_1522_zless__iff__Suc__zadd,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2))
<=> ? [N5: nat] : Z2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N5))) ) ).
% zless_iff_Suc_zadd
tff(fact_1523_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_1524_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_1525_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_1526_zless__imp__add1__zle,axiom,
! [W2: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),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),W2),one_one(int))),Z2)) ) ).
% zless_imp_add1_zle
tff(fact_1527_add1__zle__eq,axiom,
! [W2: 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),W2),one_one(int))),Z2))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ).
% add1_zle_eq
tff(fact_1528_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_1529_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_1530_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_1531_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_1532_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_1533_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_1534_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: nat,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),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_1535_int__induct,axiom,
! [P: fun(int,bool),K: int,I: int] :
( pp(aa(int,bool,P,K))
=> ( ! [I4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I4))
=> ( pp(aa(int,bool,P,I4))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int)))) ) )
=> ( ! [I4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I4),K))
=> ( pp(aa(int,bool,P,I4))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I4),one_one(int)))) ) )
=> pp(aa(int,bool,P,I)) ) ) ) ).
% int_induct
tff(fact_1536_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q4) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ~ ! [S2: 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),Q4),S2)) ) ) ).
% mod_eq_nat1E
tff(fact_1537_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q4) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ~ ! [S2: 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),Q4),S2)) ) ) ).
% mod_eq_nat2E
tff(fact_1538_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),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),X))
=> ? [Q5: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)) ) ) ).
% nat_mod_eq_lemma
tff(fact_1539_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q4: 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),Q4)) = 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)),Q4))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) ).
% mod_mult2_eq
tff(fact_1540_div__mod__decomp,axiom,
! [A5: nat,N: nat] : A5 = 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),A5),N)),N)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A5),N)) ).
% div_mod_decomp
tff(fact_1541_dbl__dec__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).
% dbl_dec_def
tff(fact_1542_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_1543_convex__bound__le,axiom,
! [A: $tType] :
( linord6961819062388156250ring_1(A)
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_le
tff(fact_1544_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),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))) ) ) ) ) ) ).
% divide_int_unfold
tff(fact_1545_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_1546_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Z2: A,X: 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),X),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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z2))),Z2) ) ) ) ).
% minus_divide_add_eq_iff
tff(fact_1547_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_1548_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_1549_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_1550_verit__le__mono__div__int,axiom,
! [A5: int,B5: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A5),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),A5),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_1551_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_1552_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_1553_incr__mult__lemma,axiom,
! [D2: int,P: fun(int,bool),K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ! [X2: int] :
( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2))) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ! [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),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).
% incr_mult_lemma
tff(fact_1554_unique__quotient__lemma__neg,axiom,
! [B2: int,Q6: int,R4: int,Q4: 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),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),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),R4))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q6)) ) ) ) ) ).
% unique_quotient_lemma_neg
tff(fact_1555_unique__quotient__lemma,axiom,
! [B2: int,Q6: int,R4: int,Q4: 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),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),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),Q6),Q4)) ) ) ) ) ).
% unique_quotient_lemma
tff(fact_1556_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q4: int,R: int,B3: int,Q6: int,R4: 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)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4) )
=> ( 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),B3),Q6)),R4)),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)),R4))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q6),Q4)) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
tff(fact_1557_zdiv__mono2__lemma,axiom,
! [B2: int,Q4: int,R: int,B3: int,Q6: int,R4: 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)),R) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4) )
=> ( 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),B3),Q6)),R4)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B3))
=> ( 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)),B3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B3),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q6)) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
tff(fact_1558_q__pos__lemma,axiom,
! [B3: int,Q6: int,R4: 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),B3),Q6)),R4)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q6)) ) ) ) ).
% q_pos_lemma
tff(fact_1559_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q4: 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),Q4)),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) = Q4 ) ) ) ) ).
% int_div_pos_eq
tff(fact_1560_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q4: 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),Q4)),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) = Q4 ) ) ) ) ).
% int_div_neg_eq
tff(fact_1561_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))
=> ! [I5: int,J3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),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),I5)),J3) ) )
=> pp(aa(int,bool,P,I5)) ) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ! [I5: int,J3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),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),I5)),J3) ) )
=> pp(aa(int,bool,P,I5)) ) ) ) ) ).
% split_zdiv
tff(fact_1562_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q4: 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),Q4)),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_1563_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q4: 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),Q4)),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_1564_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))
=> ! [I5: int,J3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),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),I5)),J3) ) )
=> pp(aa(int,bool,P,J3)) ) )
& ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ! [I5: int,J3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),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),I5)),J3) ) )
=> pp(aa(int,bool,P,J3)) ) ) ) ) ).
% split_zmod
tff(fact_1565_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)))
<=> ! [I5: int,J3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),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),I5)),J3) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I5),J3)) ) ) ) ).
% split_pos_lemma
tff(fact_1566_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)))
<=> ! [I5: int,J3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),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),I5)),J3) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I5),J3)) ) ) ) ).
% split_neg_lemma
tff(fact_1567_frac__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),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,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).
% frac_add
tff(fact_1568_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_1569_Suc__as__int,axiom,
! [X3: nat] : aa(nat,nat,suc,X3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).
% Suc_as_int
tff(fact_1570_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) )
=> ! [I5: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),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),I5)),J3) )
=> pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).
% split_mod
tff(fact_1571_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_1572_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_1573_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) )
=> ! [I5: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),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),I5)),J3) )
=> pp(aa(nat,bool,P,I5)) ) ) ) ) ) ).
% split_div
tff(fact_1574_dvd__minus__add,axiom,
! [Q4: nat,N: nat,R: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q4),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),Q4)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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)),Q4)))) ) ) ) ).
% dvd_minus_add
tff(fact_1575_convex__bound__lt,axiom,
! [A: $tType] :
( linord715952674999750819strict(A)
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3)) ) ) ) ) ) ) ).
% convex_bound_lt
tff(fact_1576_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_1577_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_1578_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,M: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).
% gbinomial_trinomial_revision
tff(fact_1579_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_1580_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_1581_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_1582_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_1583_div__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R: A,Q4: 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),Q4),B2)),R) = A3 )
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q4 ) ) ) ) ) ) ).
% div_eqI
tff(fact_1584_div__bounded,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R: A,Q4: 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),Q4),B2)),R)),B2) = Q4 ) ) ) ) ) ).
% div_bounded
tff(fact_1585_mod__eqI,axiom,
! [A: $tType] :
( euclid3128863361964157862miring(A)
=> ! [B2: A,R: A,Q4: 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),Q4),B2)),R) = A3 )
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) = R ) ) ) ) ) ) ).
% mod_eqI
tff(fact_1586_verit__le__mono__div,axiom,
! [A5: nat,B5: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A5),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),A5),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_1587_division__decomp,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
=> ? [B6: A,C5: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),C5) )
& pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B6),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C5),C2)) ) ) ) ).
% division_decomp
tff(fact_1588_dvd__productE,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [P3: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
=> ~ ! [X2: A,Y4: A] :
( ( P3 = aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y4) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X2),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y4),B2)) ) ) ) ) ).
% dvd_productE
tff(fact_1589_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
=> ( A3 = zero_zero(nat) ) ) ).
% gcd_nat.extremum_uniqueI
tff(fact_1590_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero(nat) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat)))
& ( A3 != zero_zero(nat) ) ) ) ).
% gcd_nat.not_eq_extremum
tff(fact_1591_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
<=> ( A3 = zero_zero(nat) ) ) ).
% gcd_nat.extremum_unique
tff(fact_1592_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
& ( zero_zero(nat) != A3 ) ) ).
% gcd_nat.extremum_strict
tff(fact_1593_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: 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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_squares_gt_zero_iff
tff(fact_1594_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: 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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_le_zero_iff
tff(fact_1595_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( linord4710134922213307826strict(A)
=> ! [X: 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),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_squares_eq_zero_iff
tff(fact_1596_add__scale__eq__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [R: A,A3: A,B2: A,C2: A,D2: A] :
( ( R != zero_zero(A) )
=> ( ( ( A3 = B2 )
& ( C2 != D2 ) )
=> ( 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),D2)) ) ) ) ) ).
% add_scale_eq_noteq
tff(fact_1597_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),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K)))))) ) ) ) ) ) ).
% divide_int_def
tff(fact_1598_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_1599_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_1600_abs__abs,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).
% abs_abs
tff(fact_1601_abs__0,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).
% abs_0
tff(fact_1602_abs__zero,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).
% abs_zero
tff(fact_1603_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_1604_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_1605_abs__mult__self__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% abs_mult_self_eq
tff(fact_1606_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_1607_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_1608_abs__minus,axiom,
! [A: $tType] :
( idom_abs_sgn(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
tff(fact_1609_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_1610_dvd__abs__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: A,K: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),aa(A,A,abs_abs(A),K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),K)) ) ) ).
% dvd_abs_iff
tff(fact_1611_abs__dvd__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: A,K: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,abs_abs(A),M)),K))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),K)) ) ) ).
% abs_dvd_iff
tff(fact_1612_abs__divide,axiom,
! [A: $tType] :
( field_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_divide
tff(fact_1613_abs__of__nat,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).
% abs_of_nat
tff(fact_1614_abs__bool__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P: bool] : aa(A,A,abs_abs(A),aa(bool,A,zero_neq_one_of_bool(A),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).
% abs_bool_eq
tff(fact_1615_pochhammer__1,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).
% pochhammer_1
tff(fact_1616_nat__of__bool,axiom,
! [P: bool] : aa(int,nat,nat2,aa(bool,int,zero_neq_one_of_bool(int),P)) = aa(bool,nat,zero_neq_one_of_bool(nat),P) ).
% nat_of_bool
tff(fact_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_pochhammer__Suc0,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).
% pochhammer_Suc0
tff(fact_1624_zdvd1__eq,axiom,
! [X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),one_one(int)))
<=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).
% zdvd1_eq
tff(fact_1625_abs__division__segment,axiom,
! [K: int] : aa(int,int,abs_abs(int),aa(int,int,euclid7384307370059645450egment(int),K)) = one_one(int) ).
% abs_division_segment
tff(fact_1626_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_1627_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
| ( B2 = zero_zero(A) ) ) ) ) ).
% zero_le_divide_abs_iff
tff(fact_1628_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( linordered_field(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),divide_divide(A),A3),aa(A,A,abs_abs(A),B2))),zero_zero(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A)))
| ( B2 = zero_zero(A) ) ) ) ) ).
% divide_le_0_abs_iff
tff(fact_1629_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_1630_zabs__less__one__iff,axiom,
! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z2)),one_one(int)))
<=> ( Z2 = zero_zero(int) ) ) ).
% zabs_less_one_iff
tff(fact_1631_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).
% idom_abs_sgn_class.abs_sgn
tff(fact_1632_sgn__abs,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).
% sgn_abs
tff(fact_1633_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ) ).
% nat_abs_dvd_iff
tff(fact_1634_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ).
% dvd_nat_abs_iff
tff(fact_1635_pochhammer__of__nat,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: nat,N: nat] : comm_s3205402744901411588hammer(A,aa(nat,A,semiring_1_of_nat(A),X),N) = aa(nat,A,semiring_1_of_nat(A),comm_s3205402744901411588hammer(nat,X,N)) ) ).
% pochhammer_of_nat
tff(fact_1636_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_1637_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_1638_abs__eq__0__iff,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] :
( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% abs_eq_0_iff
tff(fact_1639_abs__mult,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A,B2: A] : aa(A,A,abs_abs(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),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).
% abs_mult
tff(fact_1640_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).
% abs_one
tff(fact_1641_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_1642_abs__eq__iff,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y) )
<=> ( ( X = Y )
| ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).
% abs_eq_iff
tff(fact_1643_dvd__if__abs__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [L: A,K: A] :
( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),K)) ) ) ).
% dvd_if_abs_eq
tff(fact_1644_zdvd__antisym__abs,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),B2))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A3))
=> ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B2) ) ) ) ).
% zdvd_antisym_abs
tff(fact_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_abs__mult__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,C2: A,B2: A,D2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D2))
=> 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,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))) ) ) ) ).
% abs_mult_less
tff(fact_1653_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_1654_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_1655_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_1656_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_1657_abs__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A3)),B2))
<=> ( 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,uminus_uminus(A),A3)),B2)) ) ) ) ).
% abs_less_iff
tff(fact_1658_nonzero__abs__divide,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).
% nonzero_abs_divide
tff(fact_1659_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).
% linordered_idom_class.abs_sgn
tff(fact_1660_abs__mult__sgn,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).
% abs_mult_sgn
tff(fact_1661_sgn__mult__abs,axiom,
! [A: $tType] :
( idom_abs_sgn(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,abs_abs(A),A3)) = A3 ) ).
% sgn_mult_abs
tff(fact_1662_mult__sgn__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),X)),aa(A,A,abs_abs(A),X)) = X ) ).
% mult_sgn_abs
tff(fact_1663_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_1664_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = one_one(int) )
=> ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).
% abs_zmult_eq_1
tff(fact_1665_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_1666_pochhammer__pos,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).
% pochhammer_pos
tff(fact_1667_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,M: nat,N: nat] :
( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( comm_s3205402744901411588hammer(A,A3,N) != zero_zero(A) ) ) ) ) ).
% pochhammer_neq_0_mono
tff(fact_1668_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).
% pochhammer_eq_0_mono
tff(fact_1669_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_1670_dense__eq0__I,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs(A)
& dense_linorder(A) )
=> ! [X: A] :
( ! [E2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E2)) )
=> ( X = zero_zero(A) ) ) ) ).
% dense_eq0_I
tff(fact_1671_abs__eq__mult,axiom,
! [A: $tType] :
( ordered_ring_abs(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),A3),zero_zero(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),zero_zero(A))) ) )
=> ( aa(A,A,abs_abs(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),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).
% abs_eq_mult
tff(fact_1672_abs__mult__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).
% abs_mult_pos
tff(fact_1673_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_1674_eq__abs__iff_H,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,abs_abs(A),B2) )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
& ( ( B2 = A3 )
| ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).
% eq_abs_iff'
tff(fact_1675_abs__eq__iff_H,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,abs_abs(A),A3) = B2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
& ( ( A3 = B2 )
| ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).
% abs_eq_iff'
tff(fact_1676_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_1677_abs__if__raw,axiom,
! [A: $tType] :
( abs_if(A)
=> ! [X3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),X3) = aa(A,A,uminus_uminus(A),X3) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),X3) = X3 ) ) ) ) ).
% abs_if_raw
tff(fact_1678_abs__if,axiom,
! [A: $tType] :
( abs_if(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) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
=> ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ) ).
% abs_if
tff(fact_1679_abs__diff__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),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)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).
% abs_diff_le_iff
tff(fact_1680_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2)))),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),D2))))) ) ).
% abs_diff_triangle_ineq
tff(fact_1681_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_1682_abs__div__pos,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y)) ) ) ) ).
% abs_div_pos
tff(fact_1683_abs__diff__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),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)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).
% abs_diff_less_iff
tff(fact_1684_div__noneq__sgn__abs,axiom,
! [L: int,K: 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(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa(bool,int,zero_neq_one_of_bool(int),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K)))) ) ) ) ).
% div_noneq_sgn_abs
tff(fact_1685_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_1686_zabs__def,axiom,
! [I: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
=> ( aa(int,int,abs_abs(int),I) = aa(int,int,uminus_uminus(int),I) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
=> ( aa(int,int,abs_abs(int),I) = I ) ) ) ).
% zabs_def
tff(fact_1687_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I != zero_zero(int) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),I))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I))) ) ) ).
% dvd_imp_le_int
tff(fact_1688_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] : aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),W2))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Z2))) ).
% nat_abs_mult_distrib
tff(fact_1689_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L != zero_zero(int) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))),aa(int,int,abs_abs(int),L))) ) ).
% abs_mod_less
tff(fact_1690_pochhammer__nonneg,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).
% pochhammer_nonneg
tff(fact_1691_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_1692_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_1693_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X)))) ) ).
% abs_add_one_gt_zero
tff(fact_1694_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M != zero_zero(int) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)),M))
<=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).
% zdvd_mult_cancel1
tff(fact_1695_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V != zero_zero(int) )
=> ( 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),V)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).
% div_sgn_abs_cancel
tff(fact_1696_div__abs__eq__div__nat,axiom,
! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),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)))) ).
% div_abs_eq_div_nat
tff(fact_1697_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).
% mod_abs_eq_div_nat
tff(fact_1698_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_1699_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_1700_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_1701_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(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),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).
% div_dvd_sgn_abs
tff(fact_1702_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( idom(A)
=> ! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
=> ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
tff(fact_1703_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0(A)
& idom(A) )
=> ! [N: nat,K: nat] :
( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ) ).
% pochhammer_of_nat_eq_0_iff
tff(fact_1704_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
<=> ? [K4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K4),N))
& ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K4)) ) ) ) ) ).
% pochhammer_eq_0_iff
tff(fact_1705_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ring_char_0(A)
& idom(A) )
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
tff(fact_1706_modulo__int__def,axiom,
! [L: int,K: int] :
( ( ( L = zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) )
& ( ( 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),modulo_modulo(int),K),L) = 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),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(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),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa(bool,int,zero_neq_one_of_bool(int),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))))) ) ) ) ) ) ).
% modulo_int_def
tff(fact_1707_incr__lemma,axiom,
! [D2: int,Z2: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),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),X),Z2))),one_one(int))),D2)))) ) ).
% incr_lemma
tff(fact_1708_decr__lemma,axiom,
! [D2: int,X: int,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X),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),X),Z2))),one_one(int))),D2))),Z2)) ) ).
% decr_lemma
tff(fact_1709_nat__abs__int__diff,axiom,
! [A3: nat,B2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(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),minus_minus(nat),B2),A3) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(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),minus_minus(nat),A3),B2) ) ) ) ).
% nat_abs_int_diff
tff(fact_1710_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_1711_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: fun(nat,int),K: int] :
( ! [I4: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I4))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),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,suc,I4))),aa(nat,int,F,I4)))),one_one(int))) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F,M)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F,N)))
=> ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I4))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
& ( aa(nat,int,F,I4) = K ) ) ) ) ) ) ).
% nat_intermed_int_val
tff(fact_1712_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_1713_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_1714_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( A3 != B2 )
& ( C2 != D2 ) )
<=> ( 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),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% crossproduct_noteq
tff(fact_1715_crossproduct__eq,axiom,
! [A: $tType] :
( semiri1453513574482234551roduct(A)
=> ! [W2: A,Y: A,X: 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),W2),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Z2)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
<=> ( ( W2 = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
tff(fact_1716_nat__ivt__aux,axiom,
! [N: nat,F: fun(nat,int),K: int] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),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,suc,I4))),aa(nat,int,F,I4)))),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)))
=> ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
& ( aa(nat,int,F,I4) = K ) ) ) ) ) ).
% nat_ivt_aux
tff(fact_1717_nat0__intermed__int__val,axiom,
! [N: nat,F: fun(nat,int),K: int] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),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),I4),one_one(nat)))),aa(nat,int,F,I4)))),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)))
=> ? [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
& ( aa(nat,int,F,I4) = K ) ) ) ) ) ).
% nat0_intermed_int_val
tff(fact_1718_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_1719_int__power__div__base,axiom,
! [M: nat,K: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
=> ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).
% int_power_div_base
tff(fact_1720_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_1721_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_1722_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_1723_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_1724_frac__unique__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: A] :
( ( archimedean_frac(A,X) = A3 )
<=> ( pp(member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),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_1725_power__Suc__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).
% power_Suc_0
tff(fact_1726_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( M = zero_zero(nat) )
| ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% nat_power_eq_Suc_0_iff
tff(fact_1727_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_1728_nat__zero__less__power__iff,axiom,
! [X: 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),power_power(nat),X),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
| ( N = zero_zero(nat) ) ) ) ).
% nat_zero_less_power_iff
tff(fact_1729_of__nat__power,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),power_power(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M)),N) ) ).
% of_nat_power
tff(fact_1730_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [B2: nat,W2: nat,X: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2) = aa(nat,A,semiring_1_of_nat(A),X) )
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2) = X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1731_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [X: nat,B2: nat,W2: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),X) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2) )
<=> ( X = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1732_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_1733_frac__in__Ints__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(member(A,archimedean_frac(A,X),ring_1_Ints(A)))
<=> pp(member(A,X,ring_1_Ints(A))) ) ) ).
% frac_in_Ints_iff
tff(fact_1734_triangle__0,axiom,
nat_triangle(zero_zero(nat)) = zero_zero(nat) ).
% triangle_0
tff(fact_1735_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_1736_power__0__Suc,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).
% power_0_Suc
tff(fact_1737_power__Suc0__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).
% power_Suc0_right
tff(fact_1738_abs__power__minus,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).
% abs_power_minus
tff(fact_1739_frac__eq__0__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( archimedean_frac(A,X) = zero_zero(A) )
<=> pp(member(A,X,ring_1_Ints(A))) ) ) ).
% frac_eq_0_iff
tff(fact_1740_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_1741_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_1742_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X: 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),X)),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),X),Y)) ) ) ) ).
% power_strict_increasing_iff
tff(fact_1743_power__eq__0__iff,axiom,
! [A: $tType] :
( semiri2026040879449505780visors(A)
=> ! [A3: A,N: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = zero_zero(A) )
<=> ( ( A3 = zero_zero(A) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).
% power_eq_0_iff
tff(fact_1744_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: nat,B2: nat,W2: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
tff(fact_1745_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: nat,W2: nat,X: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),X)) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
tff(fact_1746_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: nat,B2: nat,W2: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2))) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
tff(fact_1747_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: nat,W2: nat,X: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),X)))
<=> 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),W2)),X)) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
tff(fact_1748_pow__divides__pow__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [N: nat,A3: A,B2: A] :
( 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),dvd_dvd(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),B2),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).
% pow_divides_pow_iff
tff(fact_1749_frac__gt__0__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
<=> ~ pp(member(A,X,ring_1_Ints(A))) ) ) ).
% frac_gt_0_iff
tff(fact_1750_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_1751_power__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [B2: A,X: 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),X)),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),X),Y)) ) ) ) ).
% power_increasing_iff
tff(fact_1752_power__mono__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,B2: 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),zero_zero(A)),B2))
=> ( 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),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).
% power_mono_iff
tff(fact_1753_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N)))
<=> ( ( A3 != zero_zero(A) )
| ( N = zero_zero(nat) ) ) ) ) ).
% zero_less_power_abs_iff
tff(fact_1754_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_1755_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
| ( N = zero_zero(nat) ) ) ) ) ).
% of_nat_zero_less_power_iff
tff(fact_1756_Ints__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A,N: nat] :
( pp(member(A,A3,ring_1_Ints(A)))
=> pp(member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),ring_1_Ints(A))) ) ) ).
% Ints_power
tff(fact_1757_power__not__zero,axiom,
! [A: $tType] :
( semiri2026040879449505780visors(A)
=> ! [A3: A,N: nat] :
( ( A3 != zero_zero(A) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) != zero_zero(A) ) ) ) ).
% power_not_zero
tff(fact_1758_power__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: 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),A3),N)),A3) = 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_commutes
tff(fact_1759_power__mult__distrib,axiom,
! [A: $tType] :
( comm_monoid_mult(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),times_times(A),A3),B2)),N) = aa(A,A,aa(A,fun(A,A),times_times(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),B2),N)) ) ).
% power_mult_distrib
tff(fact_1760_power__commuting__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N: nat] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).
% power_commuting_commutes
tff(fact_1761_power__mod,axiom,
! [A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A3: A,B2: A,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2)),N)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),B2) ) ).
% power_mod
tff(fact_1762_power__mult,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),times_times(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),N) ) ).
% power_mult
tff(fact_1763_Ints__0,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(member(A,zero_zero(A),ring_1_Ints(A))) ) ).
% Ints_0
tff(fact_1764_Ints__mult,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),times_times(A),A3),B2),ring_1_Ints(A))) ) ) ) ).
% Ints_mult
tff(fact_1765_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_1766_Ints__1,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(member(A,one_one(A),ring_1_Ints(A))) ) ).
% Ints_1
tff(fact_1767_Ints__diff,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),minus_minus(A),A3),B2),ring_1_Ints(A))) ) ) ) ).
% Ints_diff
tff(fact_1768_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A] :
( pp(member(A,aa(A,A,uminus_uminus(A),X),ring_1_Ints(A)))
<=> pp(member(A,X,ring_1_Ints(A))) ) ) ).
% minus_in_Ints_iff
tff(fact_1769_Ints__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A] :
( pp(member(A,A3,ring_1_Ints(A)))
=> pp(member(A,aa(A,A,uminus_uminus(A),A3),ring_1_Ints(A))) ) ) ).
% Ints_minus
tff(fact_1770_Ints__of__nat,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] : pp(member(A,aa(nat,A,semiring_1_of_nat(A),N),ring_1_Ints(A))) ) ).
% Ints_of_nat
tff(fact_1771_Ints__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(member(A,A3,ring_1_Ints(A)))
=> pp(member(A,aa(A,A,abs_abs(A),A3),ring_1_Ints(A))) ) ) ).
% Ints_abs
tff(fact_1772_fact__in__Ints,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& ring_1(A) )
=> ! [N: nat] : pp(member(A,semiring_char_0_fact(A,N),ring_1_Ints(A))) ) ).
% fact_in_Ints
tff(fact_1773_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z2),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z2)),N) ) ) ).
% nat_power_eq
tff(fact_1774_power__mono,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,B2: A,N: nat] :
( 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),zero_zero(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),B2),N))) ) ) ) ).
% power_mono
tff(fact_1775_zero__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),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).
% zero_le_power
tff(fact_1776_zero__less__power,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),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).
% zero_less_power
tff(fact_1777_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_1778_left__right__inverse__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Y: A,N: nat] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),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),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).
% left_right_inverse_power
tff(fact_1779_power__Suc2,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) ) ).
% power_Suc2
tff(fact_1780_power__Suc,axiom,
! [A: $tType] :
( power(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,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_Suc
tff(fact_1781_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_1782_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_1783_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_1784_le__imp__power__dvd,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: nat,N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(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))) ) ) ).
% le_imp_power_dvd
tff(fact_1785_power__le__dvd,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,N: nat,B2: A,M: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),B2)) ) ) ) ).
% power_le_dvd
tff(fact_1786_dvd__power__le,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A,Y: A,N: nat,M: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M))) ) ) ) ).
% dvd_power_le
tff(fact_1787_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(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),M),N)) ) ) ).
% nat_power_less_imp_less
tff(fact_1788_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_1789_power__less__imp__less__base,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat,B2: 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),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
=> ( 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),A3),B2)) ) ) ) ).
% power_less_imp_less_base
tff(fact_1790_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_1791_power__inject__base,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat,B2: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
=> ( 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))
=> ( A3 = B2 ) ) ) ) ) ).
% power_inject_base
tff(fact_1792_power__le__imp__le__base,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat,B2: 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))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
=> ( 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),A3),B2)) ) ) ) ).
% power_le_imp_le_base
tff(fact_1793_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_1794_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_1795_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_1796_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_1797_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_1798_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_1799_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N6: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
=> ( 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),N6))) ) ) ) ).
% power_strict_increasing
tff(fact_1800_power__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N6: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
=> ( 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),N6))) ) ) ) ).
% power_increasing
tff(fact_1801_zero__le__power__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N))) ) ).
% zero_le_power_abs
tff(fact_1802_zero__power,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).
% zero_power
tff(fact_1803_is__unit__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
| ( N = zero_zero(nat) ) ) ) ) ).
% is_unit_power_iff
tff(fact_1804_power__gt__expt,axiom,
! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K))) ) ).
% power_gt_expt
tff(fact_1805_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))) ) ).
% nat_one_le_power
tff(fact_1806_fact__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N)))) ) ).
% fact_le_power
tff(fact_1807_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_1808_bounded__Max__nat,axiom,
! [P: fun(nat,bool),X: nat,M7: nat] :
( pp(aa(nat,bool,P,X))
=> ( ! [X2: nat] :
( pp(aa(nat,bool,P,X2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),M7)) )
=> ~ ! [M5: nat] :
( pp(aa(nat,bool,P,M5))
=> ~ ! [X3: nat] :
( pp(aa(nat,bool,P,X3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M5)) ) ) ) ) ).
% bounded_Max_nat
tff(fact_1809_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_1810_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_1811_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_1812_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N6: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N6))
=> ( 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),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_strict_decreasing
tff(fact_1813_power__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,N6: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N6))
=> ( 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),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).
% power_decreasing
tff(fact_1814_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_1815_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,N: nat,B2: 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),B2),N) )
=> ( 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(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
tff(fact_1816_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A3: A,B2: A] :
( 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),zero_zero(A)),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
<=> ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
tff(fact_1817_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_1818_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_1819_dvd__power__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,M: nat,N: nat] :
( ( X != zero_zero(A) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ) ).
% dvd_power_iff
tff(fact_1820_dvd__power,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
| ( X = one_one(A) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).
% dvd_power
tff(fact_1821_power__diff,axiom,
! [A: $tType] :
( semidom_divide(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(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = 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)) ) ) ) ) ).
% power_diff
tff(fact_1822_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [K: nat,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),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),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A3),K))) ) ) ).
% gbinomial_ge_n_over_k_pow_k
tff(fact_1823_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_1824_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_1825_fact__div__fact__le__pow,axiom,
! [R: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),R)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R))) ) ).
% fact_div_fact_le_pow
tff(fact_1826_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(member(A,X,ring_1_Ints(A)))
=> ( ( X != 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),X))) ) ) ) ).
% Ints_nonzero_abs_ge1
tff(fact_1827_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(member(A,X,ring_1_Ints(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
=> ( X = zero_zero(A) ) ) ) ) ).
% Ints_nonzero_abs_less1
tff(fact_1828_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(member(A,X,ring_1_Ints(A)))
=> ( pp(member(A,Y,ring_1_Ints(A)))
=> ( ( X = 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),X),Y))),one_one(A))) ) ) ) ) ).
% Ints_eq_abs_less1
tff(fact_1829_power__strict__mono,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A,B2: A,N: nat] :
( 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),zero_zero(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),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).
% power_strict_mono
tff(fact_1830_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_1831_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_1832_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_1833_frac__neg,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( pp(member(A,X,ring_1_Ints(A)))
=> ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
& ( ~ pp(member(A,X,ring_1_Ints(A)))
=> ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).
% frac_neg
tff(fact_1834_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_1835_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_1836_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_1837_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_1838_power__minus_H,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A,N: nat] :
( nO_MATCH(A,A,one_one(A),X)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),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),X),N)) ) ) ) ).
% power_minus'
tff(fact_1839_ceiling__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q4: A,P3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
=> 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),Q4)))),one_one(A))),Q4)),P3)) ) ) ).
% ceiling_divide_lower
tff(fact_1840_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_1841_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( 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),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(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),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
& ( ~ 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),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
tff(fact_1842_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( 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),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
<=> ( ( 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),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),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(A),C2),zero_zero(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),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),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),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
tff(fact_1843_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( archim2362893244070406136eiling(B)
& linordered_idom(A) )
=> ! [A3: B,B2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
=> ( pp(member(B,A3,ring_1_Ints(B)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2)))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A3)),archimedean_ceiling(B,B2))))) ) ) ) ).
% mult_ceiling_le_Ints
tff(fact_1844_numeral__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [M: num,N: num] :
( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),N) )
<=> ( M = N ) ) ) ).
% numeral_eq_iff
tff(fact_1845_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
<=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).
% int_eq_iff_numeral
tff(fact_1846_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [W2: int,Z2: int] :
( ( aa(int,A,ring_1_of_int(A),W2) = aa(int,A,ring_1_of_int(A),Z2) )
<=> ( W2 = Z2 ) ) ) ).
% of_int_eq_iff
tff(fact_1847_nat__numeral,axiom,
! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).
% nat_numeral
tff(fact_1848_numeral__le__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: num,N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).
% numeral_le_iff
tff(fact_1849_numeral__less__iff,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [M: num,N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).
% numeral_less_iff
tff(fact_1850_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [V: num,W2: num,Z2: 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),times_times(A),aa(num,A,numeral_numeral(A),W2)),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2))),Z2) ) ).
% mult_numeral_left_semiring_numeral
tff(fact_1851_numeral__times__numeral,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(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),times_times(num),M),N)) ) ).
% numeral_times_numeral
tff(fact_1852_add__numeral__left,axiom,
! [A: $tType] :
( numeral(A)
=> ! [V: num,W2: 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),W2)),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),W2))),Z2) ) ).
% add_numeral_left
tff(fact_1853_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_1854_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [M: num,N: num] :
( ( 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)) )
<=> ( M = N ) ) ) ).
% neg_numeral_eq_iff
tff(fact_1855_power__zero__numeral,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).
% power_zero_numeral
tff(fact_1856_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_1857_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_1858_of__nat__numeral,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),N)) = aa(num,A,numeral_numeral(A),N) ) ).
% of_nat_numeral
tff(fact_1859_abs__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),N) ) ).
% abs_numeral
tff(fact_1860_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int,N: num] :
( ( aa(int,A,ring_1_of_int(A),Z2) = aa(num,A,numeral_numeral(A),N) )
<=> ( Z2 = aa(num,int,numeral_numeral(int),N) ) ) ) ).
% of_int_eq_numeral_iff
tff(fact_1861_of__int__numeral,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).
% of_int_numeral
tff(fact_1862_ceiling__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).
% ceiling_numeral
tff(fact_1863_euclidean__size__numeral,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [K: num] : aa(A,nat,euclid6346220572633701492n_size(A),aa(num,A,numeral_numeral(A),K)) = aa(num,nat,numeral_numeral(nat),K) ) ).
% euclidean_size_numeral
tff(fact_1864_of__int__fact,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& ring_1(A) )
=> ! [N: nat] : aa(int,A,ring_1_of_int(A),semiring_char_0_fact(int,N)) = semiring_char_0_fact(A,N) ) ).
% of_int_fact
tff(fact_1865_ceiling__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int] : archimedean_ceiling(A,aa(int,A,ring_1_of_int(A),Z2)) = Z2 ) ).
% ceiling_of_int
tff(fact_1866_round__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [N: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),N)) = aa(num,int,numeral_numeral(int),N) ) ).
% round_numeral
tff(fact_1867_round__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [N: int] : archimedean_round(A,aa(int,A,ring_1_of_int(A),N)) = N ) ).
% round_of_int
tff(fact_1868_neg__numeral__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num,N: 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),aa(num,A,numeral_numeral(A),N))))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).
% neg_numeral_le_iff
tff(fact_1869_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_1870_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_1871_neg__numeral__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num,N: 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),aa(num,A,numeral_numeral(A),N))))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).
% neg_numeral_less_iff
tff(fact_1872_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(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),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(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)) ) ).
% right_diff_distrib_numeral
tff(fact_1873_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( numeral(A)
& ring(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),minus_minus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),minus_minus(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))) ) ).
% left_diff_distrib_numeral
tff(fact_1874_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(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(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).
% mult_neg_numeral_simps(1)
tff(fact_1875_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(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),times_times(num),M),N))) ) ).
% mult_neg_numeral_simps(2)
tff(fact_1876_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(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(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).
% mult_neg_numeral_simps(3)
tff(fact_1877_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2))),Y) ) ).
% semiring_norm(172)
tff(fact_1878_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W2: num,Y: 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),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2)))),Y) ) ).
% semiring_norm(171)
tff(fact_1879_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [V: num,W2: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W2)))),Y) ) ).
% semiring_norm(170)
tff(fact_1880_semiring__norm_I168_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [V: num,W2: 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),W2))),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),W2)))),Y) ) ).
% semiring_norm(168)
tff(fact_1881_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_1882_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_1883_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_1884_abs__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),N) ) ).
% abs_neg_numeral
tff(fact_1885_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( aa(int,A,ring_1_of_int(A),zero_zero(int)) = zero_zero(A) ) ) ).
% of_int_0
tff(fact_1886_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z2) )
<=> ( Z2 = zero_zero(int) ) ) ) ).
% of_int_0_eq_iff
tff(fact_1887_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( aa(int,A,ring_1_of_int(A),Z2) = zero_zero(A) )
<=> ( Z2 = zero_zero(int) ) ) ) ).
% of_int_eq_0_iff
tff(fact_1888_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [W2: int,Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W2),Z2)) ) ) ).
% of_int_le_iff
tff(fact_1889_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [W2: int,Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W2),Z2)) ) ) ).
% of_int_less_iff
tff(fact_1890_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_1891_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_1892_of__int__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W2),Z2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_mult
tff(fact_1893_of__int__add,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),Z2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_add
tff(fact_1894_of__int__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z2)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_minus
tff(fact_1895_nat__neg__numeral,axiom,
! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).
% nat_neg_numeral
tff(fact_1896_of__int__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W2: int,Z2: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),minus_minus(int),W2),Z2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z2)) ) ).
% of_int_diff
tff(fact_1897_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_1898_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).
% of_int_of_nat_eq
tff(fact_1899_diff__nat__numeral,axiom,
! [V: num,V2: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V2)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V2))) ).
% diff_nat_numeral
tff(fact_1900_of__int__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),X)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),X)) ) ).
% of_int_abs
tff(fact_1901_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [X: int,B2: int,W2: nat] :
( ( aa(int,A,ring_1_of_int(A),X) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2) )
<=> ( X = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
tff(fact_1902_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [B2: int,W2: nat,X: int] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2) = aa(int,A,ring_1_of_int(A),X) )
<=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2) = X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
tff(fact_1903_of__int__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int,N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z2)),N) ) ).
% of_int_power
tff(fact_1904_frac__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z2)) = zero_zero(A) ) ).
% frac_of_int
tff(fact_1905_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( aa(int,nat,nat2,Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) )
<=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ).
% nat_eq_numeral_power_cancel_iff
tff(fact_1906_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = aa(int,nat,nat2,Y) )
<=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ).
% numeral_power_eq_nat_cancel_iff
tff(fact_1907_of__int__of__bool,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: bool] : aa(int,A,ring_1_of_int(A),aa(bool,int,zero_neq_one_of_bool(int),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).
% of_int_of_bool
tff(fact_1908_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))))
<=> 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),aa(num,A,numeral_numeral(A),W2))),B2)) ) ) ).
% le_divide_eq_numeral1(1)
tff(fact_1909_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,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),aa(num,A,numeral_numeral(A),W2))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).
% divide_le_eq_numeral1(1)
tff(fact_1910_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,W2: num,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) = A3 )
<=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) ) )
& ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
tff(fact_1911_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,W2: num] :
( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) )
<=> ( ( ( aa(num,A,numeral_numeral(A),W2) != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) = B2 ) )
& ( ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
tff(fact_1912_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))))
<=> 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(num,A,numeral_numeral(A),W2))),B2)) ) ) ).
% less_divide_eq_numeral1(1)
tff(fact_1913_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,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),aa(num,A,numeral_numeral(A),W2))),A3))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)))) ) ) ).
% divide_less_eq_numeral1(1)
tff(fact_1914_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int,N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),aa(num,int,numeral_numeral(int),N))) ) ) ).
% of_int_le_numeral_iff
tff(fact_1915_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num,Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z2)) ) ) ).
% of_int_numeral_le_iff
tff(fact_1916_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num,Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z2)) ) ) ).
% of_int_numeral_less_iff
tff(fact_1917_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int,N: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),aa(num,A,numeral_numeral(A),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),aa(num,int,numeral_numeral(int),N))) ) ) ).
% of_int_less_numeral_iff
tff(fact_1918_ceiling__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V))) ) ) ).
% ceiling_le_numeral
tff(fact_1919_numeral__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V)),X)) ) ) ).
% numeral_less_ceiling
tff(fact_1920_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [Y: nat,X: num,N: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
<=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1921_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [X: num,N: nat,Y: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(nat,A,semiring_1_of_nat(A),Y) )
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
tff(fact_1922_ceiling__add__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).
% ceiling_add_numeral
tff(fact_1923_ceiling__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).
% ceiling_neg_numeral
tff(fact_1924_ceiling__diff__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).
% ceiling_diff_numeral
tff(fact_1925_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_1926_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [X: num,N: nat,Y: int] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(int,A,ring_1_of_int(A),Y) )
<=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
tff(fact_1927_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Y: int,X: num,N: nat] :
( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
<=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
tff(fact_1928_ceiling__add__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z2) ) ).
% ceiling_add_of_int
tff(fact_1929_ceiling__diff__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),Z2) ) ).
% ceiling_diff_of_int
tff(fact_1930_round__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [N: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).
% round_neg_numeral
tff(fact_1931_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).
% le_divide_eq_numeral1(2)
tff(fact_1932_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,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),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),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),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).
% divide_le_eq_numeral1(2)
tff(fact_1933_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,W2: num,A3: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A3 )
<=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) )
& ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
tff(fact_1934_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A,W2: num] :
( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
<=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2 ) )
& ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
tff(fact_1935_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))) ) ) ).
% less_divide_eq_numeral1(2)
tff(fact_1936_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,W2: num,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),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3))
<=> 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(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2)) ) ) ).
% divide_less_eq_numeral1(2)
tff(fact_1937_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).
% dbl_dec_simps(1)
tff(fact_1938_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).
% dbl_inc_simps(1)
tff(fact_1939_of__int__le__0__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)),zero_zero(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),zero_zero(int))) ) ) ).
% of_int_le_0_iff
tff(fact_1940_of__int__0__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2)) ) ) ).
% of_int_0_le_iff
tff(fact_1941_of__int__less__0__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)),zero_zero(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int))) ) ) ).
% of_int_less_0_iff
tff(fact_1942_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2)) ) ) ).
% of_int_0_less_iff
tff(fact_1943_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_1944_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_1945_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_1946_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_1947_of__nat__nat,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z2)) = aa(int,A,ring_1_of_int(A),Z2) ) ) ) ).
% of_nat_nat
tff(fact_1948_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [B2: int,W2: nat,X: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).
% of_int_le_of_int_power_cancel_iff
tff(fact_1949_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: int,B2: int,W2: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).
% of_int_power_le_of_int_cancel_iff
tff(fact_1950_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: int,B2: int,W2: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2))) ) ) ).
% of_int_power_less_of_int_cancel_iff
tff(fact_1951_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [B2: int,W2: nat,X: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),X)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),X)) ) ) ).
% of_int_less_of_int_power_cancel_iff
tff(fact_1952_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_1953_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).
% nat_less_numeral_power_cancel_iff
tff(fact_1954_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A3)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).
% numeral_power_less_nat_cancel_iff
tff(fact_1955_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),aa(int,nat,nat2,A3)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ).
% numeral_power_le_nat_cancel_iff
tff(fact_1956_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).
% nat_le_numeral_power_cancel_iff
tff(fact_1957_ceiling__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),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_1958_numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
<=> 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))),X)) ) ) ).
% numeral_le_ceiling
tff(fact_1959_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [I: num,N: nat,X: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).
% numeral_power_less_of_nat_cancel_iff
tff(fact_1960_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: nat,I: num,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).
% of_nat_less_numeral_power_cancel_iff
tff(fact_1961_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: nat,I: num,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N))) ) ) ).
% of_nat_le_numeral_power_cancel_iff
tff(fact_1962_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [I: num,N: nat,X: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),X)) ) ) ).
% numeral_power_le_of_nat_cancel_iff
tff(fact_1963_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),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),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)))) ) ) ).
% ceiling_le_neg_numeral
tff(fact_1964_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: 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))),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X)) ) ) ).
% neg_numeral_less_ceiling
tff(fact_1965_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: int,X: num,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).
% of_int_le_numeral_power_cancel_iff
tff(fact_1966_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: num,N: nat,A3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).
% numeral_power_le_of_int_cancel_iff
tff(fact_1967_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: num,N: nat,A3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).
% numeral_power_less_of_int_cancel_iff
tff(fact_1968_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: int,X: num,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).
% of_int_less_numeral_power_cancel_iff
tff(fact_1969_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Y: int,X: num,N: nat] :
( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) )
<=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_1970_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [X: num,N: nat,Y: int] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) = aa(int,A,ring_1_of_int(A),Y) )
<=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) = Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_1971_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),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),X),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_1972_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: 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,X)))
<=> 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))),X)) ) ) ).
% neg_numeral_le_ceiling
tff(fact_1973_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: num,N: nat,A3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
tff(fact_1974_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: int,X: num,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
tff(fact_1975_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: num,N: nat,A3: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
tff(fact_1976_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: int,X: num,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
tff(fact_1977_of__int__neg__numeral,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).
% of_int_neg_numeral
tff(fact_1978_nat__numeral__as__int,axiom,
! [X3: num] : aa(num,nat,numeral_numeral(nat),X3) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X3)) ).
% nat_numeral_as_int
tff(fact_1979_int__ops_I3_J,axiom,
! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).
% int_ops(3)
tff(fact_1980_ex__le__of__int,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).
% ex_le_of_int
tff(fact_1981_ex__of__int__less,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),X)) ) ).
% ex_of_int_less
tff(fact_1982_ex__less__of__int,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [Z3: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z3))) ) ).
% ex_less_of_int
tff(fact_1983_mult__of__int__commute,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),X)) ) ).
% mult_of_int_commute
tff(fact_1984_zero__neq__numeral,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).
% zero_neq_numeral
tff(fact_1985_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [M: num,N: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).
% numeral_neq_neg_numeral
tff(fact_1986_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [M: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),N) ) ).
% neg_numeral_neq_numeral
tff(fact_1987_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).
% div_mult2_numeral_eq
tff(fact_1988_pochhammer__of__int,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X: int,N: nat] : comm_s3205402744901411588hammer(A,aa(int,A,ring_1_of_int(A),X),N) = aa(int,A,ring_1_of_int(A),comm_s3205402744901411588hammer(int,X,N)) ) ).
% pochhammer_of_int
tff(fact_1989_Ints__of__int,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int] : pp(member(A,aa(int,A,ring_1_of_int(A),Z2),ring_1_Ints(A))) ) ).
% Ints_of_int
tff(fact_1990_Ints__induct,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Q4: A,P: fun(A,bool)] :
( pp(member(A,Q4,ring_1_Ints(A)))
=> ( ! [Z3: int] : pp(aa(A,bool,P,aa(int,A,ring_1_of_int(A),Z3)))
=> pp(aa(A,bool,P,Q4)) ) ) ) ).
% Ints_induct
tff(fact_1991_Ints__cases,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Q4: A] :
( pp(member(A,Q4,ring_1_Ints(A)))
=> ~ ! [Z3: int] : Q4 != aa(int,A,ring_1_of_int(A),Z3) ) ) ).
% Ints_cases
tff(fact_1992_Ints__numeral,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: num] : pp(member(A,aa(num,A,numeral_numeral(A),N),ring_1_Ints(A))) ) ).
% Ints_numeral
tff(fact_1993_not__numeral__le__zero,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)),zero_zero(A))) ) ).
% not_numeral_le_zero
tff(fact_1994_zero__le__numeral,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).
% zero_le_numeral
tff(fact_1995_not__numeral__less__zero,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)),zero_zero(A))) ) ).
% not_numeral_less_zero
tff(fact_1996_zero__less__numeral,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).
% zero_less_numeral
tff(fact_1997_le__of__int__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ).
% le_of_int_ceiling
tff(fact_1998_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_1999_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_2000_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num,N: 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),aa(num,A,numeral_numeral(A),N)))) ) ).
% not_numeral_le_neg_numeral
tff(fact_2001_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num,N: 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(num,A,numeral_numeral(A),N))) ) ).
% neg_numeral_le_numeral
tff(fact_2002_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).
% zero_neq_neg_numeral
tff(fact_2003_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num,N: 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),aa(num,A,numeral_numeral(A),N)))) ) ).
% not_numeral_less_neg_numeral
tff(fact_2004_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [M: num,N: 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(num,A,numeral_numeral(A),N))) ) ).
% neg_numeral_less_numeral
tff(fact_2005_one__plus__numeral__commute,axiom,
! [A: $tType] :
( numeral(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).
% one_plus_numeral_commute
tff(fact_2006_numeral__times__minus__swap,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [W2: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).
% numeral_times_minus_swap
tff(fact_2007_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_2008_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_2009_power_Opower_Opower__Suc,axiom,
! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A,N: nat] : power2(A,One,Times,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),Times,A3),power2(A,One,Times,A3,N)) ).
% power.power.power_Suc
tff(fact_2010_power_Opower_Opower__0,axiom,
! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A] : power2(A,One,Times,A3,zero_zero(nat)) = One ).
% power.power.power_0
tff(fact_2011_ceiling__le__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).
% ceiling_le_iff
tff(fact_2012_less__ceiling__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archimedean_ceiling(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ) ).
% less_ceiling_iff
tff(fact_2013_neg__numeral__le__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: 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),N))),zero_zero(A))) ) ).
% neg_numeral_le_zero
tff(fact_2014_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).
% not_zero_le_neg_numeral
tff(fact_2015_neg__numeral__less__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: 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),N))),zero_zero(A))) ) ).
% neg_numeral_less_zero
tff(fact_2016_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).
% not_zero_less_neg_numeral
tff(fact_2017_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_2018_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_2019_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_2020_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_2021_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_2022_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,W2: num] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
tff(fact_2023_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num,B2: A,C2: A] :
( ( aa(num,A,numeral_numeral(A),W2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2) = B2 ) )
& ( ( C2 = zero_zero(A) )
=> ( aa(num,A,numeral_numeral(A),W2) = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
tff(fact_2024_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_2025_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_2026_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_2027_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_2028_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_2029_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( idom_divide(A)
=> ! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A3))
=> pp(member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2)),ring_1_Ints(A))) ) ) ).
% of_int_divide_in_Ints
tff(fact_2030_of__int__nonneg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).
% of_int_nonneg
tff(fact_2031_of__int__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).
% of_int_pos
tff(fact_2032_of__int__leD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X: 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))),X))
=> ( ( N = zero_zero(int) )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).
% of_int_leD
tff(fact_2033_of__int__lessD,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: int,X: 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))),X))
=> ( ( N = zero_zero(int) )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).
% of_int_lessD
tff(fact_2034_floor__exists1,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
? [X2: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X2)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int)))))
& ! [Y3: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int))))) )
=> ( Y3 = X2 ) ) ) ) ).
% floor_exists1
tff(fact_2035_floor__exists,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: 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)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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_2036_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,X: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),X)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),X)) ) ) ).
% of_nat_less_of_int_iff
tff(fact_2037_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W2)))
<=> ( ( 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(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2)),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(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
tff(fact_2038_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(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(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
& ( ~ 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(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
tff(fact_2039_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [B2: A,C2: A,W2: num] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) ) )
& ( ( C2 = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
tff(fact_2040_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num,B2: A,C2: A] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
<=> ( ( ( C2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2) = B2 ) )
& ( ( C2 = zero_zero(A) )
=> ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
tff(fact_2041_round__diff__minimal,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: A,M: int] : 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),Z2),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z2))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z2),aa(int,A,ring_1_of_int(A),M))))) ) ).
% round_diff_minimal
tff(fact_2042_ceiling__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X))),one_one(A))),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,X)))) ) ) ).
% ceiling_correct
tff(fact_2043_ceiling__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: 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))),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),Z2)))
=> ( archimedean_ceiling(A,X) = Z2 ) ) ) ) ).
% ceiling_unique
tff(fact_2044_ceiling__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: int] :
( ( archimedean_ceiling(A,X) = 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))),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(int,A,ring_1_of_int(A),A3))) ) ) ) ).
% ceiling_eq_iff
tff(fact_2045_ceiling__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T2: A] :
( pp(aa(int,bool,P,archimedean_ceiling(A,T2)))
<=> ! [I5: 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),I5)),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),I5))) )
=> pp(aa(int,bool,P,I5)) ) ) ) ).
% ceiling_split
tff(fact_2046_ceiling__less__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),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_2047_le__ceiling__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archimedean_ceiling(A,X)))
<=> 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))),X)) ) ) ).
% le_ceiling_iff
tff(fact_2048_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( 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),C2)),aa(num,A,numeral_numeral(A),W2)))
<=> ( ( 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),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),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(A),C2),zero_zero(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),aa(num,A,numeral_numeral(A),W2)),C2)),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),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
tff(fact_2049_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(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),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C2))) )
& ( ~ 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),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
tff(fact_2050_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [B2: A,C2: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
<=> ( ( 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(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),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(A),C2),zero_zero(A)))
=> 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,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2)),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(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
tff(fact_2051_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [W2: num,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(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(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),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(A),C2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C2))) )
& ( ~ 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(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
tff(fact_2052_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X: B,Y: B,A3: A,B2: A,C2: A] :
( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),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_2053_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( semiring(A)
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),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_2054_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ring(A)
=> ! [X: B,Y: B,C2: A,A3: A,B2: A] :
( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),C2)
=> ( aa(A,A,aa(A,fun(A,A),times_times(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),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).
% left_diff_distrib_NO_MATCH
tff(fact_2055_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ring(A)
=> ! [X: B,Y: B,A3: A,B2: A,C2: A] :
( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y),A3)
=> ( aa(A,A,aa(A,fun(A,A),times_times(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),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).
% right_diff_distrib_NO_MATCH
tff(fact_2056_ceiling__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q4: A,P3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q4)))),Q4))) ) ) ).
% ceiling_divide_upper
tff(fact_2057_of__int__of__nat,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( aa(int,A,ring_1_of_int(A),K) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
=> ( aa(int,A,ring_1_of_int(A),K) = aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K)) ) ) ) ) ).
% of_int_of_nat
tff(fact_2058_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel(A)
& field(B) )
=> ! [X: B,B2: A,A3: A] :
( nO_MATCH(B,A,X,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_2059_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel(A)
& field(B) )
=> ! [X: B,B2: A,A3: A] :
( nO_MATCH(B,A,X,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_2060_floor__le__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),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),X),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_2061_neg__numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: 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,X)))
<=> 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))),X)) ) ) ).
% neg_numeral_less_floor
tff(fact_2062_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_2063_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_2064_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_2065_floor__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int] : archim6421214686448440834_floor(A,aa(int,A,ring_1_of_int(A),Z2)) = Z2 ) ).
% floor_of_int
tff(fact_2066_floor__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).
% floor_zero
tff(fact_2067_floor__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).
% floor_numeral
tff(fact_2068_floor__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).
% floor_one
tff(fact_2069_floor__of__nat,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [N: nat] : archim6421214686448440834_floor(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,int,semiring_1_of_nat(int),N) ) ).
% floor_of_nat
tff(fact_2070_floor__uminus__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,uminus_uminus(int),Z2) ) ).
% floor_uminus_of_int
tff(fact_2071_floor__diff__of__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),Z2))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),Z2) ) ).
% floor_diff_of_int
tff(fact_2072_floor__add2,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( ( pp(member(A,X,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),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).
% floor_add2
tff(fact_2073_zero__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).
% zero_le_floor
tff(fact_2074_floor__less__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).
% floor_less_zero
tff(fact_2075_numeral__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),X)) ) ) ).
% numeral_le_floor
tff(fact_2076_zero__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).
% zero_less_floor
tff(fact_2077_floor__le__zero,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).
% floor_le_zero
tff(fact_2078_floor__less__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V))) ) ) ).
% floor_less_numeral
tff(fact_2079_one__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).
% one_le_floor
tff(fact_2080_floor__less__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).
% floor_less_one
tff(fact_2081_floor__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).
% floor_neg_numeral
tff(fact_2082_floor__diff__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).
% floor_diff_numeral
tff(fact_2083_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_2084_floor__diff__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ).
% floor_diff_one
tff(fact_2085_numeral__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
<=> 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))),X)) ) ) ).
% numeral_less_floor
tff(fact_2086_floor__le__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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_2087_neg__numeral__le__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [V: num,X: 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))),archim6421214686448440834_floor(A,X)))
<=> 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),V))),X)) ) ) ).
% neg_numeral_le_floor
tff(fact_2088_floor__less__neg__numeral,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,V: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),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),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)))) ) ) ).
% floor_less_neg_numeral
tff(fact_2089_add__inc,axiom,
! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) ).
% add_inc
tff(fact_2090_mult__inc,axiom,
! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).
% mult_inc
tff(fact_2091_floor__mono,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).
% floor_mono
tff(fact_2092_floor__less__cancel,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Y: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
% floor_less_cancel
tff(fact_2093_of__int__floor__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X))),X)) ) ).
% of_int_floor_le
tff(fact_2094_floor__le__ceiling,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X))) ) ).
% floor_le_ceiling
tff(fact_2095_floor__le__round,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X))) ) ).
% floor_le_round
tff(fact_2096_le__floor__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z2),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ) ).
% le_floor_iff
tff(fact_2097_floor__less__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ) ).
% floor_less_iff
tff(fact_2098_le__floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)))) ) ).
% le_floor_add
tff(fact_2099_int__add__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z2)),X)) ) ).
% int_add_floor
tff(fact_2100_floor__add__int,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z2) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z2))) ) ).
% floor_add_int
tff(fact_2101_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [K: int,L: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).
% floor_divide_of_int_eq
tff(fact_2102_ceiling__def,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,X) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X))) ) ).
% ceiling_def
tff(fact_2103_floor__minus,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archimedean_ceiling(A,X)) ) ).
% floor_minus
tff(fact_2104_ceiling__minus,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),X)) = aa(int,int,uminus_uminus(int),archim6421214686448440834_floor(A,X)) ) ).
% ceiling_minus
tff(fact_2105_frac__def,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_frac(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X))) ) ).
% frac_def
tff(fact_2106_of__nat__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [R: A] :
( 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),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R)))),R)) ) ) ).
% of_nat_floor
tff(fact_2107_one__add__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).
% one_add_floor
tff(fact_2108_numeral__inc,axiom,
! [A: $tType] :
( numeral(A)
=> ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).
% numeral_inc
tff(fact_2109_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [M: nat,N: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) ).
% floor_divide_of_nat_eq
tff(fact_2110_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).
% ceiling_diff_floor_le_1
tff(fact_2111_ceiling__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( ( ( X = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
=> ( archimedean_ceiling(A,X) = archim6421214686448440834_floor(A,X) ) )
& ( ( X != aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,X)) )
=> ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ) ) ) ).
% ceiling_altdef
tff(fact_2112_floor__split,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [P: fun(int,bool),T2: A] :
( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T2)))
<=> ! [I5: int] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I5)),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),I5)),one_one(A)))) )
=> pp(aa(int,bool,P,I5)) ) ) ) ).
% floor_split
tff(fact_2113_floor__eq__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,A3: int] :
( ( archim6421214686448440834_floor(A,X) = A3 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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_2114_floor__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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,X) = Z2 ) ) ) ) ).
% floor_unique
tff(fact_2115_le__mult__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(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(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ).
% le_mult_floor
tff(fact_2116_less__floor__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Z2: int,X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),archim6421214686448440834_floor(A,X)))
<=> 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))),X)) ) ) ).
% less_floor_iff
tff(fact_2117_floor__le__iff,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A,Z2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),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_2118_floor__correct,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X))),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).
% floor_correct
tff(fact_2119_floor__divide__lower,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q4: A,P3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q4)))),Q4)),P3)) ) ) ).
% floor_divide_lower
tff(fact_2120_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( archim2362893244070406136eiling(B)
& linordered_idom(A) )
=> ! [A3: B,B2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A3))
=> ( pp(member(B,A3,ring_1_Ints(B)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A3)),archim6421214686448440834_floor(B,B2)))),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B2))))) ) ) ) ).
% le_mult_floor_Ints
tff(fact_2121_floor__add,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),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,X)),archimedean_frac(A,Y))),one_one(A)))
=> ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).
% floor_add
tff(fact_2122_floor__divide__upper,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Q4: A,P3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q4))
=> 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),Q4)))),one_one(A))),Q4))) ) ) ).
% floor_divide_upper
tff(fact_2123_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_2124_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_2125_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))
=> ( ? [Z4: int] :
! [X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Z4))
=> ( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P2,X2)) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,B5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4))) ) )
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P2,X2))
<=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
=> ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
<=> ( ? [X5: int] :
( pp(member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& pp(aa(int,bool,P2,X5)) )
| ? [X5: int] :
( pp(member(int,X5,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),X5))) ) ) ) ) ) ) ) ) ).
% cpmi
tff(fact_2126_cppi,axiom,
! [D4: int,P: fun(int,bool),P2: fun(int,bool),A5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( ? [Z4: int] :
! [X2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),X2))
=> ( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P2,X2)) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,A5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) )
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P2,X2))
<=> pp(aa(int,bool,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
=> ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
<=> ( ? [X5: int] :
( pp(member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& pp(aa(int,bool,P2,X5)) )
| ? [X5: int] :
( pp(member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D4)))
& ? [Xa2: int] :
( pp(member(int,Xa2,A5))
& pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa2),X5))) ) ) ) ) ) ) ) ) ).
% cppi
tff(fact_2127_aset_I8_J,axiom,
! [D4: int,A5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X3))
=> 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),X3),D4))) ) ) ) ).
% aset(8)
tff(fact_2128_aset_I6_J,axiom,
! [D4: int,T2: int,A5: 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)),A5))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X3),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),X3),D4)),T2)) ) ) ) ) ).
% aset(6)
tff(fact_2129_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))
=> ! [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,aa(int,fun(int,bool),ord_less_eq(int),T2),X3))
=> 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),X3),D4))) ) ) ) ) ).
% bset(8)
tff(fact_2130_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( ~ 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),D2)) ) ) ) ) ).
% atLeastatMost_subset_iff
tff(fact_2131_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_2132_Icc__eq__Icc,axiom,
! [A: $tType] :
( order(A)
=> ! [L: A,H2: A,L3: A,H: A] :
( ( set_or1337092689740270186AtMost(A,L,H2) = set_or1337092689740270186AtMost(A,L3,H) )
<=> ( ( ( L = L3 )
& ( H2 = H ) )
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H2))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L3),H)) ) ) ) ) ).
% Icc_eq_Icc
tff(fact_2133_semiring__norm_I12_J,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N) = N ).
% semiring_norm(12)
tff(fact_2134_semiring__norm_I11_J,axiom,
! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).
% semiring_norm(11)
tff(fact_2135_semiring__norm_I68_J,axiom,
! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),one2),N)) ).
% semiring_norm(68)
tff(fact_2136_semiring__norm_I75_J,axiom,
! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),one2)) ).
% semiring_norm(75)
tff(fact_2137_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_2138_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_2139_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_2140_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_2141_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_2142_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_2143_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_2144_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_2145_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_2146_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_2147_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_2148_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_2149_num__induct,axiom,
! [P: fun(num,bool),X: num] :
( pp(aa(num,bool,P,one2))
=> ( ! [X2: num] :
( pp(aa(num,bool,P,X2))
=> pp(aa(num,bool,P,inc(X2))) )
=> pp(aa(num,bool,P,X)) ) ) ).
% num_induct
tff(fact_2150_le__num__One__iff,axiom,
! [X: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),X),one2))
<=> ( X = one2 ) ) ).
% le_num_One_iff
tff(fact_2151_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( ( ~ 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),D2))
& ( 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),D2)) ) ) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).
% atLeastatMost_psubset_iff
tff(fact_2152_mult__numeral__1,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).
% mult_numeral_1
tff(fact_2153_mult__numeral__1__right,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).
% mult_numeral_1_right
tff(fact_2154_numeral__One,axiom,
! [A: $tType] :
( numeral(A)
=> ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).
% numeral_One
tff(fact_2155_divide__numeral__1,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).
% divide_numeral_1
tff(fact_2156_numerals_I1_J,axiom,
aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).
% numerals(1)
tff(fact_2157_add__One,axiom,
! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).
% add_One
tff(fact_2158_bset_I1_J,axiom,
! [D4: int,B5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,B5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,B5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),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,P,X3))
& pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)))
& pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) ) ) ) ) ).
% bset(1)
tff(fact_2159_bset_I2_J,axiom,
! [D4: int,B5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,B5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),D4))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,B5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),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,P,X3))
| pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)))
| pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) ) ) ) ) ).
% bset(2)
tff(fact_2160_aset_I1_J,axiom,
! [D4: int,A5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,A5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,A5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) )
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( ( pp(aa(int,bool,P,X3))
& pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)))
& pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) ) ) ) ) ).
% aset(1)
tff(fact_2161_aset_I2_J,axiom,
! [D4: int,A5: set(int),P: fun(int,bool),Q: fun(int,bool)] :
( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,A5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,P,X2))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) )
=> ( ! [X2: int] :
( ! [Xa: int] :
( pp(member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb: int] :
( pp(member(int,Xb,A5))
=> ( X2 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb),Xa) ) ) )
=> ( pp(aa(int,bool,Q,X2))
=> pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) )
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( ( pp(aa(int,bool,P,X3))
| pp(aa(int,bool,Q,X3)) )
=> ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)))
| pp(aa(int,bool,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) ) ) ) ) ).
% aset(2)
tff(fact_2162_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).
% mult_1s_ring_1(1)
tff(fact_2163_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).
% mult_1s_ring_1(2)
tff(fact_2164_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_2165_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).
% cong_exp_iff_simps(1)
tff(fact_2166_numeral__1__eq__Suc__0,axiom,
aa(num,nat,numeral_numeral(nat),one2) = aa(nat,nat,suc,zero_zero(nat)) ).
% numeral_1_eq_Suc_0
tff(fact_2167_bset_I9_J,axiom,
! [D2: int,D4: int,B5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),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,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T2))) ) ) ) ).
% bset(9)
tff(fact_2168_bset_I10_J,axiom,
! [D2: int,D4: int,B5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),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,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
=> ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T2))) ) ) ) ).
% bset(10)
tff(fact_2169_aset_I9_J,axiom,
! [D2: int,D4: int,A5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D4))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T2))) ) ) ) ).
% aset(9)
tff(fact_2170_aset_I10_J,axiom,
! [D2: int,D4: int,A5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),D4))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),T2)))
=> ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T2))) ) ) ) ).
% aset(10)
tff(fact_2171_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_2172_periodic__finite__ex,axiom,
! [D2: int,P: fun(int,bool)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
=> ( ! [X2: int,K2: int] :
( pp(aa(int,bool,P,X2))
<=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
=> ( ? [X_13: int] : pp(aa(int,bool,P,X_13))
<=> ? [X5: int] :
( pp(member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D2)))
& pp(aa(int,bool,P,X5)) ) ) ) ) ).
% periodic_finite_ex
tff(fact_2173_aset_I7_J,axiom,
! [D4: int,A5: set(int),T2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) ) ) ).
% aset(7)
tff(fact_2174_aset_I5_J,axiom,
! [D4: int,T2: int,A5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,T2,A5))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),T2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4)),T2)) ) ) ) ) ).
% aset(5)
tff(fact_2175_aset_I4_J,axiom,
! [D4: int,T2: int,A5: set(int)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
=> ( pp(member(int,T2,A5))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( ( X3 != T2 )
=> ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4) != T2 ) ) ) ) ) ).
% aset(4)
tff(fact_2176_aset_I3_J,axiom,
! [D4: int,T2: int,A5: 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)),A5))
=> ! [X3: int] :
( ! [Xa3: int] :
( pp(member(int,Xa3,set_or1337092689740270186AtMost(int,one_one(int),D4)))
=> ! [Xb2: int] :
( pp(member(int,Xb2,A5))
=> ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb2),Xa3) ) ) )
=> ( ( X3 = T2 )
=> ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4) = T2 ) ) ) ) ) ).
% aset(3)
tff(fact_2177_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))
=> ! [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,aa(int,fun(int,bool),ord_less(int),T2),X3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4))) ) ) ) ) ).
% bset(7)
tff(fact_2178_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))
=> ! [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,aa(int,fun(int,bool),ord_less(int),X3),T2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4)),T2)) ) ) ) ).
% bset(5)
tff(fact_2179_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))
=> ! [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) ) ) )
=> ( ( X3 != T2 )
=> ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4) != T2 ) ) ) ) ) ).
% bset(4)
tff(fact_2180_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))
=> ! [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) ) ) )
=> ( ( X3 = T2 )
=> ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D4) = T2 ) ) ) ) ) ).
% bset(3)
tff(fact_2181_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))
=> ! [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,aa(int,fun(int,bool),ord_less_eq(int),X3),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),X3),D4)),T2)) ) ) ) ).
% bset(6)
tff(fact_2182_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),bit0(one2))),Z2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),N)) ) ).
% pochhammer_double
tff(fact_2183_even__succ__mod__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),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),bit0(one2))),N))) ) ) ) ) ).
% even_succ_mod_exp
tff(fact_2184_even__succ__div__exp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),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),bit0(one2))),N)) ) ) ) ) ).
% even_succ_div_exp
tff(fact_2185_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),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),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ).
% fact_double
tff(fact_2186_round__unique,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),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),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))))
=> ( archimedean_round(A,X) = Y ) ) ) ) ).
% round_unique
tff(fact_2187_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
| ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),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),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))))) ) ) ) ) ).
% even_mult_exp_div_exp_iff
tff(fact_2188_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),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),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),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_2189_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y2: num] :
( ( bit0(X22) = bit0(Y2) )
<=> ( X22 = Y2 ) ) ).
% verit_eq_simplify(8)
tff(fact_2190_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( bit0(M) = bit0(N) )
<=> ( M = N ) ) ).
% semiring_norm(87)
tff(fact_2191_semiring__norm_I85_J,axiom,
! [M: num] : bit0(M) != one2 ).
% semiring_norm(85)
tff(fact_2192_semiring__norm_I83_J,axiom,
! [N: num] : one2 != bit0(N) ).
% semiring_norm(83)
tff(fact_2193_semiring__norm_I6_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% semiring_norm(6)
tff(fact_2194_semiring__norm_I13_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(N)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ).
% semiring_norm(13)
tff(fact_2195_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),bit0(M)),bit0(N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).
% semiring_norm(78)
tff(fact_2196_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),bit0(M)),bit0(N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).
% semiring_norm(71)
tff(fact_2197_semiring__norm_I2_J,axiom,
aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = bit0(one2) ).
% semiring_norm(2)
tff(fact_2198_num__double,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),N) = bit0(N) ).
% num_double
tff(fact_2199_semiring__norm_I69_J,axiom,
! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),bit0(M)),one2)) ).
% semiring_norm(69)
tff(fact_2200_semiring__norm_I76_J,axiom,
! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),bit0(N))) ).
% semiring_norm(76)
tff(fact_2201_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),bit0(one2)) ) ) ).
% one_add_one
tff(fact_2202_even__mult__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
| pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).
% even_mult_iff
tff(fact_2203_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),bit0(one2))) = one_one(A) ) ) ).
% bits_one_mod_two_eq_one
tff(fact_2204_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),bit0(one2))) = one_one(A) ) ) ).
% one_mod_two_eq_one
tff(fact_2205_zero__eq__power2,axiom,
! [A: $tType] :
( semiri2026040879449505780visors(A)
=> ! [A3: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% zero_eq_power2
tff(fact_2206_power2__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).
% power2_minus
tff(fact_2207_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),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
% add_2_eq_Suc'
tff(fact_2208_add__2__eq__Suc,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
% add_2_eq_Suc
tff(fact_2209_Suc__1,axiom,
aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).
% Suc_1
tff(fact_2210_even__Suc,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
<=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).
% even_Suc
tff(fact_2211_even__Suc__Suc__iff,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,N))))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).
% even_Suc_Suc_iff
tff(fact_2212_mod2__Suc__Suc,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% mod2_Suc_Suc
tff(fact_2213_div2__Suc__Suc,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).
% div2_Suc_Suc
tff(fact_2214_fact__2,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).
% fact_2
tff(fact_2215_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),bit0(one2))) = M ).
% add_self_div_2
tff(fact_2216_zmod__numeral__Bit0,axiom,
! [V: num,W2: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit0(V))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),W2))) ).
% zmod_numeral_Bit0
tff(fact_2217_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),bit0(one2))) = zero_zero(A) ) ) ).
% one_div_two_eq_zero
tff(fact_2218_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),bit0(one2))) = zero_zero(A) ) ) ).
% bits_1_div_2
tff(fact_2219_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),bit0(one2))) ) ) ).
% add_neg_numeral_special(9)
tff(fact_2220_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),bit0(one2)) ) ) ).
% diff_numeral_special(11)
tff(fact_2221_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),bit0(one2))) ) ) ).
% diff_numeral_special(10)
tff(fact_2222_even__plus__one__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ).
% even_plus_one_iff
tff(fact_2223_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),bit0(one2))) != zero_zero(A) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).
% not_mod_2_eq_0_eq_1
tff(fact_2224_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),bit0(one2))) != one_one(A) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).
% not_mod_2_eq_1_eq_0
tff(fact_2225_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),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).
% minus_1_div_2_eq
tff(fact_2226_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: 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(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)))
<=> ( A3 = zero_zero(A) ) ) ) ).
% power2_less_eq_zero_iff
tff(fact_2227_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
<=> ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
tff(fact_2228_zero__less__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
<=> ( A3 != zero_zero(A) ) ) ) ).
% zero_less_power2
tff(fact_2229_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),bit0(one2))) = one_one(A) ) ) ).
% bits_minus_1_mod_2_eq
tff(fact_2230_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),bit0(one2))) = one_one(A) ) ) ).
% minus_1_mod_2_eq
tff(fact_2231_of__bool__half__eq__0,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [B2: bool] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(bool,A,zero_neq_one_of_bool(A),B2)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).
% of_bool_half_eq_0
tff(fact_2232_Power_Oring__1__class_Opower__minus__even,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)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).
% Power.ring_1_class.power_minus_even
tff(fact_2233_power__minus__odd,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat,A3: A] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).
% power_minus_odd
tff(fact_2234_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).
% Parity.ring_1_class.power_minus_even
tff(fact_2235_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).
% not_mod2_eq_Suc_0_eq_0
tff(fact_2236_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),bit0(one2))) = zero_zero(nat) ).
% add_self_mod_2
tff(fact_2237_half__nonnegative__int__iff,axiom,
! [K: 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),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% half_nonnegative_int_iff
tff(fact_2238_half__negative__int__iff,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% half_negative_int_iff
tff(fact_2239_even__Suc__div__two,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).
% even_Suc_div_two
tff(fact_2240_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).
% odd_Suc_div_two
tff(fact_2241_even__succ__div__2,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).
% even_succ_div_2
tff(fact_2242_even__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).
% even_succ_div_two
tff(fact_2243_odd__succ__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),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),bit0(one2)))),one_one(A)) ) ) ) ).
% odd_succ_div_two
tff(fact_2244_even__power,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).
% even_power
tff(fact_2245_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
| ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).
% zero_le_power_eq_numeral
tff(fact_2246_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,W2: num] :
( 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(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
<=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% power_less_zero_eq_numeral
tff(fact_2247_power__less__zero__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( 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)),zero_zero(A)))
<=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% power_less_zero_eq
tff(fact_2248_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),bit0(one2))),N)) = one_one(A) ) ).
% power_minus1_even
tff(fact_2249_neg__one__even__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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_2250_neg__one__odd__power,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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_2251_even__of__nat,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ).
% even_of_nat
tff(fact_2252_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).
% odd_Suc_minus_one
tff(fact_2253_one__less__floor,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)) ) ) ).
% one_less_floor
tff(fact_2254_floor__le__one,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).
% floor_le_one
tff(fact_2255_even__diff__nat,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).
% even_diff_nat
tff(fact_2256_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),bit0(one2)))))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).
% mod2_gr_0
tff(fact_2257_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A3 ) ) ) ).
% odd_two_times_div_two_succ
tff(fact_2258_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),N)),one_one(A))))
<=> ( N = zero_zero(nat) ) ) ) ).
% semiring_parity_class.even_mask_iff
tff(fact_2259_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,W2: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W2))))
<=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
& ( A3 != zero_zero(A) ) )
| ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).
% zero_less_power_eq_numeral
tff(fact_2260_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),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_2261_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),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_2262_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),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_2263_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,W2: num] :
( 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(num,nat,numeral_numeral(nat),W2))),zero_zero(A)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2)))
& ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2)))
& ( A3 = zero_zero(A) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
tff(fact_2264_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),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_2265_verit__eq__simplify_I10_J,axiom,
! [X22: num] : one2 != bit0(X22) ).
% verit_eq_simplify(10)
tff(fact_2266_ex__nat__less,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [M6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N))
& pp(aa(nat,bool,P,M6)) )
<=> ? [X5: nat] :
( pp(member(nat,X5,set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
& pp(aa(nat,bool,P,X5)) ) ) ).
% ex_nat_less
tff(fact_2267_all__nat__less,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [M6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N))
=> pp(aa(nat,bool,P,M6)) )
<=> ! [X5: nat] :
( pp(member(nat,X5,set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,P,X5)) ) ) ).
% all_nat_less
tff(fact_2268_numeral__Bit0,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),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_2269_inc_Osimps_I1_J,axiom,
inc(one2) = bit0(one2) ).
% inc.simps(1)
tff(fact_2270_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num,Q4: num] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(Q4))) = zero_zero(A) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).
% cong_exp_iff_simps(2)
tff(fact_2271_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),N) ) ).
% numeral_Bit0_div_2
tff(fact_2272_power__minus__Bit0,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).
% power_minus_Bit0
tff(fact_2273_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),bit0(one2))),Z2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2
tff(fact_2274_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),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z2),Z2) ) ).
% mult_2_right
tff(fact_2275_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),bit0(one2))),A3)),B2) ) ).
% left_add_twice
tff(fact_2276_even__zero,axiom,
! [A: $tType] :
( semiring_parity(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A))) ) ).
% even_zero
tff(fact_2277_evenE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ~ ! [B4: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B4) ) ) ).
% evenE
tff(fact_2278_odd__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),one_one(A))) ) ).
% odd_one
tff(fact_2279_even__minus,axiom,
! [A: $tType] :
( ring_parity(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ).
% even_minus
tff(fact_2280_zero__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).
% zero_power2
tff(fact_2281_power2__eq__square,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).
% power2_eq_square
tff(fact_2282_power4__eq__xxxx,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(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(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).
% power4_eq_xxxx
tff(fact_2283_numeral__2__eq__2,axiom,
aa(num,nat,numeral_numeral(nat),bit0(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).
% numeral_2_eq_2
tff(fact_2284_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),bit0(one2))) = one_one(A) ) ) ).
% one_power2
tff(fact_2285_power2__eq__iff,axiom,
! [A: $tType] :
( idom(A)
=> ! [X: A,Y: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
<=> ( ( X = Y )
| ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).
% power2_eq_iff
tff(fact_2286_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).
% double_not_eq_Suc_double
tff(fact_2287_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).
% Suc_double_not_eq_double
tff(fact_2288_power__even__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).
% power_even_eq
tff(fact_2289_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),bit0(one2)) ).
% nat_1_add_1
tff(fact_2290_less__exp,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).
% less_exp
tff(fact_2291_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M))) ) ).
% self_le_ge2_pow
tff(fact_2292_power2__nat__le__eq__le,axiom,
! [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),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% power2_nat_le_eq_le
tff(fact_2293_power2__nat__le__imp__le,axiom,
! [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),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% power2_nat_le_imp_le
tff(fact_2294_even__even__mod__4__iff,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))))) ) ).
% even_even_mod_4_iff
tff(fact_2295_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Y)) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
tff(fact_2296_half__gt__zero__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),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% half_gt_zero_iff
tff(fact_2297_half__gt__zero,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),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ) ).
% half_gt_zero
tff(fact_2298_even__two__times__div__two,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = A3 ) ) ) ).
% even_two_times_div_two
tff(fact_2299_zero__le__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).
% zero_le_power2
tff(fact_2300_power2__eq__imp__eq,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: A,Y: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
tff(fact_2301_power2__le__imp__le,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: A,Y: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
=> ( 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),X),Y)) ) ) ) ).
% power2_le_imp_le
tff(fact_2302_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).
% even_iff_mod_2_eq_zero
tff(fact_2303_power2__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: 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(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))) ) ).
% power2_less_0
tff(fact_2304_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),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),bit0(one2))),M) != zero_zero(A) ) ) ) ).
% exp_add_not_zero_imp_left
tff(fact_2305_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),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),bit0(one2))),N) != zero_zero(A) ) ) ) ).
% exp_add_not_zero_imp_right
tff(fact_2306_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).
% odd_iff_mod_2_eq_one
tff(fact_2307_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_power2_eq_zero_iff
tff(fact_2308_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat,M: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) != zero_zero(A) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2309_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),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_2310_less__2__cases__iff,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
<=> ( ( N = zero_zero(nat) )
| ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% less_2_cases_iff
tff(fact_2311_less__2__cases,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
=> ( ( N = zero_zero(nat) )
| ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% less_2_cases
tff(fact_2312_abs__le__square__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).
% abs_le_square_iff
tff(fact_2313_power__mono__odd,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A3: A,B2: A] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( 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(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).
% power_mono_odd
tff(fact_2314_abs__square__eq__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
<=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).
% abs_square_eq_1
tff(fact_2315_odd__pos,axiom,
! [N: nat] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% odd_pos
tff(fact_2316_uminus__power__if,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat,A3: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).
% uminus_power_if
tff(fact_2317_nat__2,axiom,
aa(int,nat,nat2,aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).
% nat_2
tff(fact_2318_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),bit0(one2))))) )
=> pp(aa(nat,bool,P,N)) ) ) ) ).
% nat_induct2
tff(fact_2319_not__exp__less__eq__0__int,axiom,
! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),zero_zero(int))) ).
% not_exp_less_eq_0_int
tff(fact_2320_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))) ) ).
% diff_le_diff_pow
tff(fact_2321_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
% dvd_power_iff_le
tff(fact_2322_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( 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),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),bit0(one2))),B2))),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),bit0(one2))),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ) ) ) ).
% divmod_digit_0(2)
tff(fact_2323_oddE,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ~ ! [B4: 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),bit0(one2))),B4)),one_one(A)) ) ) ).
% oddE
tff(fact_2324_power2__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
=> ( 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(A),X),Y)) ) ) ) ).
% power2_less_imp_less
tff(fact_2325_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
% sum_power2_le_zero_iff
tff(fact_2326_sum__power2__ge__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).
% sum_power2_ge_zero
tff(fact_2327_mod2__eq__if,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ) ).
% mod2_eq_if
tff(fact_2328_parity__cases,axiom,
! [A: $tType] :
( semiring_parity(A)
=> ! [A3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
=> ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).
% parity_cases
tff(fact_2329_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),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),bit0(one2)))) = zero_zero(A) ) ) ) ).
% bits_stable_imp_add_self
tff(fact_2330_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))) ) ).
% not_sum_power2_lt_zero
tff(fact_2331_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
% sum_power2_gt_zero_iff
tff(fact_2332_power2__sum,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: 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),X),Y)),aa(num,nat,numeral_numeral(nat),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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),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),bit0(one2))),X)),Y)) ) ).
% power2_sum
tff(fact_2333_square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))) ) ) ) ).
% square_le_1
tff(fact_2334_zero__le__even__power_H,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ) ).
% zero_le_even_power'
tff(fact_2335_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( 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),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).
% power2_le_iff_abs_le
tff(fact_2336_zero__le__power__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
| ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).
% zero_le_power_eq
tff(fact_2337_zero__le__odd__power,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A3: A] :
( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).
% zero_le_odd_power
tff(fact_2338_zero__le__even__power,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).
% zero_le_even_power
tff(fact_2339_abs__square__le__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: 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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).
% abs_square_le_1
tff(fact_2340_abs__square__less__1,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).
% abs_square_less_1
tff(fact_2341_bits__induct,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [P: fun(A,bool),A3: A] :
( ! [A6: A] :
( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A6),aa(num,A,numeral_numeral(A),bit0(one2))) = A6 )
=> pp(aa(A,bool,P,A6)) )
=> ( ! [A6: A,B4: bool] :
( pp(aa(A,bool,P,A6))
=> ( ( 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),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A6))),aa(num,A,numeral_numeral(A),bit0(one2))) = A6 )
=> 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),B4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A6)))) ) )
=> pp(aa(A,bool,P,A3)) ) ) ) ).
% bits_induct
tff(fact_2342_minus__power__mult__self,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [A3: 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),A3)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).
% minus_power_mult_self
tff(fact_2343_power__odd__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).
% power_odd_eq
tff(fact_2344_nat__bit__induct,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,N3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))) ) )
=> ( ! [N3: nat] :
( pp(aa(nat,bool,P,N3))
=> pp(aa(nat,bool,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)))) )
=> pp(aa(nat,bool,P,N)) ) ) ) ).
% nat_bit_induct
tff(fact_2345_minus__one__power__iff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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_2346_power__mono__even,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A3: A,B2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(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),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).
% power_mono_even
tff(fact_2347_Suc__n__div__2__gt__zero,axiom,
! [N: 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)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).
% Suc_n_div_2_gt_zero
tff(fact_2348_div__2__gt__zero,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).
% div_2_gt_zero
tff(fact_2349_minus__1__div__exp__eq__int,axiom,
! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).
% minus_1_div_exp_eq_int
tff(fact_2350_triangle__def,axiom,
! [N: nat] : nat_triangle(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),bit0(one2))) ).
% triangle_def
tff(fact_2351_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [B2: A,A3: A] :
( 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),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),bit0(one2))),B2))),B2))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).
% divmod_digit_0(1)
tff(fact_2352_power2__diff,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [X: 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),X),Y)),aa(num,nat,numeral_numeral(nat),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),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),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),bit0(one2))),X)),Y)) ) ).
% power2_diff
tff(fact_2353_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% odd_0_le_power_imp_0_le
tff(fact_2354_odd__power__less__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( 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(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),zero_zero(A))) ) ) ).
% odd_power_less_zero
tff(fact_2355_zero__less__power__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
<=> ( ( N = zero_zero(nat) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
& ( A3 != zero_zero(A) ) )
| ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).
% zero_less_power_eq
tff(fact_2356_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(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),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).
% mult_exp_mod_exp_eq
tff(fact_2357_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),bit0(one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% power_minus1_odd
tff(fact_2358_exp__mod__exp,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).
% exp_mod_exp
tff(fact_2359_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))))
=> ( ! [K2: int] :
( pp(aa(int,bool,P,K2))
=> ( ( K2 != zero_zero(int) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
=> ( ! [K2: int] :
( pp(aa(int,bool,P,K2))
=> ( ( K2 != 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),K2),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
=> pp(aa(int,bool,P,K)) ) ) ) ) ).
% int_bit_induct
tff(fact_2360_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),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_2361_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),bit0(one2))),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),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_2362_round__def,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% round_def
tff(fact_2363_mod__double__modulus,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: A,X: 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)),X))
=> ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),M) )
| ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),X),M)),M) ) ) ) ) ) ).
% mod_double_modulus
tff(fact_2364_divmod__digit__1_I2_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),bit0(one2))),B2))))
=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),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),bit0(one2))),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2) ) ) ) ) ) ).
% divmod_digit_1(2)
tff(fact_2365_even__mask__div__iff_H,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).
% even_mask_div_iff'
tff(fact_2366_power__le__zero__eq,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,N: nat] :
( 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)),zero_zero(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),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
& ( A3 = zero_zero(A) ) ) ) ) ) ) ).
% power_le_zero_eq
tff(fact_2367_of__int__round__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% of_int_round_le
tff(fact_2368_of__int__round__ge,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).
% of_int_round_ge
tff(fact_2369_of__int__round__gt,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).
% of_int_round_gt
tff(fact_2370_even__mod__4__div__2,axiom,
! [N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).
% even_mod_4_div_2
tff(fact_2371_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),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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_2372_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),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A3) ) ) ).
% pos_zdiv_mult_2
tff(fact_2373_even__nat__iff,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(int,nat,nat2,K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)) ) ) ).
% even_nat_iff
tff(fact_2374_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),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),A3))) ) ) ).
% pos_zmod_mult_2
tff(fact_2375_even__mask__div__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
<=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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_2376_exp__div__exp__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).
% exp_div_exp_eq
tff(fact_2377_of__int__round__abs__le,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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,X))),X))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% of_int_round_abs_le
tff(fact_2378_round__unique_H,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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),X),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),bit0(one2)))))
=> ( archimedean_round(A,X) = N ) ) ) ).
% round_unique'
tff(fact_2379_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),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),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_2380_round__altdef,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [X: 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),bit0(one2)))),archimedean_frac(A,X)))
=> ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
& ( ~ 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),bit0(one2)))),archimedean_frac(A,X)))
=> ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).
% round_altdef
tff(fact_2381_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),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% flip_bit_0
tff(fact_2382_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),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).
% set_bit_0
tff(fact_2383_psubsetI,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ( A5 != B5 )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5)) ) ) ).
% psubsetI
tff(fact_2384_unset__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% unset_bit_0
tff(fact_2385_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),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2)))))) ) ).
% unset_bit_Suc
tff(fact_2386_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),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2)))))) ) ).
% flip_bit_Suc
tff(fact_2387_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),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),bit0(one2)))))) ) ).
% set_bit_Suc
tff(fact_2388_Compl__subset__Compl__iff,axiom,
! [A: $tType,A5: set(A),B5: 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)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B5)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5)) ) ).
% Compl_subset_Compl_iff
tff(fact_2389_Compl__anti__mono,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> 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)),B5)),aa(set(A),set(A),uminus_uminus(set(A)),A5))) ) ).
% Compl_anti_mono
tff(fact_2390_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% unset_bit_nonnegative_int_iff
tff(fact_2391_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% set_bit_nonnegative_int_iff
tff(fact_2392_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% flip_bit_nonnegative_int_iff
tff(fact_2393_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% unset_bit_negative_int_iff
tff(fact_2394_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% set_bit_negative_int_iff
tff(fact_2395_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),N),K)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% flip_bit_negative_int_iff
tff(fact_2396_unset__bit__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),N) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),N))) ).
% unset_bit_nat_def
tff(fact_2397_unset__bit__less__eq,axiom,
! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),K)) ).
% unset_bit_less_eq
tff(fact_2398_set__bit__greater__eq,axiom,
! [K: int,N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K))) ).
% set_bit_greater_eq
tff(fact_2399_psubset__imp__ex__mem,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> ? [B4: A] : pp(member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ).
% psubset_imp_ex_mem
tff(fact_2400_psubset__trans,axiom,
! [A: $tType,A5: set(A),B5: set(A),C6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B5),C6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),C6)) ) ) ).
% psubset_trans
tff(fact_2401_psubsetD,axiom,
! [A: $tType,A5: set(A),B5: set(A),C2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> ( pp(member(A,C2,A5))
=> pp(member(A,C2,B5)) ) ) ).
% psubsetD
tff(fact_2402_psubsetE,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5)) ) ) ).
% psubsetE
tff(fact_2403_psubset__eq,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
& ( A5 != B5 ) ) ) ).
% psubset_eq
tff(fact_2404_psubset__imp__subset,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5)) ) ).
% psubset_imp_subset
tff(fact_2405_psubset__subset__trans,axiom,
! [A: $tType,A5: set(A),B5: set(A),C6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),C6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),C6)) ) ) ).
% psubset_subset_trans
tff(fact_2406_subset__not__subset__eq,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
& ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5)) ) ) ).
% subset_not_subset_eq
tff(fact_2407_subset__psubset__trans,axiom,
! [A: $tType,A5: set(A),B5: set(A),C6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B5),C6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),C6)) ) ) ).
% subset_psubset_trans
tff(fact_2408_subset__iff__psubset__eq,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
| ( A5 = B5 ) ) ) ).
% subset_iff_psubset_eq
tff(fact_2409_even__set__bit__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
& ( M != zero_zero(nat) ) ) ) ) ).
% even_set_bit_iff
tff(fact_2410_even__flip__bit__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A3)))
<=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
<=> ( M = zero_zero(nat) ) ) ) ) ).
% even_flip_bit_iff
tff(fact_2411_even__unset__bit__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A3)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
| ( M = zero_zero(nat) ) ) ) ) ).
% even_unset_bit_iff
tff(fact_2412_signed__take__bit__rec,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] :
( ( ( N = zero_zero(nat) )
=> ( aa(A,A,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),bit0(one2)))) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(A,A,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),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,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),bit0(one2)))))) ) ) ) ) ).
% signed_take_bit_rec
tff(fact_2413_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),bit0(one2))) ) ) ).
% dbl_simps(4)
tff(fact_2414_dbl__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).
% dbl_simps(3)
tff(fact_2415_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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_concat_bit(N,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))),L))) ).
% concat_bit_Suc
tff(fact_2416_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( pp(member(nat,aa(nat,nat,suc,N),nat_set_decode(X)))
<=> pp(member(nat,N,nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).
% set_decode_Suc
tff(fact_2417_signed__take__bit__Suc,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : aa(A,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),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).
% signed_take_bit_Suc
tff(fact_2418_Compl__eq__Compl__iff,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( ( aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),uminus_uminus(set(A)),B5) )
<=> ( A5 = B5 ) ) ).
% Compl_eq_Compl_iff
tff(fact_2419_Compl__iff,axiom,
! [A: $tType,C2: A,A5: set(A)] :
( pp(member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A5)))
<=> ~ pp(member(A,C2,A5)) ) ).
% Compl_iff
tff(fact_2420_ComplI,axiom,
! [A: $tType,C2: A,A5: set(A)] :
( ~ pp(member(A,C2,A5))
=> pp(member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A5))) ) ).
% ComplI
tff(fact_2421_signed__take__bit__of__0,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).
% signed_take_bit_of_0
tff(fact_2422_concat__bit__0,axiom,
! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).
% concat_bit_0
tff(fact_2423_dbl__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_dbl(A,zero_zero(A)) = zero_zero(A) ) ) ).
% dbl_simps(2)
tff(fact_2424_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,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_2425_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).
% signed_take_bit_Suc_1
tff(fact_2426_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [K: num] : aa(A,A,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_2427_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_concat_bit(N,K,L)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ).
% concat_bit_nonnegative_iff
tff(fact_2428_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_concat_bit(N,K,L)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ).
% concat_bit_negative_iff
tff(fact_2429_dbl__simps_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).
% dbl_simps(5)
tff(fact_2430_dbl__simps_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).
% dbl_simps(1)
tff(fact_2431_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_Suc_bit0
tff(fact_2432_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_Suc_minus_bit0
tff(fact_2433_set__decode__0,axiom,
! [X: nat] :
( pp(member(nat,zero_zero(nat),nat_set_decode(X)))
<=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X)) ) ).
% set_decode_0
tff(fact_2434_signed__take__bit__0,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),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),bit0(one2)))) ) ).
% signed_take_bit_0
tff(fact_2435_double__complement,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = A5 ).
% double_complement
tff(fact_2436_ComplD,axiom,
! [A: $tType,C2: A,A5: set(A)] :
( pp(member(A,C2,aa(set(A),set(A),uminus_uminus(set(A)),A5)))
=> ~ pp(member(A,C2,A5)) ) ).
% ComplD
tff(fact_2437_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).
% signed_take_bit_mult
tff(fact_2438_signed__take__bit__minus,axiom,
! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).
% signed_take_bit_minus
tff(fact_2439_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% subset_decode_imp_le
tff(fact_2440_dbl__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).
% dbl_def
tff(fact_2441_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).
% signed_take_bit_int_less_exp
tff(fact_2442_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).
% signed_take_bit_int_greater_eq_self_iff
tff(fact_2443_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K)) ) ).
% signed_take_bit_int_less_self_iff
tff(fact_2444_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ).
% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2445_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K)) ) ).
% signed_take_bit_int_less_eq_self_iff
tff(fact_2446_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
<=> 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),bit0(one2))),N)))) ) ).
% signed_take_bit_int_greater_self_iff
tff(fact_2447_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N))))) ) ).
% signed_take_bit_int_less_eq
tff(fact_2448_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).
% signed_take_bit_int_eq_self
tff(fact_2449_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).
% signed_take_bit_int_eq_self_iff
tff(fact_2450_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),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),bit0(one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ) ).
% signed_take_bit_int_greater_eq
tff(fact_2451_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa(int,int,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),aa(int,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),bit0(one2)))),one_one(int)) ).
% signed_take_bit_Suc_minus_bit1
tff(fact_2452_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),bit0(one2)),N,one_one(nat))) ) ).
% fact_code
tff(fact_2453_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),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).
% take_bit_rec
tff(fact_2454_num_Osize__gen_I2_J,axiom,
! [X22: num] : size_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_2455_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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).
% and_int_unfold
tff(fact_2456_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),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).
% xor_Suc_0_eq
tff(fact_2457_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( aa(num,num,bit1,M) = aa(num,num,bit1,N) )
<=> ( M = N ) ) ).
% semiring_norm(90)
tff(fact_2458_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
<=> ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
tff(fact_2459_take__bit__of__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),zero_zero(A)) = zero_zero(A) ) ).
% take_bit_of_0
tff(fact_2460_and__zero__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% and_zero_eq
tff(fact_2461_zero__and__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% zero_and_eq
tff(fact_2462_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).
% bit.conj_zero_left
tff(fact_2463_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).
% bit.conj_zero_right
tff(fact_2464_semiring__norm_I88_J,axiom,
! [M: num,N: num] : bit0(M) != aa(num,num,bit1,N) ).
% semiring_norm(88)
tff(fact_2465_semiring__norm_I89_J,axiom,
! [M: num,N: num] : aa(num,num,bit1,M) != bit0(N) ).
% semiring_norm(89)
tff(fact_2466_semiring__norm_I84_J,axiom,
! [N: num] : one2 != aa(num,num,bit1,N) ).
% semiring_norm(84)
tff(fact_2467_semiring__norm_I86_J,axiom,
! [M: num] : aa(num,num,bit1,M) != one2 ).
% semiring_norm(86)
tff(fact_2468_bit_Oxor__self,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).
% bit.xor_self
tff(fact_2469_xor__self__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).
% xor_self_eq
tff(fact_2470_xor_Oleft__neutral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).
% xor.left_neutral
tff(fact_2471_xor_Oright__neutral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).
% xor.right_neutral
tff(fact_2472_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] : bit_concat_bit(N,K,zero_zero(int)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) ).
% concat_bit_of_zero_2
tff(fact_2473_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).
% semiring_norm(73)
tff(fact_2474_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).
% semiring_norm(80)
tff(fact_2475_take__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),zero_zero(nat)),A3) = zero_zero(A) ) ).
% take_bit_0
tff(fact_2476_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_2477_bit_Oconj__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).
% bit.conj_one_right
tff(fact_2478_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_2479_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_2480_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_2481_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ).
% of_nat_nat_take_bit_eq
tff(fact_2482_and__nonnegative__int__iff,axiom,
! [K: int,L: 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),bit_se5824344872417868541ns_and(int),K),L)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
| pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).
% and_nonnegative_int_iff
tff(fact_2483_and__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(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),ord_less(int),L),zero_zero(int))) ) ) ).
% and_negative_int_iff
tff(fact_2484_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)),bit0(N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% semiring_norm(9)
tff(fact_2485_semiring__norm_I7_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(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_2486_semiring__norm_I14_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,N))) ).
% semiring_norm(14)
tff(fact_2487_semiring__norm_I15_J,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),N)) ).
% semiring_norm(15)
tff(fact_2488_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),bit0(N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).
% semiring_norm(81)
tff(fact_2489_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),bit0(M)),aa(num,num,bit1,N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).
% semiring_norm(72)
tff(fact_2490_semiring__norm_I77_J,axiom,
! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit1,N))) ).
% semiring_norm(77)
tff(fact_2491_semiring__norm_I70_J,axiom,
! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),one2)) ).
% semiring_norm(70)
tff(fact_2492_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K)) ) ).
% dbl_inc_simps(5)
tff(fact_2493_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_2494_and__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).
% and_numerals(8)
tff(fact_2495_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_2496_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)) = 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_2497_semiring__norm_I8_J,axiom,
! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).
% semiring_norm(8)
tff(fact_2498_semiring__norm_I5_J,axiom,
! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),one2) = aa(num,num,bit1,M) ).
% semiring_norm(5)
tff(fact_2499_semiring__norm_I4_J,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).
% semiring_norm(4)
tff(fact_2500_semiring__norm_I3_J,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(N)) = aa(num,num,bit1,N) ).
% semiring_norm(3)
tff(fact_2501_take__bit__of__Suc__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).
% take_bit_of_Suc_0
tff(fact_2502_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)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).
% semiring_norm(16)
tff(fact_2503_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),bit0(N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).
% semiring_norm(74)
tff(fact_2504_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),bit0(M)),aa(num,num,bit1,N)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).
% semiring_norm(79)
tff(fact_2505_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),bit0(Y))) = zero_zero(A) ) ).
% and_numerals(1)
tff(fact_2506_and__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = zero_zero(A) ) ).
% and_numerals(5)
tff(fact_2507_and__numerals_I3_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% and_numerals(3)
tff(fact_2508_xor__numerals_I3_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% xor_numerals(3)
tff(fact_2509_xor__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(X)) ) ).
% xor_numerals(8)
tff(fact_2510_xor__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).
% xor_numerals(5)
tff(fact_2511_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),bit0(Y)) ) ).
% xor_numerals(2)
tff(fact_2512_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),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% xor_numerals(1)
tff(fact_2513_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_2514_and__minus__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = one_one(int) ).
% and_minus_numerals(2)
tff(fact_2515_and__minus__numerals_I6_J,axiom,
! [N: 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)))),one_one(int)) = one_one(int) ).
% and_minus_numerals(6)
tff(fact_2516_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_2517_even__take__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)))
<=> ( ( N = zero_zero(nat) )
| pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).
% even_take_bit_eq
tff(fact_2518_and__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% and_numerals(4)
tff(fact_2519_and__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: 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,X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% and_numerals(6)
tff(fact_2520_xor__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: 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,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% xor_numerals(7)
tff(fact_2521_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_2522_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_2523_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_2524_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_2525_and__minus__numerals_I5_J,axiom,
! [N: 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),bit0(N)))),one_one(int)) = zero_zero(int) ).
% and_minus_numerals(5)
tff(fact_2526_and__minus__numerals_I1_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = zero_zero(int) ).
% and_minus_numerals(1)
tff(fact_2527_xor__nat__numerals_I4_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(X)) ).
% xor_nat_numerals(4)
tff(fact_2528_xor__nat__numerals_I3_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).
% xor_nat_numerals(3)
tff(fact_2529_xor__nat__numerals_I2_J,axiom,
! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).
% xor_nat_numerals(2)
tff(fact_2530_xor__nat__numerals_I1_J,axiom,
! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).
% xor_nat_numerals(1)
tff(fact_2531_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_2532_take__bit__Suc__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,zero_zero(nat))),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_Suc_0
tff(fact_2533_and__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: 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,X))),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% and_numerals(7)
tff(fact_2534_xor__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: 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,X))),aa(num,A,numeral_numeral(A),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% xor_numerals(6)
tff(fact_2535_xor__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% xor_numerals(4)
tff(fact_2536_take__bit__of__exp,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: nat,N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).
% take_bit_of_exp
tff(fact_2537_take__bit__of__2,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% take_bit_of_2
tff(fact_2538_zmod__numeral__Bit1,axiom,
! [V: num,W2: 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),bit0(W2))) = 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),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),W2)))),one_one(int)) ).
% zmod_numeral_Bit1
tff(fact_2539_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] : aa(int,int,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),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_Suc_bit1
tff(fact_2540_take__bit__of__nat,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),M)) ) ).
% take_bit_of_nat
tff(fact_2541_of__nat__xor__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_xor_eq
tff(fact_2542_of__nat__and__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_and_eq
tff(fact_2543_take__bit__tightened,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A,B2: A,M: nat] :
( ( 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) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A3) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),B2) ) ) ) ) ).
% take_bit_tightened
tff(fact_2544_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q4: 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),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),Q4))) ) ).
% take_bit_tightened_less_eq_nat
tff(fact_2545_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),M)),M)) ).
% take_bit_nat_less_eq_self
tff(fact_2546_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] : bit0(X22) != aa(num,num,bit1,X32) ).
% verit_eq_simplify(14)
tff(fact_2547_verit__eq__simplify_I12_J,axiom,
! [X32: num] : one2 != aa(num,num,bit1,X32) ).
% verit_eq_simplify(12)
tff(fact_2548_take__bit__mult,axiom,
! [N: nat,K: int,L: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),L))) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).
% take_bit_mult
tff(fact_2549_take__bit__minus,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,uminus_uminus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,uminus_uminus(int),K)) ).
% take_bit_minus
tff(fact_2550_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) ) ) ).
% take_bit_nat_eq
tff(fact_2551_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(int,nat,nat2,K)) ) ) ).
% nat_take_bit_eq
tff(fact_2552_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_2553_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] : Y != bit0(X23)
=> ~ ! [X33: num] : Y != aa(num,num,bit1,X33) ) ) ).
% num.exhaust
tff(fact_2554_AND__upper2_H,axiom,
! [Y: int,Z2: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z2)) ) ) ).
% AND_upper2'
tff(fact_2555_AND__upper1_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z2)) ) ) ).
% AND_upper1'
tff(fact_2556_AND__upper2,axiom,
! [Y: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y)) ) ).
% AND_upper2
tff(fact_2557_AND__upper1,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X)) ) ).
% AND_upper1
tff(fact_2558_AND__lower,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y))) ) ).
% AND_lower
tff(fact_2559_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),K))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% take_bit_int_less_eq_self_iff
tff(fact_2560_take__bit__nonnegative,axiom,
! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) ).
% take_bit_nonnegative
tff(fact_2561_not__take__bit__negative,axiom,
! [N: nat,K: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),zero_zero(int))) ).
% not_take_bit_negative
tff(fact_2562_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% take_bit_int_greater_self_iff
tff(fact_2563_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) ) ).
% take_bit_tightened_less_eq_int
tff(fact_2564_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A,B2: A] :
( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,bit_ri4674362597316999326ke_bit(A,N),B2) )
<=> ( 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_se2584673776208193580ke_bit(A),aa(nat,nat,suc,N)),B2) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
tff(fact_2565_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [M: nat,N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M),aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),bit_ri4674362597316999326ke_bit(A,M)),A3) ) ).
% signed_take_bit_take_bit
tff(fact_2566_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_2567_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,M: nat,A3: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)) ) ) ) ) ).
% take_bit_unset_bit_eq
tff(fact_2568_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,M: nat,A3: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)) ) ) ) ) ).
% take_bit_set_bit_eq
tff(fact_2569_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,M: nat,A3: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)) ) ) ) ) ).
% take_bit_flip_bit_eq
tff(fact_2570_inc_Osimps_I3_J,axiom,
! [X: num] : inc(aa(num,num,bit1,X)) = bit0(inc(X)) ).
% inc.simps(3)
tff(fact_2571_inc_Osimps_I2_J,axiom,
! [X: num] : inc(bit0(X)) = aa(num,num,bit1,X) ).
% inc.simps(2)
tff(fact_2572_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_2573_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [M: nat,N: nat,A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A3) ) ) ) ).
% take_bit_signed_take_bit
tff(fact_2574_and__less__eq,axiom,
! [L: int,K: int] :
( 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_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K)) ) ).
% and_less_eq
tff(fact_2575_AND__upper1_H_H,axiom,
! [Y: int,Z2: int,Ya: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z2)) ) ) ).
% AND_upper1''
tff(fact_2576_AND__upper2_H_H,axiom,
! [Y: int,Z2: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z2)) ) ) ).
% AND_upper2''
tff(fact_2577_eval__nat__numeral_I3_J,axiom,
! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(N))) ).
% eval_nat_numeral(3)
tff(fact_2578_power__minus__Bit1,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).
% power_minus_Bit1
tff(fact_2579_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) != zero_zero(int) )
=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),one_one(int)) ) ) ).
% take_bit_decr_eq
tff(fact_2580_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),bit0(one2)))),one_one(A)) ) ).
% take_bit_Suc_bit1
tff(fact_2581_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num,Q4: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(Q4))) != zero_zero(A) ) ).
% cong_exp_iff_simps(3)
tff(fact_2582_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),N) ) ).
% numeral_Bit1_div_2
tff(fact_2583_power3__eq__cube,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),A3) ) ).
% power3_eq_cube
tff(fact_2584_numeral__3__eq__3,axiom,
aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).
% numeral_3_eq_3
tff(fact_2585_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_2586_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),bit0(one2)))),one_one(int)) ).
% take_bit_Suc_minus_bit1
tff(fact_2587_take__bit__Suc__bit0,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),bit0(K))) = 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),bit0(one2))) ) ).
% take_bit_Suc_bit0
tff(fact_2588_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),bit0(one2))) ) ).
% and_one_eq
tff(fact_2589_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),bit0(one2))) ) ).
% one_and_eq
tff(fact_2590_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),M) = M ) ) ).
% take_bit_nat_eq_self
tff(fact_2591_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).
% take_bit_nat_less_exp
tff(fact_2592_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),M) = M )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ).
% take_bit_nat_eq_self_iff
tff(fact_2593_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Q4: num,N: num] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q4))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(Q4))) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).
% cong_exp_iff_simps(7)
tff(fact_2594_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,Q4: num] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q4))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q4))) )
<=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q4)) = zero_zero(A) ) ) ) ).
% cong_exp_iff_simps(11)
tff(fact_2595_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).
% take_bit_int_less_exp
tff(fact_2596_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType,B2: nat,A3: nat,F: fun(nat,fun(A,A)),Acc: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc) = Acc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc) = 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),Acc)) ) ) ) ).
% fold_atLeastAtMost_nat.simps
tff(fact_2597_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: fun(nat,fun(A,A)),Xa4: nat,Xb3: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X,Xa4,Xb3,Xc) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa4))
=> ( Y = Xc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa4))
=> ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa4),one_one(nat)),Xb3,aa(A,A,aa(nat,fun(A,A),X,Xa4),Xc)) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
tff(fact_2598_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_2599_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_2600_num_Osize__gen_I1_J,axiom,
size_num(one2) = zero_zero(nat) ).
% num.size_gen(1)
tff(fact_2601_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = zero_zero(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A3)) ) ) ).
% take_bit_eq_0_iff
tff(fact_2602_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),M)),M))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M)) ) ).
% take_bit_nat_less_self_iff
tff(fact_2603_take__bit__Suc__minus__bit0,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),bit0(K)))) = 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),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% take_bit_Suc_minus_bit0
tff(fact_2604_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)),K))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K)) ) ).
% take_bit_int_less_self_iff
tff(fact_2605_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).
% take_bit_int_greater_eq_self_iff
tff(fact_2606_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = K ) ) ) ).
% take_bit_int_eq_self
tff(fact_2607_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K) = K )
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).
% take_bit_int_eq_self_iff
tff(fact_2608_signed__take__bit__eq__take__bit__shift,axiom,
! [N: nat,K: int] : aa(int,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),bit0(one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).
% signed_take_bit_eq_take_bit_shift
tff(fact_2609_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),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_2610_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),bit0(one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).
% take_bit_Suc_minus_1_eq
tff(fact_2611_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),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% take_bit_Suc
tff(fact_2612_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),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).
% take_bit_numeral_minus_1_eq
tff(fact_2613_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(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),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ) ) ).
% take_bit_int_less_eq
tff(fact_2614_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),bit0(one2))),N))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),K))) ) ).
% take_bit_int_greater_eq
tff(fact_2615_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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).
% xor_nat_unfold
tff(fact_2616_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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% and_int_rec
tff(fact_2617_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),fNot(aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).
% xor_nat_rec
tff(fact_2618_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),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))) ) ).
% xor_one_eq
tff(fact_2619_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),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))) ) ).
% one_xor_eq
tff(fact_2620_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),bit0(one2))) = A3 )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = zero_zero(A) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),N)),one_one(A)) ) ) ) ) ) ).
% stable_imp_take_bit_eq
tff(fact_2621_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( 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),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),N),aa(int,int,uminus_uminus(int),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),bit0(one2))),N)),K) ) ) ) ).
% take_bit_minus_small_eq
tff(fact_2622_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).
% odd_mod_4_div_2
tff(fact_2623_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),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).
% Suc_0_xor_eq
tff(fact_2624_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa(int,int,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),aa(int,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),bit0(one2)))),one_one(int)) ).
% signed_take_bit_numeral_minus_bit1
tff(fact_2625_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),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),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_2626_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] : aa(int,int,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),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).
% signed_take_bit_numeral_bit1
tff(fact_2627_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),bit0(one2)))),one_one(int)) ).
% take_bit_numeral_minus_bit1
tff(fact_2628_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),bit0(one2)))),one_one(A)) ) ).
% take_bit_numeral_bit1
tff(fact_2629_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_numeral_minus_bit0
tff(fact_2630_xor__nonnegative__int__iff,axiom,
! [K: int,L: 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),bit_se5824344971392196577ns_xor(int),K),L)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).
% xor_nonnegative_int_iff
tff(fact_2631_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(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),ord_less(int),L),zero_zero(int))) ) ) ).
% xor_negative_int_iff
tff(fact_2632_binomial__Suc__n,axiom,
! [N: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),N) = aa(nat,nat,suc,N) ).
% binomial_Suc_n
tff(fact_2633_binomial__n__n,axiom,
! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).
% binomial_n_n
tff(fact_2634_binomial__1,axiom,
! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).
% binomial_1
tff(fact_2635_binomial__0__Suc,axiom,
! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).
% binomial_0_Suc
tff(fact_2636_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ).
% binomial_eq_0_iff
tff(fact_2637_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_2638_pred__numeral__simps_I1_J,axiom,
pred_numeral(one2) = zero_zero(nat) ).
% pred_numeral_simps(1)
tff(fact_2639_binomial__n__0,axiom,
! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).
% binomial_n_0
tff(fact_2640_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,N) )
<=> ( pred_numeral(K) = N ) ) ).
% eq_numeral_Suc
tff(fact_2641_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K) )
<=> ( N = pred_numeral(K) ) ) ).
% Suc_eq_numeral
tff(fact_2642_pred__numeral__inc,axiom,
! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).
% pred_numeral_inc
tff(fact_2643_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).
% zero_less_binomial_iff
tff(fact_2644_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),pred_numeral(K))) ) ).
% less_Suc_numeral
tff(fact_2645_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K)),N)) ) ).
% less_numeral_Suc
tff(fact_2646_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),pred_numeral(K))) ) ).
% le_Suc_numeral
tff(fact_2647_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K)),N)) ) ).
% le_numeral_Suc
tff(fact_2648_diff__numeral__Suc,axiom,
! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),pred_numeral(K)),N) ).
% diff_numeral_Suc
tff(fact_2649_diff__Suc__numeral,axiom,
! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),pred_numeral(K)) ).
% diff_Suc_numeral
tff(fact_2650_pred__numeral__simps_I3_J,axiom,
! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).
% pred_numeral_simps(3)
tff(fact_2651_and__nat__numerals_I3_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).
% and_nat_numerals(3)
tff(fact_2652_and__nat__numerals_I1_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),bit0(Y))) = zero_zero(nat) ).
% and_nat_numerals(1)
tff(fact_2653_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_2654_and__nat__numerals_I4_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).
% and_nat_numerals(4)
tff(fact_2655_Suc__0__and__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% Suc_0_and_eq
tff(fact_2656_and__Suc__0__eq,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% and_Suc_0_eq
tff(fact_2657_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% signed_take_bit_numeral_bit0
tff(fact_2658_choose__one,axiom,
! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).
% choose_one
tff(fact_2659_binomial__eq__0,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) ) ) ).
% binomial_eq_0
tff(fact_2660_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).
% Suc_times_binomial_eq
tff(fact_2661_Suc__times__binomial,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(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) ).
% Suc_times_binomial
tff(fact_2662_XOR__lower,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y))) ) ) ).
% XOR_lower
tff(fact_2663_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ).
% binomial_symmetric
tff(fact_2664_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_2665_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),R)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R))) ) ).
% binomial_le_pow
tff(fact_2666_binomial__gbinomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N: nat,K: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) ) ).
% binomial_gbinomial
tff(fact_2667_numeral__eq__Suc,axiom,
! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).
% numeral_eq_Suc
tff(fact_2668_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K))) ) ).
% zero_less_binomial
tff(fact_2669_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_2670_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( 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_eq(nat),M),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).
% choose_mult
tff(fact_2671_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_2672_binomial__Suc__Suc__eq__times,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),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K))),aa(nat,nat,suc,K)) ).
% binomial_Suc_Suc_eq_times
tff(fact_2673_and__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).
% and_nat_def
tff(fact_2674_xor__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).
% xor_nat_def
tff(fact_2675_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_2676_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_2677_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))),aa(nat,nat,binomial(N),K)) = semiring_char_0_fact(nat,N) ) ) ).
% binomial_fact_lemma
tff(fact_2678_fact__numeral,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).
% fact_numeral
tff(fact_2679_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)))) ) ) ).
% binomial_ge_n_over_k_pow_k
tff(fact_2680_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_2681_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_2682_binomial__le__pow2,axiom,
! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).
% binomial_le_pow2
tff(fact_2683_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ).
% binomial_altdef_nat
tff(fact_2684_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_2685_fact__binomial,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).
% fact_binomial
tff(fact_2686_binomial__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))) ) ) ) ).
% binomial_fact
tff(fact_2687_take__bit__numeral__bit0,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),bit0(K))) = 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),bit0(one2))) ) ).
% take_bit_numeral_bit0
tff(fact_2688_take__bit__numeral__minus__bit0,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),bit0(K)))) = 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),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).
% take_bit_numeral_minus_bit0
tff(fact_2689_XOR__upper,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ) ).
% XOR_upper
tff(fact_2690_choose__two,axiom,
! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),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),bit0(one2))) ).
% choose_two
tff(fact_2691_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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).
% and_nat_unfold
tff(fact_2692_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(fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).
% and_nat_rec
tff(fact_2693_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),fNot(aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% xor_int_rec
tff(fact_2694_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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
tff(fact_2695_signed__take__bit__eq__take__bit__minus,axiom,
! [N: nat,K: int] : aa(int,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)),K)),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),bit0(one2))),aa(nat,nat,suc,N))),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)))) ).
% signed_take_bit_eq_take_bit_minus
tff(fact_2696_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),bit0(one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),pred_numeral(N)))) ) ).
% mask_numeral
tff(fact_2697_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(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),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_2698_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),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_2699_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),bit0(one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).
% push_bit_numeral_minus_1
tff(fact_2700_mask__nat__positive__iff,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
% mask_nat_positive_iff
tff(fact_2701_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% push_bit_nonnegative_int_iff
tff(fact_2702_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% push_bit_negative_int_iff
tff(fact_2703_push__bit__of__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),zero_zero(A)) = zero_zero(A) ) ).
% push_bit_of_0
tff(fact_2704_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,A3: A] :
( ( aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% push_bit_eq_0_iff
tff(fact_2705_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] : bit_concat_bit(N,zero_zero(int),L) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L) ).
% concat_bit_of_zero_1
tff(fact_2706_mask__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ( aa(nat,A,bit_se2239418461657761734s_mask(A),zero_zero(nat)) = zero_zero(A) ) ) ).
% mask_0
tff(fact_2707_mask__eq__0__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] :
( ( aa(nat,A,bit_se2239418461657761734s_mask(A),N) = zero_zero(A) )
<=> ( N = zero_zero(nat) ) ) ) ).
% mask_eq_0_iff
tff(fact_2708_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).
% bit.conj_cancel_right
tff(fact_2709_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).
% bit.conj_cancel_left
tff(fact_2710_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_2711_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_2712_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: num,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),N)) ) ) ).
% bit_numeral_Bit0_Suc_iff
tff(fact_2713_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(num,A,numeral_numeral(A),bit0(K))) ) ).
% push_bit_Suc_numeral
tff(fact_2714_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: num,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),N)) ) ) ).
% bit_numeral_Bit1_Suc_iff
tff(fact_2715_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_2716_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.xor_cancel_right
tff(fact_2717_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.xor_cancel_left
tff(fact_2718_bit_Oxor__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).
% bit.xor_one_right
tff(fact_2719_bit_Oxor__one__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).
% bit.xor_one_left
tff(fact_2720_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_2721_not__negative__int__iff,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% not_negative_int_iff
tff(fact_2722_not__nonnegative__int__iff,axiom,
! [K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% not_nonnegative_int_iff
tff(fact_2723_minus__not__numeral__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),inc(N)) ) ).
% minus_not_numeral_eq
tff(fact_2724_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
<=> ~ pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)) ) ).
% signed_take_bit_nonnegative_iff
tff(fact_2725_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int)))
<=> pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)) ) ).
% signed_take_bit_negative_iff
tff(fact_2726_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).
% push_bit_Suc_minus_numeral
tff(fact_2727_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W2)))),aa(nat,nat,suc,N)))
<=> pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),N)) ) ).
% bit_minus_numeral_Bit0_Suc_iff
tff(fact_2728_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W2)))),aa(nat,nat,suc,N)))
<=> ~ pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W2)),N)) ) ).
% bit_minus_numeral_Bit1_Suc_iff
tff(fact_2729_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_2730_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),bit0(one2))) ) ) ).
% not_one_eq
tff(fact_2731_bit__0,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),zero_zero(nat)))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ).
% bit_0
tff(fact_2732_push__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% push_bit_Suc
tff(fact_2733_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),bit0(one2))),N) ) ).
% push_bit_of_1
tff(fact_2734_even__push__bit__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)))
<=> ( ( N != zero_zero(nat) )
| pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).
% even_push_bit_iff
tff(fact_2735_push__bit__of__Suc__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).
% push_bit_of_Suc_0
tff(fact_2736_push__bit__minus__numeral,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),pred_numeral(L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).
% push_bit_minus_numeral
tff(fact_2737_bit__minus__numeral__int_I1_J,axiom,
! [W2: num,N: num] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W2)))),aa(num,nat,numeral_numeral(nat),N)))
<=> pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),pred_numeral(N))) ) ).
% bit_minus_numeral_int(1)
tff(fact_2738_bit__minus__numeral__int_I2_J,axiom,
! [W2: num,N: num] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W2)))),aa(num,nat,numeral_numeral(nat),N)))
<=> ~ pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W2)),pred_numeral(N))) ) ).
% bit_minus_numeral_int(2)
tff(fact_2739_bit__mod__2__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),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),N))
<=> ( ( N = zero_zero(nat) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).
% bit_mod_2_iff
tff(fact_2740_of__nat__mask__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)) = aa(nat,A,bit_se2239418461657761734s_mask(A),N) ) ).
% of_nat_mask_eq
tff(fact_2741_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(nat,A,semiring_1_of_nat(A),M)),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),M),N)) ) ) ).
% bit_of_nat_iff_bit
tff(fact_2742_push__bit__of__nat,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),M)) ) ).
% push_bit_of_nat
tff(fact_2743_of__nat__push__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_push_bit
tff(fact_2744_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),M),K)),N))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& 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),N),M))) ) ) ).
% bit_push_bit_iff_int
tff(fact_2745_push__bit__minus,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)) ) ).
% push_bit_minus
tff(fact_2746_less__eq__mask,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N))) ).
% less_eq_mask
tff(fact_2747_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),K)),N))
<=> pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))),N)) ) ).
% bit_minus_int_iff
tff(fact_2748_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_2749_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_2750_unset__bit__int__def,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int)))) ).
% unset_bit_int_def
tff(fact_2751_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),N))) = zero_zero(A) ) ) ) ).
% take_bit_not_mask_eq_0
tff(fact_2752_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_2753_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_2754_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_2755_bit__take__bit__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A3)),N))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
& pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N)) ) ) ) ).
% bit_take_bit_iff
tff(fact_2756_bit__of__bool__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [B2: bool,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
<=> ( pp(B2)
& ( N = zero_zero(nat) ) ) ) ) ).
% bit_of_bool_iff
tff(fact_2757_bit__not__iff__eq,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,bit_ri4277139882892585799ns_not(A),A3)),N))
<=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
& ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N)) ) ) ) ).
% bit_not_iff_eq
tff(fact_2758_mask__nonnegative__int,axiom,
! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(nat,int,bit_se2239418461657761734s_mask(int),N))) ).
% mask_nonnegative_int
tff(fact_2759_not__mask__negative__int,axiom,
! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,bit_se2239418461657761734s_mask(int),N)),zero_zero(int))) ).
% not_mask_negative_int
tff(fact_2760_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_2761_flip__bit__int__def,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).
% flip_bit_int_def
tff(fact_2762_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),N)) ) ).
% minus_exp_eq_not_mask
tff(fact_2763_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_2764_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_2765_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_2766_not__int__def,axiom,
! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int)) ).
% not_int_def
tff(fact_2767_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),K)),one_one(int))),N))
<=> ~ pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)) ) ).
% bit_not_int_iff'
tff(fact_2768_and__not__numerals_I1_J,axiom,
aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).
% and_not_numerals(1)
tff(fact_2769_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N)) ) ).
% minus_numeral_inc_eq
tff(fact_2770_less__mask,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N))) ) ).
% less_mask
tff(fact_2771_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_2772_and__not__numerals_I4_J,axiom,
! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).
% and_not_numerals(4)
tff(fact_2773_and__not__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = one_one(int) ).
% and_not_numerals(2)
tff(fact_2774_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),K))) ) ) ).
% bit_imp_take_bit_positive
tff(fact_2775_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).
% not_numeral_Bit0_eq
tff(fact_2776_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),bit_concat_bit(M,K,L)),N))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
& pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).
% bit_concat_bit_iff
tff(fact_2777_push__bit__double,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% push_bit_double
tff(fact_2778_signed__take__bit__eq__concat__bit,axiom,
! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = bit_concat_bit(N,K,aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)))) ).
% signed_take_bit_eq_concat_bit
tff(fact_2779_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat,A3: A] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
=> ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N)) ) ) ).
% exp_eq_0_imp_not_bit
tff(fact_2780_bit__Suc,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),aa(nat,nat,suc,N)))
<=> 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),bit0(one2)))),N)) ) ) ).
% bit_Suc
tff(fact_2781_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_2782_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),M2))
=> ( pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),M2))
<=> 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_2783_and__not__numerals_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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(5)
tff(fact_2784_and__not__numerals_I7_J,axiom,
! [M: 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),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).
% and_not_numerals(7)
tff(fact_2785_and__not__numerals_I3_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = zero_zero(int) ).
% and_not_numerals(3)
tff(fact_2786_push__bit__int__def,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).
% push_bit_int_def
tff(fact_2787_push__bit__nat__def,axiom,
! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).
% push_bit_nat_def
tff(fact_2788_push__bit__eq__mult,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).
% push_bit_eq_mult
tff(fact_2789_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = zero_zero(A) )
<=> ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N)) ) ) ).
% and_exp_eq_0_iff_not_bit
tff(fact_2790_Suc__mask__eq__exp,axiom,
! [N: nat] : aa(nat,nat,suc,aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).
% Suc_mask_eq_exp
tff(fact_2791_mask__nat__less__exp,axiom,
! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2239418461657761734s_mask(nat),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).
% mask_nat_less_exp
tff(fact_2792_and__not__numerals_I9_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,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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(9)
tff(fact_2793_and__not__numerals_I6_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(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),times_times(int),aa(num,int,numeral_numeral(int),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(6)
tff(fact_2794_push__bit__minus__one,axiom,
! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).
% push_bit_minus_one
tff(fact_2795_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),N)))
<=> ( N = zero_zero(nat) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
tff(fact_2796_even__bit__succ__iff,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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_2797_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [A3: A,N: nat] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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_2798_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),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_2799_mask__int__def,axiom,
! [N: nat] : aa(nat,int,bit_se2239418461657761734s_mask(int),N) = 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),bit0(one2))),N)),one_one(int)) ).
% mask_int_def
tff(fact_2800_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),bit0(one2))),N)),one_one(nat)) ).
% mask_nat_def
tff(fact_2801_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A,N: nat] :
( ! [J2: nat] : ~ pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,suc,J2)))
=> ( 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),bit0(one2))),B2))),N))
<=> ( ( ( N = zero_zero(nat) )
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),B2)),N)) ) ) ) ) ) ).
% bit_sum_mult_2_cases
tff(fact_2802_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(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ) ).
% bit_rec
tff(fact_2803_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),bit0(one2))),N)),one_one(A)) ) ).
% mask_eq_exp_minus_1
tff(fact_2804_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),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),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_2805_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),aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))))) ).
% not_int_rec
tff(fact_2806_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),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),bit0(one2))),N))) ).
% set_bit_eq
tff(fact_2807_unset__bit__eq,axiom,
! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),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),bit0(one2))),N))) ).
% unset_bit_eq
tff(fact_2808_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_2809_num_Osize_I5_J,axiom,
! [X22: num] : aa(num,nat,size_size(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_2810_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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).
% or_int_rec
tff(fact_2811_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_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(nat,fun(int,int),bit_se4197421643247451524op_bit(int),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).
% drop_bit_numeral_minus_bit1
tff(fact_2812_or__numerals_I7_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: 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,X))),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(7)
tff(fact_2813_or__numerals_I6_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: 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,X))),aa(num,A,numeral_numeral(A),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(6)
tff(fact_2814_or_Oright__neutral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).
% or.right_neutral
tff(fact_2815_or_Oleft__neutral,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).
% or.left_neutral
tff(fact_2816_drop__bit__of__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),zero_zero(A)) = zero_zero(A) ) ).
% drop_bit_of_0
tff(fact_2817_bit_Odisj__one__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_one_left
tff(fact_2818_bit_Odisj__one__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_one_right
tff(fact_2819_drop__bit__of__bool,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,B2: bool] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat)),B2)) ) ).
% drop_bit_of_bool
tff(fact_2820_or__nonnegative__int__iff,axiom,
! [K: int,L: 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),bit_se1065995026697491101ons_or(int),K),L)))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).
% or_nonnegative_int_iff
tff(fact_2821_or__negative__int__iff,axiom,
! [K: int,L: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(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),ord_less(int),L),zero_zero(int))) ) ) ).
% or_negative_int_iff
tff(fact_2822_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).
% drop_bit_nonnegative_int_iff
tff(fact_2823_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K)),zero_zero(int)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).
% drop_bit_negative_int_iff
tff(fact_2824_drop__bit__minus__one,axiom,
! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).
% drop_bit_minus_one
tff(fact_2825_or__nat__numerals_I4_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).
% or_nat_numerals(4)
tff(fact_2826_or__nat__numerals_I2_J,axiom,
! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).
% or_nat_numerals(2)
tff(fact_2827_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(num,A,numeral_numeral(A),K)) ) ).
% drop_bit_Suc_bit0
tff(fact_2828_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_2829_or__numerals_I8_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).
% or_numerals(8)
tff(fact_2830_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_cancel_left
tff(fact_2831_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).
% bit.disj_cancel_right
tff(fact_2832_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(num,A,numeral_numeral(A),K)) ) ).
% drop_bit_Suc_bit1
tff(fact_2833_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_2834_or__numerals_I3_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).
% or_numerals(3)
tff(fact_2835_or__nat__numerals_I1_J,axiom,
! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).
% or_nat_numerals(1)
tff(fact_2836_or__nat__numerals_I3_J,axiom,
! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).
% or_nat_numerals(3)
tff(fact_2837_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),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).
% or_numerals(1)
tff(fact_2838_or__numerals_I5_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).
% or_numerals(5)
tff(fact_2839_or__minus__numerals_I6_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).
% or_minus_numerals(6)
tff(fact_2840_or__minus__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).
% or_minus_numerals(2)
tff(fact_2841_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).
% drop_bit_Suc_minus_bit0
tff(fact_2842_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).
% drop_bit_numeral_minus_bit0
tff(fact_2843_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_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(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).
% drop_bit_Suc_minus_bit1
tff(fact_2844_and__minus__minus__numerals,axiom,
! [M: num,N: 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),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).
% and_minus_minus_numerals
tff(fact_2845_or__minus__minus__numerals,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(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))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).
% or_minus_minus_numerals
tff(fact_2846_or__numerals_I4_J,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),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),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).
% or_numerals(4)
tff(fact_2847_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( size(A)
=> ! [X: A,Y: A] :
( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
tff(fact_2848_or__nat__def,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).
% or_nat_def
tff(fact_2849_of__nat__or__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_or_eq
tff(fact_2850_drop__bit__of__nat,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),M)) ) ).
% drop_bit_of_nat
tff(fact_2851_of__nat__drop__bit,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),M),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).
% of_nat_drop_bit
tff(fact_2852_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).
% bit.disj_zero_right
tff(fact_2853_or__eq__0__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) = zero_zero(A) )
<=> ( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) ) ) ) ).
% or_eq_0_iff
tff(fact_2854_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_2855_bit__Suc__0__iff,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),N))
<=> ( N = zero_zero(nat) ) ) ).
% bit_Suc_0_iff
tff(fact_2856_not__bit__Suc__0__Suc,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).
% not_bit_Suc_0_Suc
tff(fact_2857_bit__push__bit__iff__nat,axiom,
! [M: nat,Q4: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),Q4)),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),bit_se5641148757651400278ts_bit(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).
% bit_push_bit_iff_nat
tff(fact_2858_OR__lower,axiom,
! [X: int,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y))) ) ) ).
% OR_lower
tff(fact_2859_or__greater__eq,axiom,
! [L: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))) ) ).
% or_greater_eq
tff(fact_2860_set__bit__int__def,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),bit_se1065995026697491101ons_or(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).
% set_bit_int_def
tff(fact_2861_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] :
( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = A3 )
<=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = zero_zero(A) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_2862_not__bit__Suc__0__numeral,axiom,
! [N: num] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).
% not_bit_Suc_0_numeral
tff(fact_2863_or__not__numerals_I1_J,axiom,
aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).
% or_not_numerals(1)
tff(fact_2864_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_2865_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_2866_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_2867_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [A3: A,X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X) = 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)) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
tff(fact_2868_or__not__numerals_I2_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).
% or_not_numerals(2)
tff(fact_2869_or__not__numerals_I4_J,axiom,
! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).
% or_not_numerals(4)
tff(fact_2870_bit__nat__iff,axiom,
! [K: int,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(int,nat,nat2,K)),N))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
& pp(aa(nat,bool,aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),K),N)) ) ) ).
% bit_nat_iff
tff(fact_2871_num_Osize_I4_J,axiom,
aa(num,nat,size_size(num),one2) = zero_zero(nat) ).
% num.size(4)
tff(fact_2872_bit_Ocompl__unique,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
=> ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).
% bit.compl_unique
tff(fact_2873_or__not__numerals_I3_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).
% or_not_numerals(3)
tff(fact_2874_or__not__numerals_I7_J,axiom,
! [M: 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),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).
% or_not_numerals(7)
tff(fact_2875_drop__bit__Suc,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).
% drop_bit_Suc
tff(fact_2876_mask__Suc__exp,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),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),aa(nat,A,bit_se2239418461657761734s_mask(A),N)) ) ).
% mask_Suc_exp
tff(fact_2877_or__not__numerals_I6_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(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),times_times(int),aa(num,int,numeral_numeral(int),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(6)
tff(fact_2878_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),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).
% one_or_eq
tff(fact_2879_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),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).
% or_one_eq
tff(fact_2880_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),bit0(one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),N))) ) ).
% mask_Suc_double
tff(fact_2881_OR__upper,axiom,
! [X: int,N: nat,Y: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ) ).
% OR_upper
tff(fact_2882_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),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),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),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_2883_signed__take__bit__def,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),N)))) ) ).
% signed_take_bit_def
tff(fact_2884_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),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).
% or_Suc_0_eq
tff(fact_2885_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),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).
% Suc_0_or_eq
tff(fact_2886_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),bit0(one2)))) ) ) ) ) ).
% drop_bit_rec
tff(fact_2887_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),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),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_2888_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),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_2889_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(fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).
% or_nat_rec
tff(fact_2890_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,aa(int,fun(int,int),ord_max(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ) ) ) ).
% or_int_unfold
tff(fact_2891_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_2892_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(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(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(N)))) ).
% or_minus_numerals(4)
tff(fact_2893_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(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(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(N)))) ).
% or_minus_numerals(8)
tff(fact_2894_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,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).
% or_nat_unfold
tff(fact_2895_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,N,one2)),K)) ).
% xor_minus_numerals(1)
tff(fact_2896_max_Oright__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).
% max.right_idem
tff(fact_2897_max_Oleft__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ).
% max.left_idem
tff(fact_2898_max_Oidem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),A3) = A3 ) ).
% max.idem
tff(fact_2899_max_Oabsorb1,axiom,
! [A: $tType] :
( linorder(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),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb1
tff(fact_2900_max_Oabsorb2,axiom,
! [A: $tType] :
( linorder(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),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb2
tff(fact_2901_max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: 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),ord_max(A),B2),C2)),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_eq(A),C2),A3)) ) ) ) ).
% max.bounded_iff
tff(fact_2902_max_Oabsorb3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb3
tff(fact_2903_max_Oabsorb4,axiom,
! [A: $tType] :
( linorder(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),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb4
tff(fact_2904_max__less__iff__conj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2)) ) ) ) ).
% max_less_iff_conj
tff(fact_2905_of__bool__or__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fdisj(P,Q)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).
% of_bool_or_iff
tff(fact_2906_max__Suc__Suc,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)) ).
% max_Suc_Suc
tff(fact_2907_max__0R,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).
% max_0R
tff(fact_2908_max__0L,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).
% max_0L
tff(fact_2909_max__nat_Oright__neutral,axiom,
! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).
% max_nat.right_neutral
tff(fact_2910_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) )
<=> ( ( A3 = zero_zero(nat) )
& ( B2 = zero_zero(nat) ) ) ) ).
% max_nat.neutr_eq_iff
tff(fact_2911_max__nat_Oleft__neutral,axiom,
! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).
% max_nat.left_neutral
tff(fact_2912_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B2) = zero_zero(nat) )
<=> ( ( A3 = zero_zero(nat) )
& ( B2 = zero_zero(nat) ) ) ) ).
% max_nat.eq_neutr_iff
tff(fact_2913_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( numeral(A)
& ord(A) )
=> ! [U: num,V: num] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).
% max_number_of(1)
tff(fact_2914_max__0__1_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).
% max_0_1(4)
tff(fact_2915_max__0__1_I3_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).
% max_0_1(3)
tff(fact_2916_max__0__1_I2_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),zero_zero(A)) = one_one(A) ) ) ).
% max_0_1(2)
tff(fact_2917_max__0__1_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),one_one(A)) = one_one(A) ) ) ).
% max_0_1(1)
tff(fact_2918_max__0__1_I5_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).
% max_0_1(5)
tff(fact_2919_max__0__1_I6_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).
% max_0_1(6)
tff(fact_2920_sub__num__simps_I1_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ( neg_numeral_sub(A,one2,one2) = zero_zero(A) ) ) ).
% sub_num_simps(1)
tff(fact_2921_diff__numeral__simps_I1_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(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,M,N) ) ).
% diff_numeral_simps(1)
tff(fact_2922_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num,L: num] : neg_numeral_sub(A,bit0(K),bit0(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).
% sub_num_simps(6)
tff(fact_2923_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit1,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).
% sub_num_simps(9)
tff(fact_2924_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( uminus(A)
& numeral(A)
& ord(A) )
=> ! [U: num,V: num] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).
% max_number_of(2)
tff(fact_2925_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( uminus(A)
& numeral(A)
& ord(A) )
=> ! [U: num,V: 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),U))),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
& ( ~ 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),U))),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).
% max_number_of(3)
tff(fact_2926_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( uminus(A)
& numeral(A)
& ord(A) )
=> ! [U: num,V: 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),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
& ( ~ 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),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).
% max_number_of(4)
tff(fact_2927_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_2928_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_2929_semiring__norm_I166_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [V: num,W2: 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),W2))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W2)),Y) ) ).
% semiring_norm(166)
tff(fact_2930_semiring__norm_I167_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [V: num,W2: 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),W2)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W2,V)),Y) ) ).
% semiring_norm(167)
tff(fact_2931_diff__numeral__simps_I4_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(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,M) ) ).
% diff_numeral_simps(4)
tff(fact_2932_max__numeral__Suc,axiom,
! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),N)) ).
% max_numeral_Suc
tff(fact_2933_max__Suc__numeral,axiom,
! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),pred_numeral(K))) ).
% max_Suc_numeral
tff(fact_2934_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num,L: num] : neg_numeral_sub(A,bit0(K),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ).
% sub_num_simps(7)
tff(fact_2935_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),bit0(L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ).
% sub_num_simps(8)
tff(fact_2936_drop__bit__of__Suc__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ).
% drop_bit_of_Suc_0
tff(fact_2937_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_2938_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_2939_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_sub(A,aa(num,num,bit1,K),one2) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).
% sub_num_simps(5)
tff(fact_2940_not__minus__numeral__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).
% not_minus_numeral_eq
tff(fact_2941_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_2942_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_2943_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_2944_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_2945_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_2946_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_2947_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_2948_sub__num__simps_I3_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [L: num] : neg_numeral_sub(A,one2,aa(num,num,bit1,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(L))) ) ).
% sub_num_simps(3)
tff(fact_2949_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,N,one2))) ).
% xor_minus_numerals(2)
tff(fact_2950_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.strict_coboundedI2
tff(fact_2951_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.strict_coboundedI1
tff(fact_2952_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% max.strict_order_iff
tff(fact_2953_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: 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),ord_max(A),B2),C2)),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),C2),A3)) ) ) ) ).
% max.strict_boundedE
tff(fact_2954_less__max__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).
% less_max_iff_disj
tff(fact_2955_max__add__distrib__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).
% max_add_distrib_left
tff(fact_2956_max__add__distrib__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ).
% max_add_distrib_right
tff(fact_2957_max__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).
% max_diff_distrib_left
tff(fact_2958_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Y: A,Z2: A,X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_2959_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,X)) = One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_2960_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A,Z2: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Xor,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_2961_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,Compl,Y)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_2962_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),X) = One ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_2963_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,X)),Y) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,X),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_2964_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),One) = aa(A,A,Compl,X) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_2965_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),aa(A,A,aa(A,fun(A,A),Xor,X),Y)) = Y ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_2966_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,One),X) = aa(A,A,Compl,X) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_2967_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),X) = Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_2968_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y))) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_2969_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),X: A,Y: A] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> ( aa(A,A,aa(A,fun(A,A),Xor,X),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),Y)) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_2970_of__nat__max,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).
% of_nat_max
tff(fact_2971_max_Oleft__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).
% max.left_commute
tff(fact_2972_max_Ocommute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3) ) ).
% max.commute
tff(fact_2973_max_Oassoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) ) ).
% max.assoc
tff(fact_2974_of__int__max,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y)) ) ).
% of_int_max
tff(fact_2975_max__absorb2,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).
% max_absorb2
tff(fact_2976_max__absorb1,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).
% max_absorb1
tff(fact_2977_max_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,B2: A,A3: 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),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.coboundedI2
tff(fact_2978_max_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ).
% max.coboundedI1
tff(fact_2979_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( linorder(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),ord_max(A),A3),B2) = B2 ) ) ) ).
% max.absorb_iff2
tff(fact_2980_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( linorder(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),ord_max(A),A3),B2) = A3 ) ) ) ).
% max.absorb_iff1
tff(fact_2981_le__max__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),Y)) ) ) ) ).
% le_max_iff_disj
tff(fact_2982_max_Ocobounded2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ).
% max.cobounded2
tff(fact_2983_max_Ocobounded1,axiom,
! [A: $tType] :
( linorder(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),ord_max(A),A3),B2))) ) ).
% max.cobounded1
tff(fact_2984_max_Oorder__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).
% max.order_iff
tff(fact_2985_max_OboundedI,axiom,
! [A: $tType] :
( linorder(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),C2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3)) ) ) ) ).
% max.boundedI
tff(fact_2986_max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: 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),ord_max(A),B2),C2)),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_eq(A),C2),A3)) ) ) ) ).
% max.boundedE
tff(fact_2987_max_OorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% max.orderI
tff(fact_2988_max_OorderE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) ) ) ) ).
% max.orderE
tff(fact_2989_max_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C2: A,A3: A,D2: A,B2: A] :
( 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),D2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2))) ) ) ) ).
% max.mono
tff(fact_2990_max__def,axiom,
! [A: $tType] :
( ord(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),ord_max(A),A3),B2) = B2 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2) = A3 ) ) ) ) ).
% max_def
tff(fact_2991_nat__add__max__left,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q4)) ).
% nat_add_max_left
tff(fact_2992_nat__add__max__right,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(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),Q4)) ).
% nat_add_max_right
tff(fact_2993_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) ).
% nat_mult_max_left
tff(fact_2994_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)) ).
% nat_mult_max_right
tff(fact_2995_length__induct,axiom,
! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
( ! [Xs2: list(A)] :
( ! [Ys: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2)))
=> pp(aa(list(A),bool,P,Ys)) )
=> pp(aa(list(A),bool,P,Xs2)) )
=> pp(aa(list(A),bool,P,Xs)) ) ).
% length_induct
tff(fact_2996_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,aa(nat,fun(nat,nat),ord_max(nat),N),M) ).
% nat_minus_add_max
tff(fact_2997_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num,L: num] : neg_numeral_sub(A,K,L) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) ) ).
% neg_numeral_class.sub_def
tff(fact_2998_Heap_Osize__neq,axiom,
! [A: $tType,X: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),X) != zero_zero(nat) ).
% Heap.size_neq
tff(fact_2999_sub__non__negative,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num,M: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).
% sub_non_negative
tff(fact_3000_sub__non__positive,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num,M: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).
% sub_non_positive
tff(fact_3001_sub__negative,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num,M: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).
% sub_negative
tff(fact_3002_sub__positive,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: num,M: num] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).
% sub_positive
tff(fact_3003_sub__inc__One__eq,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [N: num] : neg_numeral_sub(A,inc(N),one2) = aa(num,A,numeral_numeral(A),N) ) ).
% sub_inc_One_eq
tff(fact_3004_minus__numeral__eq__not__sub__one,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),neg_numeral_sub(A,N,one2)) ) ).
% minus_numeral_eq_not_sub_one
tff(fact_3005_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] : 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))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N))) ).
% int_numeral_or_not_num_neg
tff(fact_3006_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M))) ).
% int_numeral_not_or_num_neg
tff(fact_3007_numeral__or__not__num__eq,axiom,
! [M: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N)) = aa(int,int,uminus_uminus(int),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)))) ).
% numeral_or_not_num_eq
tff(fact_3008_or__minus__numerals_I1_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).
% or_minus_numerals(1)
tff(fact_3009_or__minus__numerals_I5_J,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).
% or_minus_numerals(5)
tff(fact_3010_horner__sum__of__bool__2__less,axiom,
! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(list(bool),int,aa(int,fun(list(bool),int),aa(fun(bool,int),fun(int,fun(list(bool),int)),groups4207007520872428315er_sum(bool,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),bit0(one2))),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).
% horner_sum_of_bool_2_less
tff(fact_3011_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).
% or_minus_numerals(7)
tff(fact_3012_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).
% or_minus_numerals(3)
tff(fact_3013_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).
% dbl_dec_simps(5)
tff(fact_3014_pred__numeral__simps_I2_J,axiom,
! [K: num] : pred_numeral(bit0(K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).
% pred_numeral_simps(2)
tff(fact_3015_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_sub(A,bit0(K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).
% sub_num_simps(4)
tff(fact_3016_sub__num__simps_I2_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [L: num] : neg_numeral_sub(A,one2,bit0(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).
% sub_num_simps(2)
tff(fact_3017_semiring__norm_I26_J,axiom,
bitM(one2) = one2 ).
% semiring_norm(26)
tff(fact_3018_semiring__norm_I27_J,axiom,
! [N: num] : bitM(bit0(N)) = aa(num,num,bit1,bitM(N)) ).
% semiring_norm(27)
tff(fact_3019_semiring__norm_I28_J,axiom,
! [N: num] : bitM(aa(num,num,bit1,N)) = aa(num,num,bit1,bit0(N)) ).
% semiring_norm(28)
tff(fact_3020_inc__BitM__eq,axiom,
! [N: num] : inc(bitM(N)) = bit0(N) ).
% inc_BitM_eq
tff(fact_3021_BitM__inc__eq,axiom,
! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).
% BitM_inc_eq
tff(fact_3022_eval__nat__numeral_I2_J,axiom,
! [N: num] : aa(num,nat,numeral_numeral(nat),bit0(N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).
% eval_nat_numeral(2)
tff(fact_3023_one__plus__BitM,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = bit0(N) ).
% one_plus_BitM
tff(fact_3024_BitM__plus__one,axiom,
! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = bit0(N) ).
% BitM_plus_one
tff(fact_3025_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),bit0(N))),one_one(A)) ) ).
% numeral_BitM
tff(fact_3026_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(N))) ) ).
% not_numeral_BitM_eq
tff(fact_3027_sub__BitM__One__eq,axiom,
! [N: num] : neg_numeral_sub(int,bitM(N),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),neg_numeral_sub(int,N,one2)) ).
% sub_BitM_One_eq
tff(fact_3028_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Bs: list(bool),N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Bs)),N))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(bool),nat,size_size(list(bool)),Bs)))
& pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).
% bit_horner_sum_bit_iff
tff(fact_3029_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_3030_num__of__nat_Osimps_I2_J,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,suc,N)) = inc(aa(nat,num,num_of_nat,N)) ) )
& ( ~ 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,suc,N)) = one2 ) ) ) ).
% num_of_nat.simps(2)
tff(fact_3031_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).
% eq_numeral_iff_iszero(7)
tff(fact_3032_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_3033_num__of__nat__numeral__eq,axiom,
! [Q4: num] : aa(nat,num,num_of_nat,aa(num,nat,numeral_numeral(nat),Q4)) = Q4 ).
% num_of_nat_numeral_eq
tff(fact_3034_iszero__neg__numeral,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [W2: num] :
( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ) ).
% iszero_neg_numeral
tff(fact_3035_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),X)),aa(A,A,aa(A,fun(A,A),Disj,Z2),X)) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
tff(fact_3036_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Y: A,Z2: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Y),X)),aa(A,A,aa(A,fun(A,A),Conj,Z2),X)) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
tff(fact_3037_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,Compl,X) = aa(A,A,Compl,Y) )
<=> ( X = Y ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
tff(fact_3038_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,aa(A,fun(A,A),Conj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X),Y)),aa(A,A,aa(A,fun(A,A),Disj,X),Z2)) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
tff(fact_3039_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),aa(A,A,Compl,X)) = One ) ) ).
% abstract_boolean_algebra.disj_cancel_right
tff(fact_3040_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A,Z2: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,aa(A,fun(A,A),Disj,Y),Z2)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),Y)),aa(A,A,aa(A,fun(A,A),Conj,X),Z2)) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
tff(fact_3041_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,X)) = Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
tff(fact_3042_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,A3: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),X) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),X) = One )
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Y) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Y) = One )
=> ( X = Y ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
tff(fact_3043_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),X) = One ) ) ).
% abstract_boolean_algebra.disj_cancel_left
tff(fact_3044_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),X) = Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
tff(fact_3045_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),Zero) = X ) ) ).
% abstract_boolean_algebra.disj_zero_right
tff(fact_3046_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),Zero) = Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
tff(fact_3047_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,X),One) = One ) ) ).
% abstract_boolean_algebra.disj_one_right
tff(fact_3048_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Disj,X),Y)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
tff(fact_3049_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Conj,X),Y)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,X)),aa(A,A,Compl,Y)) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
tff(fact_3050_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,Zero),X) = Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
tff(fact_3051_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Conj,X),One) = X ) ) ).
% abstract_boolean_algebra.conj_one_right
tff(fact_3052_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,aa(A,fun(A,A),Disj,One),X) = One ) ) ).
% abstract_boolean_algebra.disj_one_left
tff(fact_3053_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,aa(A,A,Compl,X)) = X ) ) ).
% abstract_boolean_algebra.double_compl
tff(fact_3054_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,X: A,Y: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( ( aa(A,A,aa(A,fun(A,A),Conj,X),Y) = Zero )
=> ( ( aa(A,A,aa(A,fun(A,A),Disj,X),Y) = One )
=> ( aa(A,A,Compl,X) = Y ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
tff(fact_3055_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,Zero) = One ) ) ).
% abstract_boolean_algebra.compl_zero
tff(fact_3056_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( aa(A,A,Compl,One) = Zero ) ) ).
% abstract_boolean_algebra.compl_one
tff(fact_3057_iszero__def,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: A] :
( ring_1_iszero(A,Z2)
<=> ( Z2 = zero_zero(A) ) ) ) ).
% iszero_def
tff(fact_3058_iszero__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ring_1_iszero(A,zero_zero(A)) ) ).
% iszero_0
tff(fact_3059_not__iszero__numeral,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [W2: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ).
% not_iszero_numeral
tff(fact_3060_not__iszero__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,one_one(A)) ) ).
% not_iszero_1
tff(fact_3061_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A,Y: A] :
( ( X = Y )
<=> ring_1_iszero(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)) ) ) ).
% eq_iff_iszero_diff
tff(fact_3062_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ).
% abstract_boolean_algebra_sym_diff.axioms(1)
tff(fact_3063_nth__equalityI,axiom,
! [A: $tType,Xs: list(A),Ys2: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
=> ( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys2),I4) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
tff(fact_3064_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
( ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),K))
=> ? [X_13: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I5),X_13)) )
<=> ? [Xs3: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),K))
=> pp(aa(A,bool,aa(nat,fun(A,bool),P,I5),aa(nat,A,nth(A,Xs3),I5))) ) ) ) ).
% Skolem_list_nth
tff(fact_3065_list__eq__iff__nth__eq,axiom,
! [A: $tType,Xs: list(A),Ys2: list(A)] :
( ( Xs = Ys2 )
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I5) = aa(nat,A,nth(A,Ys2),I5) ) ) ) ) ).
% list_eq_iff_nth_eq
tff(fact_3066_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Y: num] :
( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).
% eq_numeral_iff_iszero(10)
tff(fact_3067_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num] :
( ( aa(num,A,numeral_numeral(A),X) = zero_zero(A) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).
% eq_numeral_iff_iszero(9)
tff(fact_3068_num__of__nat_Osimps_I1_J,axiom,
aa(nat,num,num_of_nat,zero_zero(nat)) = one2 ).
% num_of_nat.simps(1)
tff(fact_3069_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),one2)) ) ).
% not_iszero_Numeral1
tff(fact_3070_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_3071_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num,Y: num] :
( ( aa(num,A,numeral_numeral(A),X) = aa(num,A,numeral_numeral(A),Y) )
<=> ring_1_iszero(A,neg_numeral_sub(A,X,Y)) ) ) ).
% eq_numeral_iff_iszero(1)
tff(fact_3072_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Y: num] :
( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y)) ) ) ).
% eq_numeral_iff_iszero(12)
tff(fact_3073_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) )
<=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).
% eq_numeral_iff_iszero(11)
tff(fact_3074_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ring_1(A)
=> ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) ) ).
% not_iszero_neg_Numeral1
tff(fact_3075_numeral__num__of__nat,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(num,nat,numeral_numeral(nat),aa(nat,num,num_of_nat,N)) = N ) ) ).
% numeral_num_of_nat
tff(fact_3076_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_3077_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num,Y: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = 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),X),Y))) ) ) ).
% eq_numeral_iff_iszero(3)
tff(fact_3078_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num,Y: num] :
( ( aa(num,A,numeral_numeral(A),X) = 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),X),Y))) ) ) ).
% eq_numeral_iff_iszero(2)
tff(fact_3079_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num,Y: num] :
( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
<=> ring_1_iszero(A,neg_numeral_sub(A,Y,X)) ) ) ).
% eq_numeral_iff_iszero(4)
tff(fact_3080_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_3081_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)) = bit0(aa(nat,num,num_of_nat,N)) ) ) ).
% num_of_nat_double
tff(fact_3082_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_3083_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: num] :
( ( aa(num,A,numeral_numeral(A),X) = one_one(A) )
<=> ring_1_iszero(A,neg_numeral_sub(A,X,one2)) ) ) ).
% eq_numeral_iff_iszero(5)
tff(fact_3084_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_3085_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( 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,rotate1(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% nth_rotate1
tff(fact_3086_nth__update__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [A3: array(A),B2: array(B),J: nat,V: B,H2: 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,H2),A3)),I) = aa(nat,A,nth(A,array_get(A,H2,A3)),I) ) ) ) ).
% nth_update_neq
tff(fact_3087_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),bit0(one2))),N)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),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),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))),one_one(A)) ) ).
% mask_mod_exp
tff(fact_3088_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),bit0(one2))),N)),Z2)) = insert(nat,N,nat_set_decode(Z2)) ) ) ).
% set_decode_plus_power_2
tff(fact_3089_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_3090_min_Oidem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),A3) = A3 ) ).
% min.idem
tff(fact_3091_min_Oleft__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).
% min.left_idem
tff(fact_3092_min_Oright__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ).
% min.right_idem
tff(fact_3093_min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
<=> ( 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),C2)) ) ) ) ).
% min.bounded_iff
tff(fact_3094_min_Oabsorb2,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb2
tff(fact_3095_min_Oabsorb1,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb1
tff(fact_3096_min__less__iff__conj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Z2: A,X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).
% min_less_iff_conj
tff(fact_3097_min_Oabsorb4,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb4
tff(fact_3098_min_Oabsorb3,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb3
tff(fact_3099_max__min__same_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Y),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = Y ) ).
% max_min_same(4)
tff(fact_3100_max__min__same_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Y) = Y ) ).
% max_min_same(3)
tff(fact_3101_max__min__same_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),X) = X ) ).
% max_min_same(2)
tff(fact_3102_max__min__same_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = X ) ).
% max_min_same(1)
tff(fact_3103_min__Suc__Suc,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ).
% min_Suc_Suc
tff(fact_3104_min__0L,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).
% min_0L
tff(fact_3105_min__0R,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).
% min_0R
tff(fact_3106_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( numeral(A)
& ord(A) )
=> ! [U: num,V: num] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).
% min_number_of(1)
tff(fact_3107_min__0__1_I3_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).
% min_0_1(3)
tff(fact_3108_min__0__1_I4_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = zero_zero(A) ) ).
% min_0_1(4)
tff(fact_3109_min__0__1_I1_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).
% min_0_1(1)
tff(fact_3110_min__0__1_I2_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).
% min_0_1(2)
tff(fact_3111_min__0__1_I6_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = one_one(A) ) ).
% min_0_1(6)
tff(fact_3112_min__0__1_I5_J,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).
% min_0_1(5)
tff(fact_3113_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,aa(nat,nat,suc,N)),Xs) = rotate1(A,aa(list(A),list(A),rotate(A,N),Xs)) ).
% rotate_Suc
tff(fact_3114_get__update__elem__neqIndex,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,J: nat,A3: array(A),V: A,H2: heap_ext(product_unit)] :
( ( I != J )
=> ( aa(nat,A,nth(A,array_get(A,array_update(A,A3,J,V,H2),A3)),I) = aa(nat,A,nth(A,array_get(A,H2,A3)),I) ) ) ) ).
% get_update_elem_neqIndex
tff(fact_3115_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( uminus(A)
& numeral(A)
& ord(A) )
=> ! [U: num,V: 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),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
& ( ~ 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),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).
% min_number_of(4)
tff(fact_3116_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( uminus(A)
& numeral(A)
& ord(A) )
=> ! [U: num,V: 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),U))),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
& ( ~ 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),U))),aa(num,A,numeral_numeral(A),V)))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).
% min_number_of(3)
tff(fact_3117_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( uminus(A)
& numeral(A)
& ord(A) )
=> ! [U: num,V: num] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).
% min_number_of(2)
tff(fact_3118_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_3119_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
=> ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).
% rotate_id
tff(fact_3120_min__Suc__numeral,axiom,
! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),pred_numeral(K))) ).
% min_Suc_numeral
tff(fact_3121_min__numeral__Suc,axiom,
! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),N)) ).
% min_numeral_Suc
tff(fact_3122_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_3123_min__max__distrib2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) ) ).
% min_max_distrib2
tff(fact_3124_min__max__distrib1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),A3)) ) ).
% min_max_distrib1
tff(fact_3125_max__min__distrib2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C2)) ) ).
% max_min_distrib2
tff(fact_3126_max__min__distrib1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),A3)) ) ).
% max_min_distrib1
tff(fact_3127_min__def,axiom,
! [A: $tType] :
( ord(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),ord_min(A),A3),B2) = A3 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = B2 ) ) ) ) ).
% min_def
tff(fact_3128_min_Omono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A,D2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D2))) ) ) ) ).
% min.mono
tff(fact_3129_min_OorderE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).
% min.orderE
tff(fact_3130_min_OorderI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% min.orderI
tff(fact_3131_min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% min.boundedE
tff(fact_3132_min_OboundedI,axiom,
! [A: $tType] :
( linorder(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),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))) ) ) ) ).
% min.boundedI
tff(fact_3133_min_Oorder__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) ) ) ) ).
% min.order_iff
tff(fact_3134_min_Ocobounded1,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2)),A3)) ) ).
% min.cobounded1
tff(fact_3135_min_Ocobounded2,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2)),B2)) ) ).
% min.cobounded2
tff(fact_3136_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2) = A3 ) ) ) ).
% min.absorb_iff1
tff(fact_3137_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( linorder(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),ord_min(A),A3),B2) = B2 ) ) ) ).
% min.absorb_iff2
tff(fact_3138_min_OcoboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).
% min.coboundedI1
tff(fact_3139_min_OcoboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( 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),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).
% min.coboundedI2
tff(fact_3140_min__le__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% min_le_iff_disj
tff(fact_3141_min__absorb1,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).
% min_absorb1
tff(fact_3142_min__absorb2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).
% min_absorb2
tff(fact_3143_of__int__min,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [X: int,Y: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y)) ) ).
% of_int_min
tff(fact_3144_min_Oassoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).
% min.assoc
tff(fact_3145_min_Ocommute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) = aa(A,A,aa(A,fun(A,A),ord_min(A),B2),A3) ) ).
% min.commute
tff(fact_3146_min_Oleft__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B2),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)) ) ).
% min.left_commute
tff(fact_3147_of__nat__min,axiom,
! [A: $tType] :
( linord181362715937106298miring(A)
=> ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).
% of_nat_min
tff(fact_3148_min__diff__distrib__left,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z2)) ) ).
% min_diff_distrib_left
tff(fact_3149_min__add__distrib__right,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)) ) ).
% min_add_distrib_right
tff(fact_3150_min__add__distrib__left,axiom,
! [A: $tType] :
( ordere2412721322843649153imp_le(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z2)) ) ).
% min_add_distrib_left
tff(fact_3151_min__less__iff__disj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z2)) ) ) ) ).
% min_less_iff_disj
tff(fact_3152_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% min.strict_boundedE
tff(fact_3153_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% min.strict_order_iff
tff(fact_3154_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).
% min.strict_coboundedI1
tff(fact_3155_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [B2: A,C2: A,A3: A] :
( 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),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B2)),C2)) ) ) ).
% min.strict_coboundedI2
tff(fact_3156_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),Q4) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)) ).
% nat_mult_min_left
tff(fact_3157_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q4)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q4)) ).
% nat_mult_min_right
tff(fact_3158_min__diff,axiom,
! [M: nat,I: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),I) ).
% min_diff
tff(fact_3159_minus__min__eq__max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% minus_min_eq_max
tff(fact_3160_minus__max__eq__min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% minus_max_eq_min
tff(fact_3161_atLeast0__atMost__Suc,axiom,
! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,aa(nat,nat,suc,N),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).
% atLeast0_atMost_Suc
tff(fact_3162_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( insert(nat,M,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).
% atLeastAtMost_insertL
tff(fact_3163_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)) = insert(nat,aa(nat,nat,suc,N),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% atLeastAtMostSuc_conv
tff(fact_3164_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) = insert(nat,M,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).
% Icc_eq_insert_lb_nat
tff(fact_3165_length__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H2: heap_ext(product_unit),A3: array(A)] : aa(array(A),nat,array_length(A,H2),A3) = aa(list(A),nat,size_size(list(A)),array_get(A,H2,A3)) ) ).
% length_def
tff(fact_3166_max__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) ) ) ) ).
% max_mult_distrib_left
tff(fact_3167_min__mult__distrib__left,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)) ) ) ) ) ).
% min_mult_distrib_left
tff(fact_3168_max__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) ) ) ) ).
% max_mult_distrib_right
tff(fact_3169_min__mult__distrib__right,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [P3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)) ) ) ) ) ).
% min_mult_distrib_right
tff(fact_3170_min__divide__distrib__right,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [P3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) ) ) ) ).
% min_divide_distrib_right
tff(fact_3171_max__divide__distrib__right,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [P3: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P3))
=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)) ) ) ) ) ).
% max_divide_distrib_right
tff(fact_3172_and__int_Oelims,axiom,
! [X: int,Xa4: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa4) = Y )
=> ( ( ( pp(member(int,X,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,Xa4,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa4))))) ) )
& ( ~ ( pp(member(int,X,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,Xa4,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa4))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa4),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).
% and_int.elims
tff(fact_3173_and__int_Osimps,axiom,
! [K: int,L: int] :
( ( ( pp(member(int,K,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,L,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
& ( ~ ( pp(member(int,K,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,L,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).
% and_int.simps
tff(fact_3174_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: 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,cons(A,X,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_3175_get__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: array(A),S: array(B),X: list(B),H2: heap_ext(product_unit)] :
( array_noteq(A,B,R,S)
=> ( array_get(A,array_set(B,S,X,H2),R) = array_get(A,H2,R) ) ) ) ).
% get_set_neq
tff(fact_3176_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,R: A,Q4: 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),product_Pair(A,A,Q4),R)) = aa(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),bit0(one2))),Q4)),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),product_Pair(A,A,Q4),R)) = aa(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),bit0(one2))),Q4)),R) ) ) ) ) ).
% divmod_step_eq
tff(fact_3177_bot__apply,axiom,
! [D: $tType,C: $tType] :
( bot(C)
=> ! [X: D] : aa(D,C,bot_bot(fun(D,C)),X) = bot_bot(C) ) ).
% bot_apply
tff(fact_3178_set__same,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X: list(A),Y: list(A),H2: heap_ext(product_unit)] : array_set(A,R,X,array_set(A,R,Y,H2)) = array_set(A,R,X,H2) ) ).
% set_same
tff(fact_3179_max__bot2,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),bot_bot(A)) = X ) ).
% max_bot2
tff(fact_3180_max__bot,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),X) = X ) ).
% max_bot
tff(fact_3181_min__bot2,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),bot_bot(A)) = bot_bot(A) ) ).
% min_bot2
tff(fact_3182_min__bot,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),X) = bot_bot(A) ) ).
% min_bot
tff(fact_3183_get__set__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X: list(A),H2: heap_ext(product_unit)] : array_get(A,array_set(A,R,X,H2),R) = X ) ).
% get_set_eq
tff(fact_3184_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_3185_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_3186_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_3187_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,cons(A,X,Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).
% nth_Cons_Suc
tff(fact_3188_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,cons(A,X,Xs)),zero_zero(nat)) = X ).
% nth_Cons_0
tff(fact_3189_subset__Compl__singleton,axiom,
! [A: $tType,A5: set(A),B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),insert(A,B2,bot_bot(set(A))))))
<=> ~ pp(member(A,B2,A5)) ) ).
% subset_Compl_singleton
tff(fact_3190_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,cons(A,X,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_3191_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X3: A] : aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ).
% bot_fun_def
tff(fact_3192_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( ( A3 != bot_bot(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A3)) ) ) ).
% bot.not_eq_extremum
tff(fact_3193_bot_Oextremum__strict,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),bot_bot(A))) ) ).
% bot.extremum_strict
tff(fact_3194_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
=> ( A3 = bot_bot(A) ) ) ) ).
% bot.extremum_uniqueI
tff(fact_3195_bot_Oextremum__unique,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),bot_bot(A)))
<=> ( A3 = bot_bot(A) ) ) ) ).
% bot.extremum_unique
tff(fact_3196_bot_Oextremum,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A3)) ) ).
% bot.extremum
tff(fact_3197_not__psubset__empty,axiom,
! [A: $tType,A5: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),bot_bot(set(A)))) ).
% not_psubset_empty
tff(fact_3198_diff__shunt__var,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = bot_bot(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% diff_shunt_var
tff(fact_3199_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) = insert(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_3200_length__Suc__conv,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [Y5: A,Ys3: list(A)] :
( ( Xs = cons(A,Y5,Ys3) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).
% length_Suc_conv
tff(fact_3201_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
<=> ? [Y5: A,Ys3: list(A)] :
( ( Xs = cons(A,Y5,Ys3) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).
% Suc_length_conv
tff(fact_3202_impossible__Cons,axiom,
! [A: $tType,Xs: list(A),Ys2: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)))
=> ( Xs != cons(A,X,Ys2) ) ) ).
% impossible_Cons
tff(fact_3203_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)))
<=> ( A5 = bot_bot(set(A)) ) ) ).
% subset_Compl_self_eq
tff(fact_3204_set__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: array(A),R4: array(B),X: list(A),X4: list(B),H2: heap_ext(product_unit)] :
( array_noteq(A,B,R,R4)
=> ( array_set(A,R,X,array_set(B,R4,X4,H2)) = array_set(B,R4,X4,array_set(A,R,X,H2)) ) ) ) ).
% set_set_swap
tff(fact_3205_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
<=> ? [X5: A,Ys3: list(A)] :
( ( Xs = cons(A,X5,Ys3) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys3))) ) ) ).
% Suc_le_length_iff
tff(fact_3206_Compl__insert,axiom,
! [A: $tType,X: A,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),insert(A,X,A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),insert(A,X,bot_bot(set(A)))) ).
% Compl_insert
tff(fact_3207_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(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_3208_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,A,nth(A,cons(A,X,Xs)),N) = X ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,A,nth(A,cons(A,X,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_3209_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)) = insert(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_3210_psubset__insert__iff,axiom,
! [A: $tType,A5: set(A),X: A,B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),insert(A,X,B5)))
<=> ( ( pp(member(A,X,B5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5)) )
& ( ~ pp(member(A,X,B5))
=> ( ( pp(member(A,X,A5))
=> 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)),A5),insert(A,X,bot_bot(set(A))))),B5)) )
& ( ~ pp(member(A,X,A5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5)) ) ) ) ) ) ).
% psubset_insert_iff
tff(fact_3211_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
( ( X != Y )
=> ( ( aa(nat,A,nth(A,cons(A,X,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_3212_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: A,X: 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),cons(B,X,Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X)),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_3213_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),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),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).
% divmod_algorithm_code(8)
tff(fact_3214_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,bit0(M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
=> ( unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).
% divmod_algorithm_code(7)
tff(fact_3215_divides__aux__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Q4: A,R: A] :
( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q4),R))
<=> ( R = zero_zero(A) ) ) ) ).
% divides_aux_eq
tff(fact_3216_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys2: 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)),Ys2))))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys2)),N) = aa(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)),Ys2)))),aa(nat,B,nth(B,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).
% product_nth
tff(fact_3217_bit__0__eq,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ( aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).
% bit_0_eq
tff(fact_3218_set__decode__zero,axiom,
nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).
% set_decode_zero
tff(fact_3219_length__product,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys2)) = 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)),Ys2)) ).
% length_product
tff(fact_3220_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(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_3221_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N: num] : unique8689654367752047608divmod(A,one2,bit0(N)) = aa(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_3222_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),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).
% divmod_algorithm_code(4)
tff(fact_3223_divmod__BitM__2__eq,axiom,
! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(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_3224_bot__nat__def,axiom,
bot_bot(nat) = zero_zero(nat) ).
% bot_nat_def
tff(fact_3225_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),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,bit0(N))) ) ) ) ) ).
% divmod_divmod_step
tff(fact_3226_minus__one__div__numeral,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).
% minus_one_div_numeral
tff(fact_3227_one__div__minus__numeral,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).
% one_div_minus_numeral
tff(fact_3228_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: 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),product_Pair(int,int,Q4),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),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q4),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),bit0(one2))),R)),one_one(int)))) ) ) ).
% neg_eucl_rel_int_mult_2
tff(fact_3229_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R: int] : adjust_div(aa(int,product_prod(int,int),product_Pair(int,int,Q4),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q4),aa(bool,int,zero_neq_one_of_bool(int),fNot(aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))))) ).
% Divides.adjust_div_eq
tff(fact_3230_minus__numeral__div__numeral,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).
% minus_numeral_div_numeral
tff(fact_3231_numeral__div__minus__numeral,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).
% numeral_div_minus_numeral
tff(fact_3232_eucl__rel__int__by0,axiom,
! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K)) ).
% eucl_rel_int_by0
tff(fact_3233_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L != zero_zero(int) )
=> ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q4),zero_zero(int))) ) ) ).
% eucl_rel_int_dividesI
tff(fact_3234_zminus1__lemma,axiom,
! [A3: int,B2: int,Q4: int,R: int] :
( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R))
=> ( ( B2 != zero_zero(int) )
=> eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(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),Q4),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q4)),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_3235_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R: int] :
( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q4),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),Q4)),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)))
=> ( Q4 = zero_zero(int) ) ) ) ) ) ) ).
% eucl_rel_int_iff
tff(fact_3236_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q4: 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),Q4),L)),R) )
=> eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R)) ) ) ) ).
% eucl_rel_int_remainderI
tff(fact_3237_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),product_Pair(int,int,zero_zero(int)),A1) ) )
=> ( ! [Q5: int] :
( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),zero_zero(int)) )
=> ( ( A22 != zero_zero(int) )
=> ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A22) ) ) )
=> ~ ! [R3: int,Q5: int] :
( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),R3) )
=> ( ( aa(int,int,sgn_sgn(int),R3) = 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),R3)),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),Q5),A22)),R3) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
tff(fact_3238_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),product_Pair(int,int,zero_zero(int)),K4) ) )
| ? [L4: int,K4: int,Q3: int] :
( ( A1 = K4 )
& ( A22 = L4 )
& ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q3),zero_zero(int)) )
& ( L4 != zero_zero(int) )
& ( K4 = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L4) ) )
| ? [R5: int,L4: int,K4: int,Q3: int] :
( ( A1 = K4 )
& ( A22 = L4 )
& ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q3),R5) )
& ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
& 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),L4)))
& ( K4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L4)),R5) ) ) ) ) ).
% eucl_rel_int.simps
tff(fact_3239_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: 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),product_Pair(int,int,Q4),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),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q4),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),bit0(one2))),R)))) ) ) ).
% pos_eucl_rel_int_mult_2
tff(fact_3240_and__int_Opelims,axiom,
! [X: int,Xa4: int,Y: int] :
( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa4) = Y )
=> ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa4))
=> ~ ( ( ( ( pp(member(int,X,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,Xa4,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa4))))) ) )
& ( ~ ( pp(member(int,X,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,Xa4,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa4))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa4),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) )
=> ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa4)) ) ) ) ).
% and_int.pelims
tff(fact_3241_and__int_Opsimps,axiom,
! [K: int,L: int] :
( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K),L))
=> ( ( ( pp(member(int,K,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,L,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
& ( ~ ( pp(member(int,K,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,L,insert(int,zero_zero(int),insert(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(fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),fNot(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).
% and_int.psimps
tff(fact_3242_length__Cons,axiom,
! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),cons(A,X,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_Cons
tff(fact_3243_normalize__negative,axiom,
! [Q4: int,P3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q4),zero_zero(int)))
=> ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q4))) ) ) ).
% normalize_negative
tff(fact_3244_upto__aux__rec,axiom,
! [J: int,I: int,Js: list(int)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( upto_aux(I,J,Js) = Js ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( upto_aux(I,J,Js) = upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),cons(int,J,Js)) ) ) ) ).
% upto_aux_rec
tff(fact_3245_normalize__denom__zero,axiom,
! [P3: int] : normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),zero_zero(int))) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).
% normalize_denom_zero
tff(fact_3246_normalize__denom__pos,axiom,
! [R: product_prod(int,int),P3: int,Q4: int] :
( ( normalize(R) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q4)) ) ).
% normalize_denom_pos
tff(fact_3247_normalize__crossproduct,axiom,
! [Q4: int,S: int,P3: int,R: int] :
( ( Q4 != zero_zero(int) )
=> ( ( S != zero_zero(int) )
=> ( ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4)) = normalize(aa(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),Q4) ) ) ) ) ).
% normalize_crossproduct
tff(fact_3248_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
=> ( ! [K2: int,L2: int] :
( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),product_Pair(int,int,K2),L2))
=> ( ( ~ ( pp(member(int,K2,insert(int,zero_zero(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int)),bot_bot(set(int))))))
& pp(member(int,L2,insert(int,zero_zero(int),insert(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),K2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2))))) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L2)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% and_int.pinduct
tff(fact_3249_upto_Opinduct,axiom,
! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,A0),A1))
=> ( ! [I4: int,J2: int] :
( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,I4),J2))
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I4),J2))
=> pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int))),J2)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,I4),J2)) ) )
=> pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).
% upto.pinduct
tff(fact_3250_bit__cut__integer__def,axiom,
! [K: code_integer] : code_bit_cut_integer(K) = aa(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),bit0(one2)))),fNot(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),K))) ).
% bit_cut_integer_def
tff(fact_3251_bezw__0,axiom,
! [X: nat] : bezw(X,zero_zero(nat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ).
% bezw_0
tff(fact_3252_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys2: 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)),Ys2)))
=> ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys2)),I) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys2),I)) ) ) ) ).
% nth_zip
tff(fact_3253_Gcd__0__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ( gcd_Gcd(A,A5) = zero_zero(A) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),insert(A,zero_zero(A),bot_bot(set(A))))) ) ) ).
% Gcd_0_iff
tff(fact_3254_Gcd__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_empty
tff(fact_3255_divmod__integer_H__def,axiom,
! [M: num,N: num] : unique8689654367752047608divmod(code_integer,M,N) = aa(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_3256_Gcd__nat__eq__one,axiom,
! [N6: set(nat)] :
( pp(member(nat,one_one(nat),N6))
=> ( gcd_Gcd(nat,N6) = one_one(nat) ) ) ).
% Gcd_nat_eq_one
tff(fact_3257_Gcd__1,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( pp(member(A,one_one(A),A5))
=> ( gcd_Gcd(A,A5) = one_one(A) ) ) ) ).
% Gcd_1
tff(fact_3258_Gcd__int__greater__eq__0,axiom,
! [K5: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K5))) ).
% Gcd_int_greater_eq_0
tff(fact_3259_Gcd__eq__1__I,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A3: A,A5: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( pp(member(A,A3,A5))
=> ( gcd_Gcd(A,A5) = one_one(A) ) ) ) ) ).
% Gcd_eq_1_I
tff(fact_3260_Code__Numeral_Opositive__def,axiom,
code_positive = numeral_numeral(code_integer) ).
% Code_Numeral.positive_def
tff(fact_3261_integer__of__num__triv_I2_J,axiom,
aa(num,code_integer,code_integer_of_num,bit0(one2)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).
% integer_of_num_triv(2)
tff(fact_3262_card__atLeastAtMost__int,axiom,
! [L: int,U: int] : aa(set(int),nat,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_3263_possible__bit__def,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [Tyrep: itself(A),N: nat] :
( pp(bit_se6407376104438227557le_bit(A,Tyrep,N))
<=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).
% possible_bit_def
tff(fact_3264_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
=> boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor) ) ) ).
% abstract_boolean_algebra_sym_diff.intro
tff(fact_3265_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Conj: fun(A,fun(A,A)),Compl: fun(A,A)] :
( ! [X2: A,Y4: A] : aa(A,A,aa(A,fun(A,A),Xor,X2),Y4) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,Compl,Y4))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X2)),Y4))
=> boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
tff(fact_3266_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Xor: fun(A,fun(A,A))] :
( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
<=> ! [X5: A,Y5: A] : aa(A,A,aa(A,fun(A,A),Xor,X5),Y5) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X5),aa(A,A,Compl,Y5))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X5)),Y5)) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
tff(fact_3267_possible__bit__0,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [Ty: itself(A)] : pp(bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat))) ) ).
% possible_bit_0
tff(fact_3268_possible__bit__less__imp,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [Tyrep: itself(A),I: nat,J: nat] :
( pp(bit_se6407376104438227557le_bit(A,Tyrep,I))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> pp(bit_se6407376104438227557le_bit(A,Tyrep,J)) ) ) ) ).
% possible_bit_less_imp
tff(fact_3269_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_3270_integer__of__num__def,axiom,
code_integer_of_num = numeral_numeral(code_integer) ).
% integer_of_num_def
tff(fact_3271_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
=> boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
tff(fact_3272_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
<=> ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
& boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ) ).
% abstract_boolean_algebra_sym_diff_def
tff(fact_3273_card__Diff__insert,axiom,
! [A: $tType,A3: A,A5: set(A),B5: set(A)] :
( pp(member(A,A3,A5))
=> ( ~ pp(member(A,A3,B5))
=> ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,A3,B5))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),one_one(nat)) ) ) ) ).
% card_Diff_insert
tff(fact_3274_card_Oempty,axiom,
! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).
% card.empty
tff(fact_3275_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set(A),X: 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),aa(set(A),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),aa(set(A),nat,finite_card(A),insert(A,X,Y))),N)) ) ) ).
% card_insert_le_m1
tff(fact_3276_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A5: set(A)] :
( ( pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)) ) )
& ( ~ pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).
% card_Diff_singleton_if
tff(fact_3277_card__Diff__singleton,axiom,
! [A: $tType,X: A,A5: set(A)] :
( pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)) ) ) ).
% card_Diff_singleton
tff(fact_3278_card__atLeastAtMost,axiom,
! [L: nat,U: nat] : aa(set(nat),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_3279_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_3280_card__insert__le,axiom,
! [A: $tType,A5: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),insert(A,X,A5)))) ).
% card_insert_le
tff(fact_3281_card__1__singletonE,axiom,
! [A: $tType,A5: set(A)] :
( ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) )
=> ~ ! [X2: A] : A5 != insert(A,X2,bot_bot(set(A))) ) ).
% card_1_singletonE
tff(fact_3282_card__1__singleton__iff,axiom,
! [A: $tType,A5: set(A)] :
( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ? [X5: A] : A5 = insert(A,X5,bot_bot(set(A))) ) ).
% card_1_singleton_iff
tff(fact_3283_card__eq__SucD,axiom,
! [A: $tType,A5: set(A),K: nat] :
( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
=> ? [B4: A,B7: set(A)] :
( ( A5 = insert(A,B4,B7) )
& ~ pp(member(A,B4,B7))
& ( aa(set(A),nat,finite_card(A),B7) = K )
& ( ( K = zero_zero(nat) )
=> ( B7 = bot_bot(set(A)) ) ) ) ) ).
% card_eq_SucD
tff(fact_3284_card__Suc__eq,axiom,
! [A: $tType,A5: set(A),K: nat] :
( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
<=> ? [B8: A,B9: set(A)] :
( ( A5 = insert(A,B8,B9) )
& ~ pp(member(A,B8,B9))
& ( aa(set(A),nat,finite_card(A),B9) = K )
& ( ( K = zero_zero(nat) )
=> ( B9 = bot_bot(set(A)) ) ) ) ) ).
% card_Suc_eq
tff(fact_3285_card__Diff1__le,axiom,
! [A: $tType,A5: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))) ).
% card_Diff1_le
tff(fact_3286_drop__bit__exp__eq,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N),bit_se6407376104438227557le_bit(A,type2(A),N)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).
% drop_bit_exp_eq
tff(fact_3287_bit__minus__2__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),aa(num,A,numeral_numeral(A),bit0(one2)))),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).
% bit_minus_2_iff
tff(fact_3288_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),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) )
& pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ) ).
% bit_double_iff
tff(fact_3289_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),bit0(one2))),M)),one_one(A))),N))
<=> ( pp(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_3290_bit__minus__exp__iff,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).
% bit_minus_exp_iff
tff(fact_3291_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))
<=> pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).
% bit_minus_1_iff
tff(fact_3292_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_3293_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_3294_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_3295_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_3296_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_3297_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_3298_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_3299_bit__mask__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(nat,A,bit_se2239418461657761734s_mask(A),M)),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).
% bit_mask_iff
tff(fact_3300_bit__of__nat__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(nat,A,semiring_1_of_nat(A),M)),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),N))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),M),N)) ) ) ) ).
% bit_of_nat_iff
tff(fact_3301_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))
<=> ( pp(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_3302_bit__push__bit__iff,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,A3: A,N: nat] :
( pp(aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A3)),N))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
& pp(bit_se6407376104438227557le_bit(A,type2(A),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),M))) ) ) ) ).
% bit_push_bit_iff
tff(fact_3303_fold__possible__bit,axiom,
! [A: $tType] :
( bit_semiring_bits(A)
=> ! [N: nat] :
( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
<=> ~ pp(bit_se6407376104438227557le_bit(A,type2(A),N)) ) ) ).
% fold_possible_bit
tff(fact_3304_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),bit0(one2))),N))
<=> ( pp(bit_se6407376104438227557le_bit(A,type2(A),one_one(nat)))
& ( N = one_one(nat) ) ) ) ) ).
% bit_2_iff
tff(fact_3305_CHAR__eq0__iff,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
<=> ! [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
=> ( aa(nat,A,semiring_1_of_nat(A),N5) != zero_zero(A) ) ) ) ) ).
% CHAR_eq0_iff
tff(fact_3306_CHAR__eq__posI,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [C2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
=> ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
=> ( ! [X2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),C2))
=> ( aa(nat,A,semiring_1_of_nat(A),X2) != zero_zero(A) ) ) )
=> ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).
% CHAR_eq_posI
tff(fact_3307_CHAR__pos__iff,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
<=> ? [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
& ( aa(nat,A,semiring_1_of_nat(A),N5) = zero_zero(A) ) ) ) ) ).
% CHAR_pos_iff
tff(fact_3308_of__nat__CHAR,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).
% of_nat_CHAR
tff(fact_3309_CHAR__eqI,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [C2: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
=> ( ! [X2: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),X2) = zero_zero(A) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),C2),X2)) )
=> ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).
% CHAR_eqI
tff(fact_3310_CHAR__eq__0,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).
% CHAR_eq_0
tff(fact_3311_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] :
( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),N)) ) ) ).
% of_nat_eq_0_iff_char_dvd
tff(fact_3312_divmod__integer__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(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_3313_divmod__abs__def,axiom,
! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(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_3314_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] : aa(set(int),nat,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_3315_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),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_3316_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list(A)] :
( ( ( X = Y )
=> ( count_list(A,cons(A,X,Xs),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),count_list(A,Xs,Y)),one_one(nat)) ) )
& ( ( X != Y )
=> ( count_list(A,cons(A,X,Xs),Y) = count_list(A,Xs,Y) ) ) ) ).
% count_list.simps(2)
tff(fact_3317_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_3318_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_3319_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_3320_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_3321_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,cons(B,X,Xs)) = cons(product_prod(nat,B),aa(B,product_prod(nat,B),product_Pair(nat,B,N),X),enumerate(B,aa(nat,nat,suc,N),Xs)) ).
% enumerate_simps(2)
tff(fact_3322_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_3323_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),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).
% divmod_abs_code(6)
tff(fact_3324_count__le__length,axiom,
! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),count_list(A,Xs,X)),aa(list(A),nat,size_size(list(A)),Xs))) ).
% count_le_length
tff(fact_3325_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_3326_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),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_3327_sqr_Osimps_I3_J,axiom,
! [N: num] : sqr(aa(num,num,bit1,N)) = aa(num,num,bit1,bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(N)),N))) ).
% sqr.simps(3)
tff(fact_3328_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),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_3329_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa4: nat,Y: product_prod(nat,nat)] :
( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa4) = Y )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa4),X))
=> ( Y = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa4)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa4),X))
=> ( Y = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa4),aa(nat,nat,suc,X))) ) ) ) ) ).
% prod_decode_aux.elims
tff(fact_3330_prod__decode__aux_Osimps,axiom,
! [M: nat,K: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
=> ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M) = aa(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))
=> ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M) = aa(nat,product_prod(nat,nat),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_3331_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_3332_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_3333_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,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).
% of_nat_of_integer
tff(fact_3334_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] : aa(set(nat),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_3335_rat__one__code,axiom,
quotient_of(one_one(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int)) ).
% rat_one_code
tff(fact_3336_rat__zero__code,axiom,
quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ).
% rat_zero_code
tff(fact_3337_quotient__of__number_I4_J,axiom,
quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(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_3338_quotient__of__number_I3_J,axiom,
! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(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_3339_prod__decode__aux_Ocases,axiom,
! [X: product_prod(nat,nat)] :
~ ! [K2: nat,M5: nat] : X != aa(nat,product_prod(nat,nat),product_Pair(nat,nat,K2),M5) ).
% prod_decode_aux.cases
tff(fact_3340_sgn__rat__def,axiom,
! [A3: rat] :
( ( ( A3 = zero_zero(rat) )
=> ( aa(rat,rat,sgn_sgn(rat),A3) = zero_zero(rat) ) )
& ( ( A3 != zero_zero(rat) )
=> ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
=> ( aa(rat,rat,sgn_sgn(rat),A3) = one_one(rat) ) )
& ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A3))
=> ( aa(rat,rat,sgn_sgn(rat),A3) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).
% sgn_rat_def
tff(fact_3341_diff__rat__def,axiom,
! [Q4: rat,R: rat] : aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Q4),R) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q4),aa(rat,rat,uminus_uminus(rat),R)) ).
% diff_rat_def
tff(fact_3342_abs__rat__def,axiom,
! [A3: rat] :
( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
=> ( aa(rat,rat,abs_abs(rat),A3) = aa(rat,rat,uminus_uminus(rat),A3) ) )
& ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A3),zero_zero(rat)))
=> ( aa(rat,rat,abs_abs(rat),A3) = A3 ) ) ) ).
% abs_rat_def
tff(fact_3343_sqr_Osimps_I2_J,axiom,
! [N: num] : sqr(bit0(N)) = bit0(bit0(sqr(N))) ).
% sqr.simps(2)
tff(fact_3344_sqr_Osimps_I1_J,axiom,
sqr(one2) = one2 ).
% sqr.simps(1)
tff(fact_3345_sqr__conv__mult,axiom,
! [X: num] : sqr(X) = aa(num,num,aa(num,fun(num,num),times_times(num),X),X) ).
% sqr_conv_mult
tff(fact_3346_quotient__of__denom__pos,axiom,
! [R: rat,P3: int,Q4: int] :
( ( quotient_of(R) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q4)) ) ).
% quotient_of_denom_pos
tff(fact_3347_numeral__sqr,axiom,
! [A: $tType] :
( semiring_numeral(A)
=> ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).
% numeral_sqr
tff(fact_3348_quotient__of__int,axiom,
! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),product_Pair(int,int,A3),one_one(int)) ).
% quotient_of_int
tff(fact_3349_pow_Osimps_I3_J,axiom,
! [X: num,Y: num] : pow(X,aa(num,num,bit1,Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y))),X) ).
% pow.simps(3)
tff(fact_3350_Frct__code__post_I4_J,axiom,
! [K: num] : frct(aa(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_3351_Frct__code__post_I5_J,axiom,
! [K: num] : frct(aa(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_3352_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),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_3353_prod__encode__eq,axiom,
! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
( ( aa(product_prod(nat,nat),nat,nat_prod_encode,X) = aa(product_prod(nat,nat),nat,nat_prod_encode,Y) )
<=> ( X = Y ) ) ).
% prod_encode_eq
tff(fact_3354_less__eq__rat__def,axiom,
! [X: rat,Y: rat] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y))
<=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
| ( X = Y ) ) ) ).
% less_eq_rat_def
tff(fact_3355_Frct__code__post_I9_J,axiom,
! [Q4: product_prod(int,int)] : aa(rat,rat,uminus_uminus(rat),aa(rat,rat,uminus_uminus(rat),frct(Q4))) = frct(Q4) ).
% Frct_code_post(9)
tff(fact_3356_pow_Osimps_I1_J,axiom,
! [X: num] : pow(X,one2) = X ).
% pow.simps(1)
tff(fact_3357_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),product_Pair(nat,nat,A3),B2)))) ).
% le_prod_encode_2
tff(fact_3358_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),product_Pair(nat,nat,A3),B2)))) ).
% le_prod_encode_1
tff(fact_3359_Frct__code__post_I2_J,axiom,
! [A3: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,A3),zero_zero(int))) = zero_zero(rat) ).
% Frct_code_post(2)
tff(fact_3360_Frct__code__post_I1_J,axiom,
! [A3: int] : frct(aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A3)) = zero_zero(rat) ).
% Frct_code_post(1)
tff(fact_3361_Frct__code__post_I3_J,axiom,
frct(aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) = one_one(rat) ).
% Frct_code_post(3)
tff(fact_3362_pow_Osimps_I2_J,axiom,
! [X: num,Y: num] : pow(X,bit0(Y)) = sqr(pow(X,Y)) ).
% pow.simps(2)
tff(fact_3363_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] : frct(aa(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),product_Pair(int,int,A3),B2))) ).
% Frct_code_post(8)
tff(fact_3364_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] : frct(aa(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),product_Pair(int,int,A3),B2))) ).
% Frct_code_post(7)
tff(fact_3365_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_3366_rat__sgn__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P3)) = aa(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_3367_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_3368_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat),Fs: list(fun(A,nat))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),measures(A,cons(fun(A,nat),F,Fs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
| ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),measures(A,Fs))) ) ) ) ).
% in_measures(2)
tff(fact_3369_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)),product_Pair(list(A),list(A),cons(A,M,Ms)),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),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)),product_Pair(list(A),list(A),Ms),Ns),lenlex(A,R))) ) ) ) ).
% Cons_lenlex_iff
tff(fact_3370_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_3371_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_3372_Suc__0__div__numeral,axiom,
! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).
% Suc_0_div_numeral
tff(fact_3373_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)),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_3374_measures__less,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),measures(A,cons(fun(A,nat),F,Fs)))) ) ).
% measures_less
tff(fact_3375_measures__lesseq,axiom,
! [A: $tType,F: fun(A,nat),X: 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,X)),aa(A,nat,F,Y)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),measures(A,Fs)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),measures(A,cons(fun(A,nat),F,Fs)))) ) ) ).
% measures_lesseq
tff(fact_3376_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),J))
=> ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = cons(nat,aa(nat,nat,suc,I),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).
% sorted_list_of_set_greaterThanLessThan
tff(fact_3377_bezw_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero(nat) )
=> ( bezw(X,Y) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
& ( ( Y != zero_zero(nat) )
=> ( bezw(X,Y) = aa(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),X),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),X),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),X),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ) ).
% bezw.simps
tff(fact_3378_bezw_Oelims,axiom,
! [X: nat,Xa4: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa4) = Y )
=> ( ( ( Xa4 = zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
& ( ( Xa4 != zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa4,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa4,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa4,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa4))))) ) ) ) ) ).
% bezw.elims
tff(fact_3379_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
=> ( bezw(X,Y) = aa(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),X),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),X),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),X),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y))))) ) ) ).
% bezw_non_0
tff(fact_3380_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_3381_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys2: 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)),product_Pair(list(A),list(B),Xs),Ys2),listrel(A,B,R)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
& ! [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),N5)),aa(nat,B,nth(B,Ys2),N5)),R)) ) ) ) ).
% listrel_iff_nth
tff(fact_3382_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_3383_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_3384_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_3385_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_3386_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_3387_Ioc__inj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( set_or3652927894154168847AtMost(A,A3,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
<=> ( ( 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),D2),C2)) )
| ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
tff(fact_3388_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_3389_Ioc__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% Ioc_subset_iff
tff(fact_3390_divides__aux__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Qr: product_prod(A,A)] :
( unique5940410009612947441es_aux(A,Qr)
<=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).
% divides_aux_def
tff(fact_3391_quotient__of__denom__pos_H,axiom,
! [R: rat] : 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),quotient_of(R)))) ).
% quotient_of_denom_pos'
tff(fact_3392_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I)),J))
=> ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = cons(nat,aa(nat,nat,suc,I),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).
% sorted_list_of_set_greaterThanAtMost
tff(fact_3393_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_3394_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_3395_minus__one__mod__numeral,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).
% minus_one_mod_numeral
tff(fact_3396_one__mod__minus__numeral,axiom,
! [N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).
% one_mod_minus_numeral
tff(fact_3397_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N)))) ).
% numeral_mod_minus_numeral
tff(fact_3398_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N))) ).
% minus_numeral_mod_numeral
tff(fact_3399_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_3400_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_3401_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_3402_Suc__0__mod__numeral,axiom,
! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).
% Suc_0_mod_numeral
tff(fact_3403_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_3404_Divides_Oadjust__mod__def,axiom,
! [R: int,L: int] :
( ( ( R = zero_zero(int) )
=> ( adjust_mod(L,R) = zero_zero(int) ) )
& ( ( R != zero_zero(int) )
=> ( adjust_mod(L,R) = aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R) ) ) ) ).
% Divides.adjust_mod_def
tff(fact_3405_bezw_Opelims,axiom,
! [X: nat,Xa4: nat,Y: product_prod(int,int)] :
( ( bezw(X,Xa4) = Y )
=> ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa4))
=> ~ ( ( ( ( Xa4 = zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
& ( ( Xa4 != zero_zero(nat) )
=> ( Y = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa4,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa4,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa4,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa4))))) ) ) )
=> ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa4)) ) ) ) ).
% bezw.pelims
tff(fact_3406_in__set__enumerate__eq,axiom,
! [A: $tType,P3: product_prod(nat,A),N: nat,Xs: list(A)] :
( pp(member(product_prod(nat,A),P3,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_3407_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list(nat)] : aa(list(nat),nat,nat_list_encode,cons(nat,X,Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),aa(list(nat),nat,nat_list_encode,Xs)))) ).
% list_encode.simps(2)
tff(fact_3408_bezw__aux,axiom,
! [X: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),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(X,Y))),aa(nat,int,semiring_1_of_nat(int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).
% bezw_aux
tff(fact_3409_in__set__zip,axiom,
! [A: $tType,B: $tType,P3: product_prod(A,B),Xs: list(A),Ys2: list(B)] :
( pp(member(product_prod(A,B),P3,set2(product_prod(A,B),zip(A,B,Xs,Ys2))))
<=> ? [N5: nat] :
( ( aa(nat,A,nth(A,Xs),N5) = aa(product_prod(A,B),A,product_fst(A,B),P3) )
& ( aa(nat,B,nth(B,Ys2),N5) = aa(product_prod(A,B),B,product_snd(A,B),P3) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(B),nat,size_size(list(B)),Ys2))) ) ) ).
% in_set_zip
tff(fact_3410_gcd__eq__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: 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) ) ) ) ) ).
% gcd_eq_0_iff
tff(fact_3411_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_3412_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_3413_gcd__neg2,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).
% gcd_neg2
tff(fact_3414_gcd__neg1,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).
% gcd_neg1
tff(fact_3415_gcd__0__left__nat,axiom,
! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).
% gcd_0_left_nat
tff(fact_3416_gcd__0__nat,axiom,
! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).
% gcd_0_nat
tff(fact_3417_gcd__nat_Oright__neutral,axiom,
! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),zero_zero(nat)) = A3 ).
% gcd_nat.right_neutral
tff(fact_3418_gcd__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) )
<=> ( ( A3 = zero_zero(nat) )
& ( B2 = zero_zero(nat) ) ) ) ).
% gcd_nat.neutr_eq_iff
tff(fact_3419_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A3) = A3 ).
% gcd_nat.left_neutral
tff(fact_3420_gcd__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = zero_zero(nat) )
<=> ( ( A3 = zero_zero(nat) )
& ( B2 = zero_zero(nat) ) ) ) ).
% gcd_nat.eq_neutr_iff
tff(fact_3421_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_3422_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).
% gcd_neg_numeral_2
tff(fact_3423_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [N: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).
% gcd_neg_numeral_1
tff(fact_3424_is__unit__gcd__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_3425_gcd__Suc__0,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).
% gcd_Suc_0
tff(fact_3426_gcd__pos__nat,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),gcd_gcd(nat),M),N)))
<=> ( ( M != zero_zero(nat) )
| ( N != zero_zero(nat) ) ) ) ).
% gcd_pos_nat
tff(fact_3427_count__notin,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ~ pp(member(A,X,set2(A,Xs)))
=> ( count_list(A,Xs,X) = zero_zero(nat) ) ) ).
% count_notin
tff(fact_3428_list__encode__eq,axiom,
! [X: list(nat),Y: list(nat)] :
( ( aa(list(nat),nat,nat_list_encode,X) = aa(list(nat),nat,nat_list_encode,Y) )
<=> ( X = Y ) ) ).
% list_encode_eq
tff(fact_3429_gcd__mult__distrib__nat,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),gcd_gcd(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(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)) ).
% gcd_mult_distrib_nat
tff(fact_3430_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_3431_gcd__dvd__prod,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,K: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))) ) ).
% gcd_dvd_prod
tff(fact_3432_gcd__le1__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3)) ) ).
% gcd_le1_nat
tff(fact_3433_gcd__le2__nat,axiom,
! [B2: nat,A3: nat] :
( ( B2 != zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2)) ) ).
% gcd_le2_nat
tff(fact_3434_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).
% gcd_diff2_nat
tff(fact_3435_gcd__diff1__nat,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),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).
% gcd_diff1_nat
tff(fact_3436_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)) ) ) ).
% gcd_non_0_nat
tff(fact_3437_gcd__nat_Osimps,axiom,
! [Y: nat,X: nat] :
( ( ( Y = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) )
& ( ( Y != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Y)) ) ) ) ).
% gcd_nat.simps
tff(fact_3438_gcd__nat_Oelims,axiom,
! [X: nat,Xa4: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa4) = Y )
=> ( ( ( Xa4 = zero_zero(nat) )
=> ( Y = X ) )
& ( ( Xa4 != zero_zero(nat) )
=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa4),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)) ) ) ) ) ).
% gcd_nat.elims
tff(fact_3439_gcd__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_3440_gcd__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_3441_gcd__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_3442_gcd__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_3443_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3 != zero_zero(nat) )
=> ? [X2: nat,Y4: nat] : 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),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).
% bezout_nat
tff(fact_3444_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X2: nat,Y4: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)))
& ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)),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) ) )
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)))
& ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).
% bezout_gcd_nat'
tff(fact_3445_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member(A,X,set2(A,Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% length_pos_if_in_set
tff(fact_3446_card__length,axiom,
! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),set2(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% card_length
tff(fact_3447_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool)] :
( ! [X5: A] :
( pp(member(A,X5,set2(A,Xs)))
=> pp(aa(A,bool,P,X5)) )
<=> ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I5))) ) ) ).
% all_set_conv_all_nth
tff(fact_3448_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list(A),P: fun(A,bool),X: A] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) )
=> ( pp(member(A,X,set2(A,Xs)))
=> pp(aa(A,bool,P,X)) ) ) ).
% all_nth_imp_all_set
tff(fact_3449_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member(A,X,set2(A,Xs)))
<=> ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),I5) = X ) ) ) ).
% in_set_conv_nth
tff(fact_3450_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).
% list_ball_nth
tff(fact_3451_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member(A,aa(nat,A,nth(A,Xs),N),set2(A,Xs))) ) ).
% nth_mem
tff(fact_3452_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat] :
( ~ pp(member(A,X,set2(A,Xs)))
=> ( 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,cons(A,X,Xs)),N) = X )
<=> ( N = zero_zero(nat) ) ) ) ) ).
% nth_equal_first_eq
tff(fact_3453_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa4: nat,Y: product_prod(nat,nat)] :
( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa4) = Y )
=> ( accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa4))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa4),X))
=> ( Y = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa4)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa4),X))
=> ( Y = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa4),aa(nat,nat,suc,X))) ) ) )
=> ~ accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa4)) ) ) ) ).
% prod_decode_aux.pelims
tff(fact_3454_gcd__nat_Opelims,axiom,
! [X: nat,Xa4: nat,Y: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa4) = Y )
=> ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa4))
=> ~ ( ( ( ( Xa4 = zero_zero(nat) )
=> ( Y = X ) )
& ( ( Xa4 != zero_zero(nat) )
=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa4),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X),Xa4)) ) ) )
=> ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa4)) ) ) ) ).
% gcd_nat.pelims
tff(fact_3455_list__encode_Oelims,axiom,
! [X: list(nat),Y: nat] :
( ( aa(list(nat),nat,nat_list_encode,X) = Y )
=> ( ( ( X = nil(nat) )
=> ( Y != zero_zero(nat) ) )
=> ~ ! [X2: nat,Xs2: list(nat)] :
( ( X = cons(nat,X2,Xs2) )
=> ( Y != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X2),aa(list(nat),nat,nat_list_encode,Xs2)))) ) ) ) ) ).
% list_encode.elims
tff(fact_3456_zero__rat__def,axiom,
zero_zero(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))) ).
% zero_rat_def
tff(fact_3457_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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2)),fun_pair_less)) ) ) ).
% pair_lessI2
tff(fact_3458_gcd__1__int,axiom,
! [M: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),one_one(int)) = one_one(int) ).
% gcd_1_int
tff(fact_3459_gcd__neg2__int,axiom,
! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).
% gcd_neg2_int
tff(fact_3460_gcd__neg1__int,axiom,
! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).
% gcd_neg1_int
tff(fact_3461_gcd__pos__int,axiom,
! [M: int,N: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)))
<=> ( ( M != zero_zero(int) )
| ( N != zero_zero(int) ) ) ) ).
% gcd_pos_int
tff(fact_3462_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),X) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),N)),X) ).
% gcd_neg_numeral_1_int
tff(fact_3463_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(num,int,numeral_numeral(int),N)) ).
% gcd_neg_numeral_2_int
tff(fact_3464_gcd__0__left__int,axiom,
! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),zero_zero(int)),X) = aa(int,int,abs_abs(int),X) ).
% gcd_0_left_int
tff(fact_3465_gcd__0__int,axiom,
! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),zero_zero(int)) = aa(int,int,abs_abs(int),X) ).
% gcd_0_int
tff(fact_3466_gcd__int__int__eq,axiom,
! [M: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),M)),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),gcd_gcd(nat),M),N)) ).
% gcd_int_int_eq
tff(fact_3467_length__0__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
<=> ( Xs = nil(A) ) ) ).
% length_0_conv
tff(fact_3468_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_0(A)
=> ! [F: fun(B,A),A3: 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),nil(B)) = zero_zero(A) ) ).
% horner_sum_simps(1)
tff(fact_3469_length__greater__0__conv,axiom,
! [A: $tType,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
<=> ( Xs != nil(A) ) ) ).
% length_greater_0_conv
tff(fact_3470_gcd__nat__abs__right__eq,axiom,
! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).
% gcd_nat_abs_right_eq
tff(fact_3471_gcd__nat__abs__left__eq,axiom,
! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).
% gcd_nat_abs_left_eq
tff(fact_3472_pair__less__iff1,axiom,
! [X: 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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Y)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Z2)),fun_pair_less))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),Z2)) ) ).
% pair_less_iff1
tff(fact_3473_list__encode_Ocases,axiom,
! [X: list(nat)] :
( ( X != nil(nat) )
=> ~ ! [X2: nat,Xs2: list(nat)] : X != cons(nat,X2,Xs2) ) ).
% list_encode.cases
tff(fact_3474_gcd__ge__0__int,axiom,
! [X: int,Y: 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),gcd_gcd(int),X),Y))) ).
% gcd_ge_0_int
tff(fact_3475_bezout__int,axiom,
! [X: 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),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V4),Y)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) ).
% bezout_int
tff(fact_3476_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)) ).
% gcd_mult_distrib_int
tff(fact_3477_list_Osize_I3_J,axiom,
! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).
% list.size(3)
tff(fact_3478_list__encode_Osimps_I1_J,axiom,
aa(list(nat),nat,nat_list_encode,nil(nat)) = zero_zero(nat) ).
% list_encode.simps(1)
tff(fact_3479_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] : count_list(A,nil(A),Y) = zero_zero(nat) ).
% count_list.simps(1)
tff(fact_3480_gcd__le1__int,axiom,
! [A3: int,B2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),A3)) ) ).
% gcd_le1_int
tff(fact_3481_gcd__le2__int,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)),B2)) ) ).
% gcd_le2_int
tff(fact_3482_gcd__cases__int,axiom,
! [X: int,Y: int,P: fun(int,bool)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)))) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y))) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y)))) ) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ) ) ).
% gcd_cases_int
tff(fact_3483_gcd__unique__int,axiom,
! [D2: int,A3: int,B2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D2))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),A3))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),B2))
& ! [E3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E3),A3))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E3),B2)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E3),D2)) ) )
<=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2) ) ) ).
% gcd_unique_int
tff(fact_3484_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
=> ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X),Y)) ) ) ).
% gcd_non_0_int
tff(fact_3485_gcd__code__int,axiom,
! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),L) = aa(int,int,abs_abs(int),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),L),zero_zero(int)),K,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))))) ).
% gcd_code_int
tff(fact_3486_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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2)),fun_pair_less)) ) ).
% pair_lessI1
tff(fact_3487_gcd__int__def,axiom,
! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ).
% gcd_int_def
tff(fact_3488_one__rat__def,axiom,
one_one(rat) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))) ).
% one_rat_def
tff(fact_3489_plus__rat_Oabs__eq,axiom,
! [Xa4: product_prod(int,int),X: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa4),Xa4))
=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
=> ( aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(product_prod(int,int),rat,abs_Rat,Xa4)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(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),Xa4)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa4)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa4)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).
% plus_rat.abs_eq
tff(fact_3490_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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2)),fun_pair_leq)) ) ) ).
% pair_leqI2
tff(fact_3491_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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B2),T2)),fun_pair_leq)) ) ).
% pair_leqI1
tff(fact_3492_smax__insertI,axiom,
! [Y: product_prod(nat,nat),Y6: set(product_prod(nat,nat)),X: product_prod(nat,nat),X6: set(product_prod(nat,nat))] :
( pp(member(product_prod(nat,nat),Y,Y6))
=> ( 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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),X),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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),X6),Y6),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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert(product_prod(nat,nat),X,X6)),Y6),fun_max_strict)) ) ) ) ).
% smax_insertI
tff(fact_3493_smin__insertI,axiom,
! [X: 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),X,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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),X),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))),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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),XS),insert(product_prod(nat,nat),Y,YS)),fun_min_strict)) ) ) ) ).
% smin_insertI
tff(fact_3494_ratrel__iff,axiom,
! [X: product_prod(int,int),Y: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),Y))
<=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X) != zero_zero(int) )
& ( aa(product_prod(int,int),int,product_snd(int,int),Y) != zero_zero(int) )
& ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Y)),aa(product_prod(int,int),int,product_snd(int,int),X)) ) ) ) ).
% ratrel_iff
tff(fact_3495_smin__emptyI,axiom,
! [X6: set(product_prod(nat,nat))] :
( ( X6 != 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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),X6),bot_bot(set(product_prod(nat,nat)))),fun_min_strict)) ) ).
% smin_emptyI
tff(fact_3496_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),product_Pair(int,int,one_one(int)),one_one(int))),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int)))) ).
% one_rat.rsp
tff(fact_3497_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),product_Pair(int,int,zero_zero(int)),one_one(int))),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)))) ).
% zero_rat.rsp
tff(fact_3498_times__rat_Oabs__eq,axiom,
! [Xa4: product_prod(int,int),X: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Xa4),Xa4))
=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
=> ( aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(product_prod(int,int),rat,abs_Rat,Xa4)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(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),Xa4)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa4)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).
% times_rat.abs_eq
tff(fact_3499_uminus__rat_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
=> ( aa(rat,rat,uminus_uminus(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(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),X))),aa(product_prod(int,int),int,product_snd(int,int),X))) ) ) ).
% uminus_rat.abs_eq
tff(fact_3500_wmin__insertI,axiom,
! [X: 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),X,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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),X),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))),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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),XS),insert(product_prod(nat,nat),Y,YS)),fun_min_weak)) ) ) ) ).
% wmin_insertI
tff(fact_3501_wmax__insertI,axiom,
! [Y: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X: 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)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),X),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))),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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert(product_prod(nat,nat),X,XS)),YS),fun_max_weak)) ) ) ) ).
% wmax_insertI
tff(fact_3502_positive_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
=> ( pp(aa(rat,bool,positive,aa(product_prod(int,int),rat,abs_Rat,X)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ).
% positive.abs_eq
tff(fact_3503_upto_Opelims,axiom,
! [X: int,Xa4: int,Y: list(int)] :
( ( upto(X,Xa4) = Y )
=> ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa4))
=> ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa4))
=> ( Y = cons(int,X,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa4)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa4))
=> ( Y = nil(int) ) ) )
=> ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),product_Pair(int,int,X),Xa4)) ) ) ) ).
% upto.pelims
tff(fact_3504_upto_Opsimps,axiom,
! [I: int,J: int] :
( accp(product_prod(int,int),upto_rel,aa(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) = 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_3505_upto__Nil,axiom,
! [I: int,J: int] :
( ( upto(I,J) = nil(int) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I)) ) ).
% upto_Nil
tff(fact_3506_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil(int) = upto(I,J) )
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I)) ) ).
% upto_Nil2
tff(fact_3507_upto__empty,axiom,
! [J: int,I: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I))
=> ( upto(I,J) = nil(int) ) ) ).
% upto_empty
tff(fact_3508_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_3509_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_3510_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)) = 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_3511_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))) = 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_3512_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)) = 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_3513_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))) = 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_3514_less__by__empty,axiom,
! [A: $tType,A5: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( ( A5 = bot_bot(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))),A5),B5)) ) ).
% less_by_empty
tff(fact_3515_wmin__emptyI,axiom,
! [X6: 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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),X6),bot_bot(set(product_prod(nat,nat)))),fun_min_weak)) ).
% wmin_emptyI
tff(fact_3516_positive__minus,axiom,
! [X: rat] :
( ~ pp(aa(rat,bool,positive,X))
=> ( ( X != zero_zero(rat) )
=> pp(aa(rat,bool,positive,aa(rat,rat,uminus_uminus(rat),X))) ) ) ).
% positive_minus
tff(fact_3517_greaterThanAtMost__upto,axiom,
! [I: int,J: int] : set_or3652927894154168847AtMost(int,I,J) = set2(int,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ).
% greaterThanAtMost_upto
tff(fact_3518_less__rat__def,axiom,
! [X: rat,Y: rat] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
<=> pp(aa(rat,bool,positive,aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),Y),X))) ) ).
% less_rat_def
tff(fact_3519_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) = cons(int,I,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).
% upto_rec1
tff(fact_3520_upto_Oelims,axiom,
! [X: int,Xa4: int,Y: list(int)] :
( ( upto(X,Xa4) = Y )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa4))
=> ( Y = cons(int,X,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa4)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa4))
=> ( Y = nil(int) ) ) ) ) ).
% upto.elims
tff(fact_3521_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) = 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_3522_greaterThanLessThan__upto,axiom,
! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = 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_3523_positive_Orep__eq,axiom,
! [X: rat] :
( pp(aa(rat,bool,positive,X))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).
% positive.rep_eq
tff(fact_3524_inverse__rat_Oabs__eq,axiom,
! [X: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X),X))
=> ( aa(rat,rat,inverse_inverse(rat),aa(product_prod(int,int),rat,abs_Rat,X)) = 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),X)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),X)),aa(product_prod(int,int),int,product_fst(int,int),X)))) ) ) ).
% inverse_rat.abs_eq
tff(fact_3525_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))))),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_3526_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))))),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_3527_upto__rec2,axiom,
! [I: int,J: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
=> ( upto(I,J) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),cons(int,J,nil(int))) ) ) ).
% upto_rec2
tff(fact_3528_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
<=> ( A3 = B2 ) ) ) ).
% inverse_eq_iff_eq
tff(fact_3529_inverse__inverse__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ).
% inverse_inverse_eq
tff(fact_3530_inverse__zero,axiom,
! [A: $tType] :
( division_ring(A)
=> ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).
% inverse_zero
tff(fact_3531_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% inverse_nonzero_iff_nonzero
tff(fact_3532_inverse__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : aa(A,A,inverse_inverse(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),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)) ) ).
% inverse_mult_distrib
tff(fact_3533_inverse__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).
% inverse_1
tff(fact_3534_inverse__eq__1__iff,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A] :
( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
<=> ( X = one_one(A) ) ) ) ).
% inverse_eq_1_iff
tff(fact_3535_inverse__minus__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ).
% inverse_minus_eq
tff(fact_3536_inverse__divide,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) ) ).
% inverse_divide
tff(fact_3537_abs__inverse,axiom,
! [A: $tType] :
( field_abs_sgn(A)
=> ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ).
% abs_inverse
tff(fact_3538_sgn__inverse,axiom,
! [A: $tType] :
( field_abs_sgn(A)
=> ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A3)) ) ).
% sgn_inverse
tff(fact_3539_inverse__sgn,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).
% inverse_sgn
tff(fact_3540_inverse__nonnegative__iff__nonnegative,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,inverse_inverse(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).
% inverse_nonnegative_iff_nonnegative
tff(fact_3541_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) ) ) ).
% inverse_nonpositive_iff_nonpositive
tff(fact_3542_inverse__less__iff__less,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),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% inverse_less_iff_less
tff(fact_3543_inverse__less__iff__less__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),B2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3)) ) ) ) ) ).
% inverse_less_iff_less_neg
tff(fact_3544_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ).
% inverse_negative_iff_negative
tff(fact_3545_inverse__positive__iff__positive,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,inverse_inverse(A),A3)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).
% inverse_positive_iff_positive
tff(fact_3546_inverse__le__iff__le,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),zero_zero(A)),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% inverse_le_iff_le
tff(fact_3547_inverse__le__iff__le__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),B2),zero_zero(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ) ) ).
% inverse_le_iff_le_neg
tff(fact_3548_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_3549_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_3550_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W2)) ) ).
% inverse_eq_divide_numeral
tff(fact_3551_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [W2: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = 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),W2))) ) ).
% inverse_eq_divide_neg_numeral
tff(fact_3552_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Y: A,X: A] :
( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
tff(fact_3553_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
=> ( A3 = B2 ) ) ) ).
% inverse_eq_imp_eq
tff(fact_3554_field__class_Ofield__inverse__zero,axiom,
! [A: $tType] :
( field(A)
=> ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).
% field_class.field_inverse_zero
tff(fact_3555_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
=> ( A3 = zero_zero(A) ) ) ) ).
% inverse_zero_imp_zero
tff(fact_3556_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
=> ( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( A3 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
tff(fact_3557_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).
% nonzero_inverse_inverse_eq
tff(fact_3558_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).
% nonzero_imp_inverse_nonzero
tff(fact_3559_inverse__less__imp__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
=> ( 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)) ) ) ) ).
% inverse_less_imp_less
tff(fact_3560_less__imp__inverse__less,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),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% less_imp_inverse_less
tff(fact_3561_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
=> ( 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),B2),A3)) ) ) ) ).
% inverse_less_imp_less_neg
tff(fact_3562_less__imp__inverse__less__neg,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),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% less_imp_inverse_less_neg
tff(fact_3563_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)))
=> ( ( A3 != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).
% inverse_negative_imp_negative
tff(fact_3564_inverse__positive__imp__positive,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,inverse_inverse(A),A3)))
=> ( ( A3 != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ).
% inverse_positive_imp_positive
tff(fact_3565_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: 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,inverse_inverse(A),A3)),zero_zero(A))) ) ) ).
% negative_imp_inverse_negative
tff(fact_3566_positive__imp__inverse__positive,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),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))) ) ) ).
% positive_imp_inverse_positive
tff(fact_3567_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,inverse_inverse(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),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).
% nonzero_inverse_mult_distrib
tff(fact_3568_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).
% nonzero_inverse_minus_eq
tff(fact_3569_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_3570_inverse__numeral__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ( aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),one2)) = aa(num,A,numeral_numeral(A),one2) ) ) ).
% inverse_numeral_1
tff(fact_3571_divide__inverse__commute,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,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A3) ) ).
% divide_inverse_commute
tff(fact_3572_divide__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).
% divide_inverse
tff(fact_3573_field__class_Ofield__divide__inverse,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,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B2)) ) ).
% field_class.field_divide_inverse
tff(fact_3574_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_3575_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: 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),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),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,inverse_inverse(A),X)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).
% power_mult_power_inverse_commute
tff(fact_3576_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).
% power_mult_inverse_distrib
tff(fact_3577_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Xa4: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa4))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa4))) ) ).
% mult_inverse_of_nat_commute
tff(fact_3578_nonzero__abs__inverse,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ) ) ).
% nonzero_abs_inverse
tff(fact_3579_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Xa4: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa4))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa4))) ) ).
% mult_inverse_of_int_commute
tff(fact_3580_inverse__le__imp__le,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
=> ( 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)) ) ) ) ).
% inverse_le_imp_le
tff(fact_3581_le__imp__inverse__le,axiom,
! [A: $tType] :
( linordered_field(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(A),zero_zero(A)),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% le_imp_inverse_le
tff(fact_3582_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B2)))
=> ( 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_eq(A),B2),A3)) ) ) ) ).
% inverse_le_imp_le_neg
tff(fact_3583_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( linordered_field(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(A),B2),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A3))) ) ) ) ).
% le_imp_inverse_le_neg
tff(fact_3584_inverse__le__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).
% inverse_le_1_iff
tff(fact_3585_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_3586_one__less__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).
% one_less_inverse_iff
tff(fact_3587_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_3588_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_3589_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_3590_division__ring__inverse__diff,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),minus_minus(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),minus_minus(A),B2),A3))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).
% division_ring_inverse_diff
tff(fact_3591_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_3592_inverse__le__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(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),times_times(A),A3),B2)))
=> 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),times_times(A),A3),B2)),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ).
% inverse_le_iff
tff(fact_3593_inverse__less__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(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),times_times(A),A3),B2)))
=> 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_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),zero_zero(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ) ).
% inverse_less_iff
tff(fact_3594_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_3595_inverse__less__1__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).
% inverse_less_1_iff
tff(fact_3596_one__le__inverse__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).
% one_le_inverse_iff
tff(fact_3597_reals__Archimedean,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ? [N3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),X)) ) ) ).
% reals_Archimedean
tff(fact_3598_length__append__singleton,axiom,
! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,cons(A,X,nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_append_singleton
tff(fact_3599_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
<=> ? [Y5: A,Ys3: list(A)] :
( ( Xs = append(A,Ys3,cons(A,Y5,nil(A))) )
& ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).
% length_Suc_conv_rev
tff(fact_3600_nth__append,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys2: list(A)] :
( ( 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,append(A,Xs,Ys2)),N) = aa(nat,A,nth(A,Xs),N) ) )
& ( ~ 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,append(A,Xs,Ys2)),N) = aa(nat,A,nth(A,Ys2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).
% nth_append
tff(fact_3601_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) = 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_3602_upto__split1,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) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int))),upto(J,K)) ) ) ) ).
% upto_split1
tff(fact_3603_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( archim462609752435547400_field(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> ? [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N3))),X)) ) ) ) ).
% ex_inverse_of_nat_less
tff(fact_3604_power__diff__conv__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: nat,N: nat] :
( ( X != zero_zero(A) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).
% power_diff_conv_inverse
tff(fact_3605_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,A),A3: A,Xs: list(B),Ys2: 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),append(B,Xs,Ys2)) = 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),Ys2))) ) ).
% horner_sum_append
tff(fact_3606_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) = append(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(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_3607_list__encode_Opelims,axiom,
! [X: list(nat),Y: nat] :
( ( aa(list(nat),nat,nat_list_encode,X) = Y )
=> ( accp(list(nat),nat_list_encode_rel,X)
=> ( ( ( X = nil(nat) )
=> ( ( Y = zero_zero(nat) )
=> ~ accp(list(nat),nat_list_encode_rel,nil(nat)) ) )
=> ~ ! [X2: nat,Xs2: list(nat)] :
( ( X = cons(nat,X2,Xs2) )
=> ( ( Y = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X2),aa(list(nat),nat,nat_list_encode,Xs2)))) )
=> ~ accp(list(nat),nat_list_encode_rel,cons(nat,X2,Xs2)) ) ) ) ) ) ).
% list_encode.pelims
tff(fact_3608_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),F: fun(B,A)] :
( fun_reduction_pair(A,P)
=> fun_reduction_pair(B,aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(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)))),fun_rp_inv_image(A,B),P),F)) ) ).
% rp_inv_image_rp
tff(fact_3609_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( take(A,aa(nat,nat,suc,I),Xs) = append(A,take(A,I,Xs),cons(A,aa(nat,A,nth(A,Xs),I),nil(A))) ) ) ).
% take_Suc_conv_app_nth
tff(fact_3610_min__strict__def,axiom,
fun_min_strict = min_ext(product_prod(nat,nat),fun_pair_less) ).
% min_strict_def
tff(fact_3611_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
=> ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = cons(nat,aa(num,nat,numeral_numeral(nat),M),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
=> ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).
% upt_rec_numeral
tff(fact_3612_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,aa(nat,nat,suc,N),cons(A,X,Xs)) = cons(A,X,take(A,N,Xs)) ).
% take_Suc_Cons
tff(fact_3613_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( nil(A) = take(A,N,Xs) )
<=> ( ( N = zero_zero(nat) )
| ( Xs = nil(A) ) ) ) ).
% take_eq_Nil2
tff(fact_3614_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( take(A,N,Xs) = nil(A) )
<=> ( ( N = zero_zero(nat) )
| ( Xs = nil(A) ) ) ) ).
% take_eq_Nil
tff(fact_3615_take0,axiom,
! [A: $tType,X3: list(A)] : take(A,zero_zero(nat),X3) = nil(A) ).
% take0
tff(fact_3616_take__all,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)),N))
=> ( take(A,N,Xs) = Xs ) ) ).
% take_all
tff(fact_3617_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( take(A,N,Xs) = Xs )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).
% take_all_iff
tff(fact_3618_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_3619_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
=> ( aa(nat,A,nth(A,take(A,N,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).
% nth_take
tff(fact_3620_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
=> ( upt(I,J) = nil(nat) ) ) ).
% upt_conv_Nil
tff(fact_3621_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( upt(I,J) = nil(nat) )
<=> ( ( J = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I)) ) ) ).
% upt_eq_Nil_conv
tff(fact_3622_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_3623_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),cons(A,X,Xs)) = cons(A,X,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_3624_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)) = 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_3625_take__0,axiom,
! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).
% take_0
tff(fact_3626_atLeastAtMost__upt,axiom,
! [N: nat,M: nat] : set_or1337092689740270186AtMost(nat,N,M) = set2(nat,upt(N,aa(nat,nat,suc,M))) ).
% atLeastAtMost_upt
tff(fact_3627_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),cons(nat,J,nil(nat))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).
% upt_Suc
tff(fact_3628_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),cons(nat,J,nil(nat))) ) ) ).
% upt_Suc_append
tff(fact_3629_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list(nat),Q4: nat] :
( ( cons(nat,M,cons(nat,N,Ns)) = upt(M,Q4) )
<=> ( cons(nat,N,Ns) = upt(aa(nat,nat,suc,M),Q4) ) ) ).
% upt_conv_Cons_Cons
tff(fact_3630_upt__0,axiom,
! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).
% upt_0
tff(fact_3631_greaterThanAtMost__upt,axiom,
! [N: nat,M: nat] : set_or3652927894154168847AtMost(nat,N,M) = set2(nat,upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M))) ).
% greaterThanAtMost_upt
tff(fact_3632_greaterThanLessThan__upt,axiom,
! [N: nat,M: nat] : set_or5935395276787703475ssThan(nat,N,M) = set2(nat,upt(aa(nat,nat,suc,N),M)) ).
% greaterThanLessThan_upt
tff(fact_3633_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set2(A,take(A,M,Xs))),set2(A,take(A,N,Xs)))) ) ).
% set_take_subset_set_take
tff(fact_3634_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( upt(I,J) = cons(nat,I,upt(aa(nat,nat,suc,I),J)) ) ) ).
% upt_conv_Cons
tff(fact_3635_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list(A),Ys2: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys2)))
=> ( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
=> ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys2),I4) ) )
=> ( take(A,K,Xs) = take(A,K,Ys2) ) ) ) ) ).
% nth_take_lemma
tff(fact_3636_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list(nat)] :
( ( upt(I,J) = cons(nat,X,Xs) )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
& ( I = X )
& ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).
% upt_eq_Cons_conv
tff(fact_3637_upt__rec,axiom,
! [I: nat,J: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( upt(I,J) = cons(nat,I,upt(aa(nat,nat,suc,I),J)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( upt(I,J) = nil(nat) ) ) ) ).
% upt_rec
tff(fact_3638_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( take(A,N,cons(A,X,Xs)) = nil(A) ) )
& ( ( N != zero_zero(nat) )
=> ( take(A,N,cons(A,X,Xs)) = cons(A,X,take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs)) ) ) ) ).
% take_Cons'
tff(fact_3639_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [A3: A,N: nat] : aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),map(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),A3),upt(zero_zero(nat),N))) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) ) ).
% horner_sum_bit_eq_take_bit
tff(fact_3640_lexord__take__index__conv,axiom,
! [A: $tType,X: 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)),product_Pair(list(A),list(A),X),Y),lexord(A,R)))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
& ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
| ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
& ( take(A,I5,X) = take(A,I5,Y) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,X),I5)),aa(nat,A,nth(A,Y),I5)),R)) ) ) ) ).
% lexord_take_index_conv
tff(fact_3641_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys2: 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)),product_Pair(list(A),list(A),Xs),Ys2),lex(A,R)))
=> ~ ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Ys2)))
=> ( ( take(A,I4,Xs) = take(A,I4,Ys2) )
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Ys2),I4)),R)) ) ) ) ) ).
% lex_take_index
tff(fact_3642_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)),insert(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))),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_3643_sup_Oidem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).
% sup.idem
tff(fact_3644_sup__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),X) = X ) ).
% sup_idem
tff(fact_3645_sup_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).
% sup.left_idem
tff(fact_3646_sup__left__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).
% sup_left_idem
tff(fact_3647_sup_Oright__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ).
% sup.right_idem
tff(fact_3648_sup__apply,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F),G),X) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ).
% sup_apply
tff(fact_3649_le__sup__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z2)) ) ) ) ).
% le_sup_iff
tff(fact_3650_sup_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: 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),sup_sup(A),B2),C2)),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_eq(A),C2),A3)) ) ) ) ).
% sup.bounded_iff
tff(fact_3651_sup__bot__left,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),X) = X ) ).
% sup_bot_left
tff(fact_3652_sup__bot__right,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).
% sup_bot_right
tff(fact_3653_bot__eq__sup__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A,Y: A] :
( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) )
<=> ( ( X = bot_bot(A) )
& ( Y = bot_bot(A) ) ) ) ) ).
% bot_eq_sup_iff
tff(fact_3654_sup__eq__bot__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = bot_bot(A) )
<=> ( ( X = bot_bot(A) )
& ( Y = bot_bot(A) ) ) ) ) ).
% sup_eq_bot_iff
tff(fact_3655_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = bot_bot(A) )
<=> ( ( A3 = bot_bot(A) )
& ( B2 = bot_bot(A) ) ) ) ) ).
% sup_bot.eq_neutr_iff
tff(fact_3656_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).
% sup_bot.left_neutral
tff(fact_3657_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A,B2: A] :
( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
<=> ( ( A3 = bot_bot(A) )
& ( B2 = bot_bot(A) ) ) ) ) ).
% sup_bot.neutr_eq_iff
tff(fact_3658_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).
% sup_bot.right_neutral
tff(fact_3659_Compl__Diff__eq,axiom,
! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),B5) ).
% Compl_Diff_eq
tff(fact_3660_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list(A),F: fun(A,B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,B,nth(B,map(A,B,F,Xs)),N) = aa(A,B,F,aa(nat,A,nth(A,Xs),N)) ) ) ).
% nth_map
tff(fact_3661_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) ) ).
% inf_sup_aci(8)
tff(fact_3662_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% inf_sup_aci(7)
tff(fact_3663_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).
% inf_sup_aci(6)
tff(fact_3664_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).
% inf_sup_aci(5)
tff(fact_3665_sup_Oassoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).
% sup.assoc
tff(fact_3666_sup__assoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) ) ).
% sup_assoc
tff(fact_3667_sup_Ocommute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),A3) ) ).
% sup.commute
tff(fact_3668_sup__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X) ) ).
% sup_commute
tff(fact_3669_sup_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)) ) ).
% sup.left_commute
tff(fact_3670_sup__left__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% sup_left_commute
tff(fact_3671_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_sup(B)
=> ! [F: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F),G),X3) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ).
% sup_fun_def
tff(fact_3672_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.sup1
tff(fact_3673_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B5: A,K: A,B2: A,A3: A] :
( ( B5 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B5) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.sup2
tff(fact_3674_less__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% less_supI1
tff(fact_3675_less__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% less_supI2
tff(fact_3676_sup_Oabsorb3,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb3
tff(fact_3677_sup_Oabsorb4,axiom,
! [A: $tType] :
( semilattice_sup(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),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb4
tff(fact_3678_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: 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),sup_sup(A),B2),C2)),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),C2),A3)) ) ) ) ).
% sup.strict_boundedE
tff(fact_3679_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% sup.strict_order_iff
tff(fact_3680_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.strict_coboundedI1
tff(fact_3681_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.strict_coboundedI2
tff(fact_3682_sup__max,axiom,
! [A: $tType] :
( ( semilattice_sup(A)
& linorder(A) )
=> ( sup_sup(A) = ord_max(A) ) ) ).
% sup_max
tff(fact_3683_sup_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A3: 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),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.coboundedI2
tff(fact_3684_sup_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,B2: A] :
( 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),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% sup.coboundedI1
tff(fact_3685_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( semilattice_sup(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),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb_iff2
tff(fact_3686_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( semilattice_sup(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),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb_iff1
tff(fact_3687_sup_Ocobounded2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ).
% sup.cobounded2
tff(fact_3688_sup_Ocobounded1,axiom,
! [A: $tType] :
( semilattice_sup(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),sup_sup(A),A3),B2))) ) ).
% sup.cobounded1
tff(fact_3689_sup_Oorder__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).
% sup.order_iff
tff(fact_3690_sup_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(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),C2),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A3)) ) ) ) ).
% sup.boundedI
tff(fact_3691_sup_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: 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),sup_sup(A),B2),C2)),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_eq(A),C2),A3)) ) ) ) ).
% sup.boundedE
tff(fact_3692_sup__absorb2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).
% sup_absorb2
tff(fact_3693_sup__absorb1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).
% sup_absorb1
tff(fact_3694_sup_Oabsorb2,axiom,
! [A: $tType] :
( semilattice_sup(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),sup_sup(A),A3),B2) = B2 ) ) ) ).
% sup.absorb2
tff(fact_3695_sup_Oabsorb1,axiom,
! [A: $tType] :
( semilattice_sup(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),sup_sup(A),A3),B2) = A3 ) ) ) ).
% sup.absorb1
tff(fact_3696_sup__unique,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [F: fun(A,fun(A,A)),X: A,Y: A] :
( ! [X2: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F,X2),Y4)))
=> ( ! [X2: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),aa(A,A,aa(A,fun(A,A),F,X2),Y4)))
=> ( ! [X2: A,Y4: A,Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,Y4),Z3)),X2)) ) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).
% sup_unique
tff(fact_3697_sup_OorderI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3)) ) ) ).
% sup.orderI
tff(fact_3698_sup_OorderE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) ) ) ) ).
% sup.orderE
tff(fact_3699_le__iff__sup,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).
% le_iff_sup
tff(fact_3700_sup__least,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X)) ) ) ) ).
% sup_least
tff(fact_3701_sup__mono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,C2: A,B2: A,D2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2))) ) ) ) ).
% sup_mono
tff(fact_3702_sup_Omono,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A3: A,D2: A,B2: A] :
( 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),D2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ) ).
% sup.mono
tff(fact_3703_le__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% le_supI2
tff(fact_3704_le__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2))) ) ) ).
% le_supI1
tff(fact_3705_sup__ge2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% sup_ge2
tff(fact_3706_sup__ge1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% sup_ge1
tff(fact_3707_le__supI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,X: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X)) ) ) ) ).
% le_supI
tff(fact_3708_le__supE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A3: A,B2: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2)),X))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).
% le_supE
tff(fact_3709_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% inf_sup_ord(3)
tff(fact_3710_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).
% inf_sup_ord(4)
tff(fact_3711_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),bot_bot(A)) = X ) ).
% boolean_algebra.disj_zero_right
tff(fact_3712_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_3713_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_3714_card__Un__le,axiom,
! [A: $tType,A5: set(A),B5: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)))) ).
% card_Un_le
tff(fact_3715_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_3716_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_3717_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_3718_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,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_3719_lexord__sufI,axiom,
! [A: $tType,U: list(A),W2: 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)),product_Pair(list(A),list(A),U),W2),lexord(A,R)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),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)),product_Pair(list(A),list(A),append(A,U,V)),append(A,W2,Z2)),lexord(A,R))) ) ) ).
% lexord_sufI
tff(fact_3720_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)),insert(A,L,bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(5)
tff(fact_3721_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_3722_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)),insert(A,U,bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(4)
tff(fact_3723_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) )
=> ( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I4)) ) )
=> ( map(nat,A,F,upt(M,N)) = Xs ) ) ) ).
% map_upt_eqI
tff(fact_3724_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)),insert(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))),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_3725_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] : linord4507533701916653071of_set(nat,set_ord_atMost(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_atMost(nat,K)),cons(nat,aa(nat,nat,suc,K),nil(nat))) ).
% sorted_list_of_set_atMost_Suc
tff(fact_3726_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] : linord4507533701916653071of_set(nat,set_ord_lessThan(nat,aa(nat,nat,suc,K))) = append(nat,linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K)),cons(nat,K,nil(nat))) ).
% sorted_list_of_set_lessThan_Suc
tff(fact_3727_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( Xs = append(A,take(A,I,Xs),cons(A,aa(nat,A,nth(A,Xs),I),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).
% id_take_nth_drop
tff(fact_3728_atMost__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,set_ord_atMost(A,K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),K)) ) ) ).
% atMost_iff
tff(fact_3729_lessThan__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,set_ord_lessThan(A,K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),K)) ) ) ).
% lessThan_iff
tff(fact_3730_drop0,axiom,
! [A: $tType,X3: list(A)] : drop(A,zero_zero(nat),X3) = X3 ).
% drop0
tff(fact_3731_atMost__subset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atMost(A,X)),set_ord_atMost(A,Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% atMost_subset_iff
tff(fact_3732_lessThan__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_lessThan(A,X)),set_ord_lessThan(A,Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).
% lessThan_subset_iff
tff(fact_3733_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),cons(A,X,Xs)) = drop(A,N,Xs) ).
% drop_Suc_Cons
tff(fact_3734_lessThan__0,axiom,
set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).
% lessThan_0
tff(fact_3735_card__atMost,axiom,
! [U: nat] : aa(set(nat),nat,finite_card(nat),set_ord_atMost(nat,U)) = aa(nat,nat,suc,U) ).
% card_atMost
tff(fact_3736_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A,H2: A,H: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H2)),set_ord_atMost(A,H)))
<=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H2),H)) ) ) ) ).
% Icc_subset_Iic_iff
tff(fact_3737_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( nil(A) = drop(A,N,Xs) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).
% drop_eq_Nil2
tff(fact_3738_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( ( drop(A,N,Xs) = nil(A) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).
% drop_eq_Nil
tff(fact_3739_drop__all,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)),N))
=> ( drop(A,N,Xs) = nil(A) ) ) ).
% drop_all
tff(fact_3740_atMost__0,axiom,
set_ord_atMost(nat,zero_zero(nat)) = insert(nat,zero_zero(nat),bot_bot(set(nat))) ).
% atMost_0
tff(fact_3741_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),cons(A,X,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_3742_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_3743_sup__nat__def,axiom,
sup_sup(nat) = ord_max(nat) ).
% sup_nat_def
tff(fact_3744_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)),set_ord_atMost(A,A3)),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_3745_drop__0,axiom,
! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).
% drop_0
tff(fact_3746_lessThan__Suc__atMost,axiom,
! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = set_ord_atMost(nat,K) ).
% lessThan_Suc_atMost
tff(fact_3747_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)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).
% ivl_disj_un_one(4)
tff(fact_3748_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)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).
% ivl_disj_un_one(1)
tff(fact_3749_sup__int__def,axiom,
sup_sup(int) = ord_max(int) ).
% sup_int_def
tff(fact_3750_atMost__atLeast0,axiom,
! [N: nat] : set_ord_atMost(nat,N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).
% atMost_atLeast0
tff(fact_3751_map__Suc__upt,axiom,
! [M: nat,N: nat] : map(nat,nat,suc,upt(M,N)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N)) ).
% map_Suc_upt
tff(fact_3752_lessThan__Suc,axiom,
! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = insert(nat,K,set_ord_lessThan(nat,K)) ).
% lessThan_Suc
tff(fact_3753_lessThan__empty__iff,axiom,
! [N: nat] :
( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
<=> ( N = zero_zero(nat) ) ) ).
% lessThan_empty_iff
tff(fact_3754_atMost__Suc,axiom,
! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = insert(nat,aa(nat,nat,suc,K),set_ord_atMost(nat,K)) ).
% atMost_Suc
tff(fact_3755_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)),set_ord_lessThan(A,M)),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_3756_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set2(A,drop(A,M,Xs))),set2(A,drop(A,N,Xs)))) ) ).
% set_drop_subset_set_drop
tff(fact_3757_atLeast__upt,axiom,
! [N: nat] : set_ord_lessThan(nat,N) = set2(nat,upt(zero_zero(nat),N)) ).
% atLeast_upt
tff(fact_3758_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)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).
% ivl_disj_un_one(3)
tff(fact_3759_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list(A)] :
( ( ( N = zero_zero(nat) )
=> ( drop(A,N,cons(A,X,Xs)) = cons(A,X,Xs) ) )
& ( ( N != zero_zero(nat) )
=> ( drop(A,N,cons(A,X,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ) ).
% drop_Cons'
tff(fact_3760_append__eq__append__conv__if,axiom,
! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
( ( append(A,Xs_1,Xs_2) = append(A,Ys_1,Ys_2) )
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
=> ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
& ( Xs_2 = append(A,drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1),Ys_2) ) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
=> ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
& ( append(A,drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1),Xs_2) = Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
tff(fact_3761_atMost__upto,axiom,
! [N: nat] : set_ord_atMost(nat,N) = set2(nat,upt(zero_zero(nat),aa(nat,nat,suc,N))) ).
% atMost_upto
tff(fact_3762_max__strict__def,axiom,
fun_max_strict = max_ext(product_prod(nat,nat),fun_pair_less) ).
% max_strict_def
tff(fact_3763_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( cons(A,aa(nat,A,nth(A,Xs),I),drop(A,aa(nat,nat,suc,I),Xs)) = drop(A,I,Xs) ) ) ).
% Cons_nth_drop_Suc
tff(fact_3764_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)),set_ord_atMost(nat,N)),insert(nat,zero_zero(nat),bot_bot(set(nat)))) ).
% atLeast1_atMost_eq_remove0
tff(fact_3765_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( append(A,take(A,N,Xs),cons(A,hd(A,drop(A,N,Xs)),nil(A))) = take(A,aa(nat,nat,suc,N),Xs) ) ) ).
% take_hd_drop
tff(fact_3766_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list(A),A3: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( list_update(A,Xs,I,A3) = append(A,take(A,I,Xs),cons(A,A3,drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).
% upd_conv_take_nth_drop
tff(fact_3767_effect__makeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: array(A),N2: nat] :
( heap_Time_effect(array(A),array_make(A,N,F),H2,H,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,map(nat,A,F,upt(zero_zero(nat),N)),H2)) )
=> ( ( 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,map(nat,A,F,upt(zero_zero(nat),N)),H2)) )
=> ( ( array_get(A,H,R) = map(nat,A,F,upt(zero_zero(nat),N)) )
=> ( pp(aa(array(A),bool,array_present(A,H),R))
=> ( ~ pp(aa(array(A),bool,array_present(A,H2),R))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N2 ) ) ) ) ) ) ) ) ).
% effect_makeE
tff(fact_3768_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N = zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = cons(list(A),nil(A),nil(list(A))) ) )
& ( ( N != zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).
% n_lists_Nil
tff(fact_3769_hd__upt,axiom,
! [I: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
=> ( hd(nat,upt(I,J)) = I ) ) ).
% hd_upt
tff(fact_3770_list__update__beyond,axiom,
! [A: $tType,Xs: list(A),I: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
=> ( list_update(A,Xs,I,X) = Xs ) ) ).
% list_update_beyond
tff(fact_3771_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list(A),Y: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( take(A,N,list_update(A,Xs,M,Y)) = take(A,N,Xs) ) ) ).
% take_update_cancel
tff(fact_3772_drop__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( drop(A,M,list_update(A,Xs,N,X)) = drop(A,M,Xs) ) ) ).
% drop_update_cancel
tff(fact_3773_get__update__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,V: A,H2: heap_ext(product_unit)] : array_get(A,array_update(A,A3,I,V,H2),A3) = list_update(A,array_get(A,H2,A3),I,V) ) ).
% get_update_eq
tff(fact_3774_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,list_update(A,Xs,I,X)),I) = X ) ) ).
% nth_list_update_eq
tff(fact_3775_hd__take,axiom,
! [A: $tType,J: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J))
=> ( hd(A,take(A,J,Xs)) = hd(A,Xs) ) ) ).
% hd_take
tff(fact_3776_set__swap,axiom,
! [A: $tType,I: nat,Xs: list(A),J: nat] :
( 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),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( set2(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = set2(A,Xs) ) ) ) ).
% set_swap
tff(fact_3777_get__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit),Ys2: 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,H2)),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Ys2,H2))) = Xs ) ).
% get_alloc
tff(fact_3778_set,axiom,
! [A: $tType] :
( heap(A)
=> ! [Ls: list(A),H2: heap_ext(product_unit),New_ls: list(A)] : 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,H2)),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,H2))) = 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,H2)) ) ).
% set
tff(fact_3779_not__present__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [H2: heap_ext(product_unit),Xs: list(A)] : ~ pp(aa(array(A),bool,array_present(A,H2),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H2)))) ) ).
% not_present_alloc
tff(fact_3780_present__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: 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,H2))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Xs,H2)))) ) ).
% present_alloc
tff(fact_3781_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list(A),I: nat,Y: A] : list_update(A,cons(A,X,Xs),aa(nat,nat,suc,I),Y) = cons(A,X,list_update(A,Xs,I,Y)) ).
% list_update_code(3)
tff(fact_3782_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list(A),Y: A] : list_update(A,cons(A,X,Xs),zero_zero(nat),Y) = cons(A,Y,Xs) ).
% list_update_code(2)
tff(fact_3783_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member(A,X,set2(A,list_update(A,Xs,N,X)))) ) ).
% set_update_memI
tff(fact_3784_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list(A),Ys2: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( list_update(A,append(A,Xs,Ys2),I,X) = append(A,list_update(A,Xs,I,X),Ys2) ) ) ).
% list_update_append1
tff(fact_3785_length__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit),Ys2: 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,H2))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),array_alloc(A,Ys2,H2))) = aa(list(A),nat,size_size(list(A)),Xs) ) ).
% length_alloc
tff(fact_3786_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( list_update(A,Xs,I,X) = Xs )
<=> ( aa(nat,A,nth(A,Xs),I) = X ) ) ) ).
% list_update_same_conv
tff(fact_3787_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list(A),J: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( ( I = J )
=> ( aa(nat,A,nth(A,list_update(A,Xs,I,X)),J) = X ) )
& ( ( I != J )
=> ( aa(nat,A,nth(A,list_update(A,Xs,I,X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ) ) ).
% nth_list_update
tff(fact_3788_hd__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( hd(A,Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).
% hd_conv_nth
tff(fact_3789_drop__update__swap,axiom,
! [A: $tType,M: nat,N: nat,Xs: list(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( drop(A,M,list_update(A,Xs,N,X)) = list_update(A,drop(A,M,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M),X) ) ) ).
% drop_update_swap
tff(fact_3790_present__alloc__noteq,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [H2: heap_ext(product_unit),A3: array(A),Xs: list(B)] :
( pp(aa(array(A),bool,array_present(A,H2),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,H2))) ) ) ).
% present_alloc_noteq
tff(fact_3791_update__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,X: A,H2: heap_ext(product_unit)] : array_update(A,A3,I,X,H2) = array_set(A,A3,list_update(A,array_get(A,H2,A3),I,X),H2) ) ).
% update_def
tff(fact_3792_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list(A),Ys2: list(A),X: A] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( list_update(A,append(A,Xs,Ys2),N,X) = append(A,list_update(A,Xs,N,X),Ys2) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( list_update(A,append(A,Xs,Ys2),N,X) = append(A,Xs,list_update(A,Ys2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),X)) ) ) ) ).
% list_update_append
tff(fact_3793_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = cons(list(A),nil(A),nil(list(A))) ).
% n_lists.simps(1)
tff(fact_3794_effect__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit),F: fun(nat,A),N: nat,H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),product_Pair(array(A),heap_ext(product_unit),A3),H) = array_alloc(A,map(nat,A,F,upt(zero_zero(nat),N)),H2) )
=> heap_Time_effect(array(A),array_make(A,N,F),H2,H,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_makeI
tff(fact_3795_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( hd(A,drop(A,N,Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).
% hd_drop_conv_nth
tff(fact_3796_effect__of__listE,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: array(A),N2: nat] :
( heap_Time_effect(array(A),array_of_list(A,Xs),H2,H,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,H2)) )
=> ( ( 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,Xs,H2)) )
=> ( ( array_get(A,H,R) = Xs )
=> ( pp(aa(array(A),bool,array_present(A,H),R))
=> ( ~ pp(aa(array(A),bool,array_present(A,H2),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_3797_effect__of__listI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit),Xs: list(A),H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),product_Pair(array(A),heap_ext(product_unit),A3),H) = array_alloc(A,Xs,H2) )
=> heap_Time_effect(array(A),array_of_list(A,Xs),H2,H,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_3798_set__update__distinct,axiom,
! [A: $tType,Xs: list(A),N: nat,X: A] :
( distinct(A,Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( set2(A,list_update(A,Xs,N,X)) = insert(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set2(A,Xs)),insert(A,aa(nat,A,nth(A,Xs),N),bot_bot(set(A))))) ) ) ) ).
% set_update_distinct
tff(fact_3799_effect__newE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: array(A),N2: nat] :
( heap_Time_effect(array(A),array_new(A,N,X),H2,H,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,X),H2)) )
=> ( ( 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,replicate(A,N,X),H2)) )
=> ( ( array_get(A,H,R) = replicate(A,N,X) )
=> ( pp(aa(array(A),bool,array_present(A,H),R))
=> ( ~ pp(aa(array(A),bool,array_present(A,H2),R))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) != N2 ) ) ) ) ) ) ) ) ).
% effect_newE
tff(fact_3800_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y: A] :
( ( replicate(A,M,X) = replicate(A,N,Y) )
<=> ( ( M = N )
& ( ( M != zero_zero(nat) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
tff(fact_3801_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate(A,N,X) = nil(A) )
<=> ( N = zero_zero(nat) ) ) ).
% replicate_empty
tff(fact_3802_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( nil(A) = replicate(A,N,X) )
<=> ( N = zero_zero(nat) ) ) ).
% empty_replicate
tff(fact_3803_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
( ! [X5: A] :
( pp(member(A,X5,set2(A,replicate(A,N,A3))))
=> pp(aa(A,bool,P,X5)) )
<=> ( pp(aa(A,bool,P,A3))
| ( N = zero_zero(nat) ) ) ) ).
% Ball_set_replicate
tff(fact_3804_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
( ? [X5: A] :
( pp(member(A,X5,set2(A,replicate(A,N,A3))))
& pp(aa(A,bool,P,X5)) )
<=> ( pp(aa(A,bool,P,A3))
& ( N != zero_zero(nat) ) ) ) ).
% Bex_set_replicate
tff(fact_3805_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( pp(member(A,X,set2(A,replicate(A,N,Y))))
<=> ( ( X = Y )
& ( N != zero_zero(nat) ) ) ) ).
% in_set_replicate
tff(fact_3806_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),N))
=> ( aa(nat,A,nth(A,replicate(A,N,X)),I) = X ) ) ).
% nth_replicate
tff(fact_3807_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( hd(A,replicate(A,N,X)) = X ) ) ).
% hd_replicate
tff(fact_3808_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( set2(A,replicate(A,N,X)) = insert(A,X,bot_bot(set(A))) ) ) ).
% set_replicate
tff(fact_3809_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list(A),J: nat] :
( 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),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( distinct(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I)))
<=> distinct(A,Xs) ) ) ) ).
% distinct_swap
tff(fact_3810_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = cons(A,X,replicate(A,N,X)) ).
% replicate_Suc
tff(fact_3811_replicate__0,axiom,
! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).
% replicate_0
tff(fact_3812_distinct__conv__nth,axiom,
! [A: $tType,Xs: list(A)] :
( distinct(A,Xs)
<=> ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
=> ! [J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( I5 != J3 )
=> ( aa(nat,A,nth(A,Xs),I5) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).
% distinct_conv_nth
tff(fact_3813_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list(A),I: nat,J: nat] :
( distinct(A,Xs)
=> ( 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),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J) )
<=> ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
tff(fact_3814_effect__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit),N: nat,X: A,H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),product_Pair(array(A),heap_ext(product_unit),A3),H) = array_alloc(A,replicate(A,N,X),H2) )
=> heap_Time_effect(array(A),array_new(A,N,X),H2,H,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).
% effect_newI
tff(fact_3815_distinct__Ex1,axiom,
! [A: $tType,Xs: list(A),X: A] :
( distinct(A,Xs)
=> ( pp(member(A,X,set2(A,Xs)))
=> ? [X2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),X2) = X )
& ! [Y3: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y3),aa(list(A),nat,size_size(list(A)),Xs)))
& ( aa(nat,A,nth(A,Xs),Y3) = X ) )
=> ( Y3 = X2 ) ) ) ) ) ).
% distinct_Ex1
tff(fact_3816_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] : set2(A,replicate(A,aa(nat,nat,suc,N),X)) = insert(A,X,bot_bot(set(A))) ).
% set_replicate_Suc
tff(fact_3817_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N = zero_zero(nat) )
=> ( set2(A,replicate(A,N,X)) = bot_bot(set(A)) ) )
& ( ( N != zero_zero(nat) )
=> ( set2(A,replicate(A,N,X)) = insert(A,X,bot_bot(set(A))) ) ) ) ).
% set_replicate_conv_if
tff(fact_3818_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
( ( cons(A,X,Xs) = replicate(A,N,Y) )
<=> ( ( X = 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)),X) ) ) ) ).
% Cons_replicate_eq
tff(fact_3819_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)),set_ord_lessThan(nat,N)),insert(nat,zero_zero(nat),bot_bot(set(nat)))) ).
% atLeast1_lessThan_eq_remove0
tff(fact_3820_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys2: 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)),product_Pair(list(A),list(A),Xs),Ys2),listrel1(A,R)))
<=> ? [Y5: A,N5: nat] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),N5)),Y5),R))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
& ( Ys2 = list_update(A,Xs,N5,Y5) ) ) ) ).
% listrel1_iff_update
tff(fact_3821_Gcd__remove0__nat,axiom,
! [M7: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),M7))
=> ( gcd_Gcd(nat,M7) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M7),insert(nat,zero_zero(nat),bot_bot(set(nat))))) ) ) ).
% Gcd_remove0_nat
tff(fact_3822_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs: list(A)] :
( ( aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( Xs != nil(A) )
& ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),hd(A,Xs)) ) ) ) ).
% remdups_adj_singleton_iff
tff(fact_3823_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_3824_infinite__Icc__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(set(A),bool,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_3825_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_3826_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_3827_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_3828_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_3829_infinite__Ico__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(set(A),bool,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_3830_card_Oinfinite,axiom,
! [A: $tType,A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) ) ) ).
% card.infinite
tff(fact_3831_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_3832_infinite__Ioc__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(set(A),bool,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_3833_infinite__Ioo__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A] :
( ~ pp(aa(set(A),bool,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_3834_card__0__eq,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
<=> ( A5 = bot_bot(set(A)) ) ) ) ).
% card_0_eq
tff(fact_3835_card__insert__disjoint,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ~ pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),insert(A,X,A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5)) ) ) ) ).
% card_insert_disjoint
tff(fact_3836_atLeastLessThan__singleton,axiom,
! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = insert(nat,M,bot_bot(set(nat))) ).
% atLeastLessThan_singleton
tff(fact_3837_finite__psubset__induct,axiom,
! [A: $tType,A5: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ! [A7: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A7))
=> ( ! [B10: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B10),A7))
=> pp(aa(set(A),bool,P,B10)) )
=> pp(aa(set(A),bool,P,A7)) ) )
=> pp(aa(set(A),bool,P,A5)) ) ) ).
% finite_psubset_induct
tff(fact_3838_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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),D2))
=> ( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
<=> ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
tff(fact_3839_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
=> ( 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),D2))
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
tff(fact_3840_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( set_or7035219750837199246ssThan(A,A3,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
=> ( 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),D2))
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
tff(fact_3841_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))
=> ~ pp(aa(set(A),bool,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B2))) ) ) ).
% infinite_Ico
tff(fact_3842_finite__nat__set__iff__bounded,axiom,
! [N6: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),N6))
<=> ? [M6: nat] :
! [X5: nat] :
( pp(member(nat,X5,N6))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X5),M6)) ) ) ).
% finite_nat_set_iff_bounded
tff(fact_3843_bounded__nat__set__is__finite,axiom,
! [N6: set(nat),N: nat] :
( ! [X2: nat] :
( pp(member(nat,X2,N6))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),N)) )
=> pp(aa(set(nat),bool,finite_finite2(nat),N6)) ) ).
% bounded_nat_set_is_finite
tff(fact_3844_finite__nat__set__iff__bounded__le,axiom,
! [N6: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),N6))
<=> ? [M6: nat] :
! [X5: nat] :
( pp(member(nat,X5,N6))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X5),M6)) ) ) ).
% finite_nat_set_iff_bounded_le
tff(fact_3845_finite__has__minimal2,axiom,
! [A: $tType] :
( order(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> ? [X2: A] :
( pp(member(A,X2,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3))
& ! [Xa: A] :
( pp(member(A,Xa,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X2))
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
tff(fact_3846_finite__has__maximal2,axiom,
! [A: $tType] :
( order(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> ? [X2: A] :
( pp(member(A,X2,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X2))
& ! [Xa: A] :
( pp(member(A,Xa,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa))
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
tff(fact_3847_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set(nat),N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
=> pp(aa(set(nat),bool,finite_finite2(nat),N6)) ) ).
% subset_eq_atLeast0_lessThan_finite
tff(fact_3848_finite__set__decode,axiom,
! [N: nat] : pp(aa(set(nat),bool,finite_finite2(nat),nat_set_decode(N))) ).
% finite_set_decode
tff(fact_3849_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
=> ( 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),D2)) ) ) ) ) ).
% atLeastLessThan_subset_iff
tff(fact_3850_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_3851_ex__nat__less__eq,axiom,
! [N: nat,P: fun(nat,bool)] :
( ? [M6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M6),N))
& pp(aa(nat,bool,P,M6)) )
<=> ? [X5: nat] :
( pp(member(nat,X5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
& pp(aa(nat,bool,P,X5)) ) ) ).
% ex_nat_less_eq
tff(fact_3852_all__nat__less__eq,axiom,
! [N: nat,P: fun(nat,bool)] :
( ! [M6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M6),N))
=> pp(aa(nat,bool,P,M6)) )
<=> ! [X5: nat] :
( pp(member(nat,X5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,P,X5)) ) ) ).
% all_nat_less_eq
tff(fact_3853_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_3854_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_3855_finite__has__minimal,axiom,
! [A: $tType] :
( order(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ? [X2: A] :
( pp(member(A,X2,A5))
& ! [Xa: A] :
( pp(member(A,Xa,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa),X2))
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
tff(fact_3856_finite__has__maximal,axiom,
! [A: $tType] :
( order(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ? [X2: A] :
( pp(member(A,X2,A5))
& ! [Xa: A] :
( pp(member(A,Xa,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa))
=> ( X2 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
tff(fact_3857_infinite__growing,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X6: set(A)] :
( ( X6 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> ? [Xa: A] :
( pp(member(A,Xa,X6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Xa)) ) )
=> ~ pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).
% infinite_growing
tff(fact_3858_ex__min__if__finite,axiom,
! [A: $tType] :
( order(A)
=> ! [S3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> ? [X2: A] :
( pp(member(A,X2,S3))
& ~ ? [Xa: A] :
( pp(member(A,Xa,S3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa),X2)) ) ) ) ) ) ).
% ex_min_if_finite
tff(fact_3859_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))
=> ~ pp(aa(set(A),bool,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B2))) ) ) ).
% infinite_Icc
tff(fact_3860_lessThan__atLeast0,axiom,
! [N: nat] : set_ord_lessThan(nat,N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).
% lessThan_atLeast0
tff(fact_3861_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set(A),A5: set(B),R: fun(B,fun(A,bool))] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ! [A6: B] :
( pp(member(B,A6,A5))
=> ? [B11: A] :
( pp(member(A,B11,B5))
& pp(aa(A,bool,aa(B,fun(A,bool),R,A6),B11)) ) )
=> ( ! [A12: B,A23: B,B4: A] :
( pp(member(B,A12,A5))
=> ( pp(member(B,A23,A5))
=> ( pp(member(A,B4,B5))
=> ( pp(aa(A,bool,aa(B,fun(A,bool),R,A12),B4))
=> ( pp(aa(A,bool,aa(B,fun(A,bool),R,A23),B4))
=> ( A12 = A23 ) ) ) ) ) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),aa(set(A),nat,finite_card(A),B5))) ) ) ) ).
% card_le_if_inj_on_rel
tff(fact_3862_remdups__adj__length,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)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% remdups_adj_length
tff(fact_3863_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))
=> ~ pp(aa(set(A),bool,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B2))) ) ) ).
% infinite_Ioc
tff(fact_3864_atLeastLessThan0,axiom,
! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).
% atLeastLessThan0
tff(fact_3865_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))
=> ~ pp(aa(set(A),bool,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B2))) ) ) ).
% infinite_Ioo
tff(fact_3866_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_3867_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_3868_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)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).
% ivl_disj_un_one(2)
tff(fact_3869_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [S3: set(B),P: fun(set(B),bool),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( pp(aa(set(B),bool,P,bot_bot(set(B))))
=> ( ! [X2: B,S4: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),S4))
=> ( ! [Y3: B] :
( pp(member(B,Y3,S4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,Y3)),aa(B,A,F,X2))) )
=> ( pp(aa(set(B),bool,P,S4))
=> pp(aa(set(B),bool,P,insert(B,X2,S4))) ) ) )
=> pp(aa(set(B),bool,P,S3)) ) ) ) ) ).
% finite_ranking_induct
tff(fact_3870_finite__linorder__max__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [B4: A,A7: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A7))
=> ( ! [X3: A] :
( pp(member(A,X3,A7))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),B4)) )
=> ( pp(aa(set(A),bool,P,A7))
=> pp(aa(set(A),bool,P,insert(A,B4,A7))) ) ) )
=> pp(aa(set(A),bool,P,A5)) ) ) ) ) ).
% finite_linorder_max_induct
tff(fact_3871_finite__linorder__min__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [B4: A,A7: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A7))
=> ( ! [X3: A] :
( pp(member(A,X3,A7))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B4),X3)) )
=> ( pp(aa(set(A),bool,P,A7))
=> pp(aa(set(A),bool,P,insert(A,B4,A7))) ) ) )
=> pp(aa(set(A),bool,P,A5)) ) ) ) ) ).
% finite_linorder_min_induct
tff(fact_3872_card__eq__0__iff,axiom,
! [A: $tType,A5: set(A)] :
( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
<=> ( ( A5 = bot_bot(set(A)) )
| ~ pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).
% card_eq_0_iff
tff(fact_3873_atLeast0__lessThan__Suc,axiom,
! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,N,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).
% atLeast0_lessThan_Suc
tff(fact_3874_card__Suc__eq__finite,axiom,
! [A: $tType,A5: set(A),K: nat] :
( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
<=> ? [B8: A,B9: set(A)] :
( ( A5 = insert(A,B8,B9) )
& ~ pp(member(A,B8,B9))
& ( aa(set(A),nat,finite_card(A),B9) = K )
& pp(aa(set(A),bool,finite_finite2(A),B9)) ) ) ).
% card_Suc_eq_finite
tff(fact_3875_card__insert__if,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),insert(A,X,A5)) = aa(set(A),nat,finite_card(A),A5) ) )
& ( ~ pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),insert(A,X,A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5)) ) ) ) ) ).
% card_insert_if
tff(fact_3876_card__ge__0__finite,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5)))
=> pp(aa(set(A),bool,finite_finite2(A),A5)) ) ).
% card_ge_0_finite
tff(fact_3877_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S3: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),S3)))
=> ~ ! [T3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T3),S3))
=> ( ( aa(set(A),nat,finite_card(A),T3) = N )
=> ~ pp(aa(set(A),bool,finite_finite2(A),T3)) ) ) ) ).
% obtain_subset_with_card_n
tff(fact_3878_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F3: set(A),C6: nat] :
( ! [G2: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G2),F3))
=> ( pp(aa(set(A),bool,finite_finite2(A),G2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G2)),C6)) ) )
=> ( pp(aa(set(A),bool,finite_finite2(A),F3))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F3)),C6)) ) ) ).
% finite_if_finite_subsets_card_bdd
tff(fact_3879_card__seteq,axiom,
! [A: $tType,B5: set(A),A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B5)),aa(set(A),nat,finite_card(A),A5)))
=> ( A5 = B5 ) ) ) ) ).
% card_seteq
tff(fact_3880_card__mono,axiom,
! [A: $tType,B5: set(A),A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))) ) ) ).
% card_mono
tff(fact_3881_card__less__sym__Diff,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))) ) ) ) ).
% card_less_sym_Diff
tff(fact_3882_card__le__sym__Diff,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))) ) ) ) ).
% card_le_sym_Diff
tff(fact_3883_psubset__card__mono,axiom,
! [A: $tType,B5: set(A),A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))) ) ) ).
% psubset_card_mono
tff(fact_3884_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set(nat),N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
=> pp(aa(set(nat),bool,finite_finite2(nat),N6)) ) ).
% subset_eq_atLeast0_atMost_finite
tff(fact_3885_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_3886_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_3887_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_3888_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_3889_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: 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,D2)))
<=> ( 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),D2)) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_3890_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_3891_card__gt__0__iff,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5)))
<=> ( ( A5 != bot_bot(set(A)) )
& pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ).
% card_gt_0_iff
tff(fact_3892_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(nat,nat,suc,zero_zero(nat))))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> ! [Xa2: A] :
( pp(member(A,Xa2,A5))
=> ( X5 = Xa2 ) ) ) ) ) ).
% card_le_Suc0_iff_eq
tff(fact_3893_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A5: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),A5)))
<=> ? [A8: A,B9: set(A)] :
( ( A5 = insert(A,A8,B9) )
& ~ pp(member(A,A8,B9))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B9)))
& pp(aa(set(A),bool,finite_finite2(A),B9)) ) ) ).
% card_le_Suc_iff
tff(fact_3894_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_3895_finite__induct__select,axiom,
! [A: $tType,S3: set(A),P: fun(set(A),bool)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( pp(aa(set(A),bool,P,bot_bot(set(A))))
=> ( ! [T3: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T3),S3))
=> ( pp(aa(set(A),bool,P,T3))
=> ? [X3: A] :
( pp(member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S3),T3)))
& pp(aa(set(A),bool,P,insert(A,X3,T3))) ) ) )
=> pp(aa(set(A),bool,P,S3)) ) ) ) ).
% finite_induct_select
tff(fact_3896_card__psubset,axiom,
! [A: $tType,B5: set(A),A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5)) ) ) ) ).
% card_psubset
tff(fact_3897_diff__card__le__card__Diff,axiom,
! [A: $tType,B5: set(A),A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)))) ) ).
% diff_card_le_card_Diff
tff(fact_3898_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))))
=> ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I)) ) ) ).
% remdups_adj_adjacent
tff(fact_3899_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N = zero_zero(nat) )
=> ( remdups_adj(A,replicate(A,N,X)) = nil(A) ) )
& ( ( N != zero_zero(nat) )
=> ( remdups_adj(A,replicate(A,N,X)) = cons(A,X,nil(A)) ) ) ) ).
% remdups_adj_replicate
tff(fact_3900_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)) = insert(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_3901_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set(nat),N: nat] :
( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N6)),N)) ) ).
% subset_eq_atLeast0_lessThan_card
tff(fact_3902_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)),insert(A,U,bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(6)
tff(fact_3903_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_3904_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)),insert(A,L,bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).
% ivl_disj_un_singleton(3)
tff(fact_3905_card_Oremove,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ).
% card.remove
tff(fact_3906_card_Oinsert__remove,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),nat,finite_card(A),insert(A,X,A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ).
% card.insert_remove
tff(fact_3907_card__Suc__Diff1,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).
% card_Suc_Diff1
tff(fact_3908_card__Diff1__less,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))) ) ) ).
% card_Diff1_less
tff(fact_3909_card__Diff2__less,axiom,
! [A: $tType,A5: set(A),X: A,Y: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( pp(member(A,Y,A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(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)),A5),insert(A,X,bot_bot(set(A))))),insert(A,Y,bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))) ) ) ) ).
% card_Diff2_less
tff(fact_3910_card__Diff1__less__iff,axiom,
! [A: $tType,A5: set(A),X: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)))
<=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
& pp(member(A,X,A5)) ) ) ).
% card_Diff1_less_iff
tff(fact_3911_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(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)),remdups_adj(A,Xs)))) ) ).
% remdups_adj_length_ge1
tff(fact_3912_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)) = insert(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_3913_set__encode__insert,axiom,
! [A5: set(nat),N: nat] :
( pp(aa(set(nat),bool,finite_finite2(nat),A5))
=> ( ~ pp(member(nat,N,A5))
=> ( aa(set(nat),nat,nat_set_encode,insert(nat,N,A5)) = 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),bit0(one2))),N)),aa(set(nat),nat,nat_set_encode,A5)) ) ) ) ).
% set_encode_insert
tff(fact_3914_even__set__encode__iff,axiom,
! [A5: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),A5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A5)))
<=> ~ pp(member(nat,zero_zero(nat),A5)) ) ) ).
% even_set_encode_iff
tff(fact_3915_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S3: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> ~ ? [X3: A] :
( pp(member(A,X3,S3))
& pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X3)),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S3)))) ) ) ) ) ).
% arg_min_if_finite(2)
tff(fact_3916_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S3: set(A),Y: A,F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> ( pp(member(A,Y,S3))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,lattic7623131987881927897min_on(A,B,F,S3))),aa(A,B,F,Y))) ) ) ) ) ).
% arg_min_least
tff(fact_3917_set__decode__inverse,axiom,
! [N: nat] : aa(set(nat),nat,nat_set_encode,nat_set_decode(N)) = N ).
% set_decode_inverse
tff(fact_3918_set__encode__empty,axiom,
aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).
% set_encode_empty
tff(fact_3919_set__encode__inverse,axiom,
! [A5: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),A5))
=> ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A5)) = A5 ) ) ).
% set_encode_inverse
tff(fact_3920_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : pp(aa(set(int),bool,finite_finite2(int),set_or7035219750837199246ssThan(int,zero_zero(int),U))) ).
% finite_atLeastZeroLessThan_int
tff(fact_3921_finite__maxlen,axiom,
! [A: $tType,M7: set(list(A))] :
( pp(aa(set(list(A)),bool,finite_finite2(list(A)),M7))
=> ? [N3: nat] :
! [X3: list(A)] :
( pp(member(list(A),X3,M7))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X3)),N3)) ) ) ).
% finite_maxlen
tff(fact_3922_set__encode__eq,axiom,
! [A5: set(nat),B5: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),A5))
=> ( pp(aa(set(nat),bool,finite_finite2(nat),B5))
=> ( ( aa(set(nat),nat,nat_set_encode,A5) = aa(set(nat),nat,nat_set_encode,B5) )
<=> ( A5 = B5 ) ) ) ) ).
% set_encode_eq
tff(fact_3923_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_3924_card__atLeastZeroLessThan__int,axiom,
! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = aa(int,nat,nat2,U) ).
% card_atLeastZeroLessThan_int
tff(fact_3925_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_3926_set__encode__inf,axiom,
! [A5: set(nat)] :
( ~ pp(aa(set(nat),bool,finite_finite2(nat),A5))
=> ( aa(set(nat),nat,nat_set_encode,A5) = zero_zero(nat) ) ) ).
% set_encode_inf
tff(fact_3927_atLeastLessThan__upto,axiom,
! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = set2(int,upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))) ).
% atLeastLessThan_upto
tff(fact_3928_wmax__emptyI,axiom,
! [X6: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),X6))
=> 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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),bot_bot(set(product_prod(nat,nat)))),X6),fun_max_weak)) ) ).
% wmax_emptyI
tff(fact_3929_smax__emptyI,axiom,
! [Y6: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),Y6))
=> ( ( Y6 != 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))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat)),bot_bot(set(product_prod(nat,nat)))),Y6),fun_max_strict)) ) ) ).
% smax_emptyI
tff(fact_3930_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S3: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> pp(member(A,lattic7623131987881927897min_on(A,B,F,S3),S3)) ) ) ) ).
% arg_min_if_finite(1)
tff(fact_3931_nth__image,axiom,
! [A: $tType,L: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( image(nat,A,nth(A,Xs),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = set2(A,take(A,L,Xs)) ) ) ).
% nth_image
tff(fact_3932_fact__split,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),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),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K))) ) ) ) ).
% fact_split
tff(fact_3933_sum__list__update,axiom,
! [A: $tType] :
( ordere1170586879665033532d_diff(A)
=> ! [K: nat,Xs: list(A),X: 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,X)) = 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)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).
% sum_list_update
tff(fact_3934_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list(A),I: nat,J: nat] :
( distinct(A,Vs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set2(A,take(A,I,Vs))),set2(A,drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).
% set_take_disj_set_drop_if_distinct
tff(fact_3935_inf_Oidem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),A3) = A3 ) ).
% inf.idem
tff(fact_3936_inf__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),X) = X ) ).
% inf_idem
tff(fact_3937_inf_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).
% inf.left_idem
tff(fact_3938_inf__left__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).
% inf_left_idem
tff(fact_3939_inf_Oright__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ).
% inf.right_idem
tff(fact_3940_inf__right__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).
% inf_right_idem
tff(fact_3941_inf__apply,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F: fun(A,B),G: fun(A,B),X: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F),G),X) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ).
% inf_apply
tff(fact_3942_le__inf__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2)) ) ) ) ).
% le_inf_iff
tff(fact_3943_inf_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
<=> ( 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),C2)) ) ) ) ).
% inf.bounded_iff
tff(fact_3944_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).
% boolean_algebra.conj_zero_right
tff(fact_3945_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).
% boolean_algebra.conj_zero_left
tff(fact_3946_inf__bot__right,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),bot_bot(A)) = bot_bot(A) ) ).
% inf_bot_right
tff(fact_3947_inf__bot__left,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),X) = bot_bot(A) ) ).
% inf_bot_left
tff(fact_3948_inf__sup__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = X ) ).
% inf_sup_absorb
tff(fact_3949_sup__inf__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = X ) ).
% sup_inf_absorb
tff(fact_3950_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] : image(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_3951_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] : image(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_3952_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult(B)
& ring_1(B) )
=> ! [A5: set(A),F: fun(A,B)] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(member(B,aa(A,B,F,X2),ring_1_Ints(B))) )
=> pp(member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A5),ring_1_Ints(B))) ) ) ).
% Ints_prod
tff(fact_3953_image__add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(A)] : image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),S3) = S3 ) ).
% image_add_0
tff(fact_3954_inf__compl__bot__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = bot_bot(A) ) ).
% inf_compl_bot_left1
tff(fact_3955_inf__compl__bot__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),Y)) = bot_bot(A) ) ).
% inf_compl_bot_left2
tff(fact_3956_inf__compl__bot__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),X))) = bot_bot(A) ) ).
% inf_compl_bot_right
tff(fact_3957_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),X) = bot_bot(A) ) ).
% boolean_algebra.conj_cancel_left
tff(fact_3958_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),X)) = bot_bot(A) ) ).
% boolean_algebra.conj_cancel_right
tff(fact_3959_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% boolean_algebra.de_Morgan_conj
tff(fact_3960_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).
% boolean_algebra.de_Morgan_disj
tff(fact_3961_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A] : image(A,A,uminus_uminus(A),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_atLeastAtMost
tff(fact_3962_sum__list_ONil,axiom,
! [A: $tType] :
( monoid_add(A)
=> ( aa(list(A),A,groups8242544230860333062m_list(A),nil(A)) = zero_zero(A) ) ) ).
% sum_list.Nil
tff(fact_3963_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [Ns: list(A)] :
( ( aa(list(A),A,groups8242544230860333062m_list(A),Ns) = zero_zero(A) )
<=> ! [X5: A] :
( pp(member(A,X5,set2(A,Ns)))
=> ( X5 = zero_zero(A) ) ) ) ) ).
% sum_list_eq_0_iff
tff(fact_3964_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A] : image(A,A,uminus_uminus(A),set_or5935395276787703475ssThan(A,X,Y)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_greaterThanLessThan
tff(fact_3965_Compl__disjoint,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = bot_bot(set(A)) ).
% Compl_disjoint
tff(fact_3966_Compl__disjoint2,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),A5) = bot_bot(set(A)) ).
% Compl_disjoint2
tff(fact_3967_Diff__Compl,axiom,
! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ).
% Diff_Compl
tff(fact_3968_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A] : image(A,A,uminus_uminus(A),set_or3652927894154168847AtMost(A,X,Y)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_greaterThanAtMost
tff(fact_3969_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A,Y: A] : image(A,A,uminus_uminus(A),set_or7035219750837199246ssThan(A,X,Y)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_atLeastLessThan
tff(fact_3970_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),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),set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% prod.atMost_Suc
tff(fact_3971_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),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),set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).
% prod.lessThan_Suc
tff(fact_3972_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
=> ( image(A,A,aa(A,fun(A,A),times_times(A),D2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).
% image_mult_atLeastAtMost
tff(fact_3973_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_3974_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_3975_distrib__imp1,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y4),Z3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y4)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Z3))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ) ) ).
% distrib_imp1
tff(fact_3976_distrib__imp2,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] :
( ! [X2: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y4),Z3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y4)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Z3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ) ) ).
% distrib_imp2
tff(fact_3977_inf__sup__distrib1,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% inf_sup_distrib1
tff(fact_3978_inf__sup__distrib2,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).
% inf_sup_distrib2
tff(fact_3979_sup__inf__distrib1,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% sup_inf_distrib1
tff(fact_3980_sup__inf__distrib2,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).
% sup_inf_distrib2
tff(fact_3981_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% boolean_algebra.conj_disj_distrib
tff(fact_3982_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)) ) ).
% boolean_algebra.disj_conj_distrib
tff(fact_3983_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z2),X)) ) ).
% boolean_algebra.conj_disj_distrib2
tff(fact_3984_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [Y: A,Z2: A,X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)),X) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z2),X)) ) ).
% boolean_algebra.disj_conj_distrib2
tff(fact_3985_inf__min,axiom,
! [A: $tType] :
( ( semilattice_inf(A)
& linorder(A) )
=> ( inf_inf(A) = ord_min(A) ) ) ).
% inf_min
tff(fact_3986_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) ) ).
% inf_sup_aci(4)
tff(fact_3987_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% inf_sup_aci(3)
tff(fact_3988_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).
% inf_sup_aci(2)
tff(fact_3989_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X) ) ).
% inf_sup_aci(1)
tff(fact_3990_inf_Oassoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).
% inf.assoc
tff(fact_3991_inf__assoc,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) ) ).
% inf_assoc
tff(fact_3992_inf_Ocommute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),A3) ) ).
% inf.commute
tff(fact_3993_inf__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),X) ) ).
% inf_commute
tff(fact_3994_inf_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)) ) ).
% inf.left_commute
tff(fact_3995_inf__left__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2)) ) ).
% inf_left_commute
tff(fact_3996_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( semilattice_inf(B)
=> ! [F: fun(A,B),G: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F),G),X3) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,X3)),aa(A,B,G,X3)) ) ).
% inf_fun_def
tff(fact_3997_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A3) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.inf1
tff(fact_3998_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B5: A,K: A,B2: A,A3: A] :
( ( B5 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B2) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B5) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)) ) ) ) ).
% boolean_algebra_cancel.inf2
tff(fact_3999_less__infI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,X: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% less_infI1
tff(fact_4000_less__infI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,X: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% less_infI2
tff(fact_4001_inf_Oabsorb3,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb3
tff(fact_4002_inf_Oabsorb4,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb4
tff(fact_4003_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% inf.strict_boundedE
tff(fact_4004_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
tff(fact_4005_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).
% inf.strict_coboundedI1
tff(fact_4006_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C2: A,A3: A] :
( 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),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).
% inf.strict_coboundedI2
tff(fact_4007_inf_OcoboundedI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,C2: A,A3: A] :
( 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),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).
% inf.coboundedI2
tff(fact_4008_inf_OcoboundedI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),C2)) ) ) ).
% inf.coboundedI1
tff(fact_4009_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb_iff2
tff(fact_4010_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb_iff1
tff(fact_4011_inf_Ocobounded2,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2)),B2)) ) ).
% inf.cobounded2
tff(fact_4012_inf_Ocobounded1,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2)),A3)) ) ).
% inf.cobounded1
tff(fact_4013_inf_Oorder__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).
% inf.order_iff
tff(fact_4014_inf__greatest,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))) ) ) ) ).
% inf_greatest
tff(fact_4015_inf_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(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),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))) ) ) ) ).
% inf.boundedI
tff(fact_4016_inf_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
=> ~ ( 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),C2)) ) ) ) ).
% inf.boundedE
tff(fact_4017_inf__absorb2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).
% inf_absorb2
tff(fact_4018_inf__absorb1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).
% inf_absorb1
tff(fact_4019_inf_Oabsorb2,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2) = B2 ) ) ) ).
% inf.absorb2
tff(fact_4020_inf_Oabsorb1,axiom,
! [A: $tType] :
( semilattice_inf(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),inf_inf(A),A3),B2) = A3 ) ) ) ).
% inf.absorb1
tff(fact_4021_le__iff__inf,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
<=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).
% le_iff_inf
tff(fact_4022_inf__unique,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [F: fun(A,fun(A,A)),X: A,Y: A] :
( ! [X2: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),Y4)),X2))
=> ( ! [X2: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),Y4)),Y4))
=> ( ! [X2: A,Y4: A,Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(A,A,aa(A,fun(A,A),F,Y4),Z3))) ) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F,X),Y) ) ) ) ) ) ).
% inf_unique
tff(fact_4023_inf_OorderI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ).
% inf.orderI
tff(fact_4024_inf_OorderE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) ) ) ) ).
% inf.orderE
tff(fact_4025_le__infI2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [B2: A,X: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% le_infI2
tff(fact_4026_le__infI1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,X: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),X)) ) ) ).
% le_infI1
tff(fact_4027_inf__mono,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A3: A,C2: A,B2: A,D2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2))) ) ) ) ).
% inf_mono
tff(fact_4028_le__infI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2))) ) ) ) ).
% le_infI
tff(fact_4029_le__infE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A3))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).
% le_infE
tff(fact_4030_inf__le2,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).
% inf_le2
tff(fact_4031_inf__le1,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).
% inf_le1
tff(fact_4032_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).
% inf_sup_ord(1)
tff(fact_4033_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).
% inf_sup_ord(2)
tff(fact_4034_distrib__sup__le,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z2)))) ) ).
% distrib_sup_le
tff(fact_4035_distrib__inf__le,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X: A,Y: A,Z2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)))) ) ).
% distrib_inf_le
tff(fact_4036_diff__eq,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y)) ) ).
% diff_eq
tff(fact_4037_inf__cancel__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),B2)) = bot_bot(A) ) ).
% inf_cancel_left1
tff(fact_4038_inf__cancel__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),B2)) = bot_bot(A) ) ).
% inf_cancel_left2
tff(fact_4039_zero__notin__Suc__image,axiom,
! [A5: set(nat)] : ~ pp(member(nat,zero_zero(nat),image(nat,nat,suc,A5))) ).
% zero_notin_Suc_image
tff(fact_4040_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] : image(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_4041_member__le__sum__list,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [X: A,Xs: list(A)] :
( pp(member(A,X,set2(A,Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).
% member_le_sum_list
tff(fact_4042_Compl__Un,axiom,
! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).
% Compl_Un
tff(fact_4043_Compl__Int,axiom,
! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).
% Compl_Int
tff(fact_4044_Diff__eq,axiom,
! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).
% Diff_eq
tff(fact_4045_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& comm_monoid_mult(A) )
=> ! [A3: B,C2: B,B2: B,D2: B,G: fun(B,A),H2: fun(B,A)] :
( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X2))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),D2))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) ) )
=> ( 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),H2),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).
% prod.ivl_cong
tff(fact_4046_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_4047_inf__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).
% inf_shunt
tff(fact_4048_shunt1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z2))) ) ) ).
% shunt1
tff(fact_4049_shunt2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2))) ) ) ).
% shunt2
tff(fact_4050_sup__neg__inf,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [P3: A,Q4: A,R: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q4),R)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q4))),R)) ) ) ).
% sup_neg_inf
tff(fact_4051_card__image__le,axiom,
! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),image(A,B,F,A5))),aa(set(A),nat,finite_card(A),A5))) ) ).
% card_image_le
tff(fact_4052_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [Xs: list(A)] :
( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> 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)),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).
% Groups_List.sum_list_nonneg
tff(fact_4053_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [Xs: list(A)] :
( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X2)) )
=> ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
<=> ! [X5: A] :
( pp(member(A,X5,set2(A,Xs)))
=> ( X5 = zero_zero(A) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
tff(fact_4054_sum__list__nonpos,axiom,
! [A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [Xs: list(A)] :
( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> 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),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A))) ) ) ).
% sum_list_nonpos
tff(fact_4055_Ioc__disjoint,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = 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),D2),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),D2),A3)) ) ) ) ).
% Ioc_disjoint
tff(fact_4056_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5))) ) ).
% disjoint_eq_subset_Compl
tff(fact_4057_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] : image(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_4058_sum__list__abs,axiom,
! [A: $tType] :
( ordere166539214618696060dd_abs(A)
=> ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(A,A,abs_abs(A),Xs)))) ) ).
% sum_list_abs
tff(fact_4059_sum__list__replicate,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat,C2: A] : aa(list(A),A,groups8242544230860333062m_list(A),replicate(A,N,C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),C2) ) ).
% sum_list_replicate
tff(fact_4060_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_4061_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_4062_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_4063_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_4064_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_4065_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_4066_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_4067_surj__card__le,axiom,
! [B: $tType,A: $tType,A5: set(A),B5: set(B),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),image(A,B,F,A5)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B5)),aa(set(A),nat,finite_card(A),A5))) ) ) ).
% surj_card_le
tff(fact_4068_prod__Suc__fact,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).
% prod_Suc_fact
tff(fact_4069_prod__Suc__Suc__fact,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).
% prod_Suc_Suc_fact
tff(fact_4070_image__Suc__atMost,axiom,
! [N: nat] : image(nat,nat,suc,set_ord_atMost(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).
% image_Suc_atMost
tff(fact_4071_Iio__Int__singleton,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,K: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),insert(A,X,bot_bot(set(A)))) = insert(A,X,bot_bot(set(A))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).
% Iio_Int_singleton
tff(fact_4072_image__Suc__lessThan,axiom,
! [N: nat] : image(nat,nat,suc,set_ord_lessThan(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).
% image_Suc_lessThan
tff(fact_4073_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).
% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4074_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).
% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4075_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_4076_atMost__Suc__eq__insert__0,axiom,
! [N: nat] : set_ord_atMost(nat,aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_ord_atMost(nat,N))) ).
% atMost_Suc_eq_insert_0
tff(fact_4077_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,N)) = insert(nat,zero_zero(nat),image(nat,nat,suc,set_ord_lessThan(nat,N))) ).
% lessThan_Suc_eq_insert_0
tff(fact_4078_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& strict9044650504122735259up_add(B) )
=> ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ( Xs != nil(A) )
=> ( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,G,X2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs)))) ) ) ) ).
% sum_list_strict_mono
tff(fact_4079_elem__le__sum__list,axiom,
! [A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [K: nat,Ns: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),aa(list(A),A,groups8242544230860333062m_list(A),Ns))) ) ) ).
% elem_le_sum_list
tff(fact_4080_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_4081_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_4082_fact__prod__Suc,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),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% fact_prod_Suc
tff(fact_4083_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) = image(nat,int,semiring_1_of_nat(int),set_ord_lessThan(nat,aa(int,nat,nat2,U))) ) ) ).
% image_atLeastZeroLessThan_int
tff(fact_4084_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C2: A,X: A,Y: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),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),X),Y))
=> ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,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),X)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( image(A,A,aa(A,fun(A,A),times_times(A),C2),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_4085_prod__pos__nat__iff,axiom,
! [A: $tType,A5: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F,X5))) ) ) ) ).
% prod_pos_nat_iff
tff(fact_4086_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ~ pp(member(B,X,A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),insert(B,X,A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5)) ) ) ) ) ).
% prod.insert
tff(fact_4087_prod__eq__1__iff,axiom,
! [A: $tType,A5: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A5) = one_one(nat) )
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> ( aa(A,nat,F,X5) = one_one(nat) ) ) ) ) ).
% prod_eq_1_iff
tff(fact_4088_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A)] :
( ~ pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = one_one(A) ) ) ) ).
% prod.infinite
tff(fact_4089_prod__zero__iff,axiom,
! [B: $tType,A: $tType] :
( semidom(A)
=> ! [A5: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5) = zero_zero(A) )
<=> ? [X5: B] :
( pp(member(B,X5,A5))
& ( aa(B,A,F,X5) = zero_zero(A) ) ) ) ) ) ).
% prod_zero_iff
tff(fact_4090_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_4091_inf__nat__def,axiom,
inf_inf(nat) = ord_min(nat) ).
% inf_nat_def
tff(fact_4092_inf__int__def,axiom,
inf_inf(int) = ord_min(int) ).
% inf_int_def
tff(fact_4093_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = one_one(A) ) ) ) ).
% prod.neutral
tff(fact_4094_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A5: set(B)] :
( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) != one_one(A) )
=> ~ ! [A6: B] :
( pp(member(B,A6,A5))
=> ( aa(B,A,G,A6) = one_one(A) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
tff(fact_4095_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5))) ) ) ).
% prod_nonneg
tff(fact_4096_prod__mono,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(A)
=> ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I4)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),aa(B,A,G,I4))) ) )
=> 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),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5))) ) ) ).
% prod_mono
tff(fact_4097_prod__pos,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5))) ) ) ).
% prod_pos
tff(fact_4098_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( linord181362715937106298miring(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F,X2))) )
=> 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),A5))) ) ) ).
% prod_ge_1
tff(fact_4099_prod__zero,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_1(A)
=> ! [A5: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ? [X3: B] :
( pp(member(B,X3,A5))
& ( aa(B,A,F,X3) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5) = zero_zero(A) ) ) ) ) ).
% prod_zero
tff(fact_4100_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( linord181362715937106298miring(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X2)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),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),A5)),one_one(A))) ) ) ).
% prod_le_1
tff(fact_4101_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [R2: fun(A,fun(A,bool)),S3: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),R2,one_one(A)),one_one(A)))
=> ( ! [X1: A,Y1: A,X23: A,Y22: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X1),X23))
& pp(aa(A,bool,aa(A,fun(A,bool),R2,Y1),Y22)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),times_times(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y22))) )
=> ( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H2,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3))) ) ) ) ) ) ).
% prod.related
tff(fact_4102_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( pp(member(B,X,A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),insert(B,X,A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) ) )
& ( ~ pp(member(B,X,A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),insert(B,X,A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5)) ) ) ) ) ) ).
% prod.insert_if
tff(fact_4103_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S5: set(B),T4: set(C),S3: set(B),I: fun(C,B),J: fun(B,C),T5: set(C),G: fun(B,A),H2: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S5))
=> ( pp(aa(set(C),bool,finite_finite2(C),T4))
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5)))
=> ( aa(C,B,I,aa(B,C,J,A6)) = A6 ) )
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5)))
=> pp(member(C,aa(B,C,J,A6),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))) )
=> ( ! [B4: C] :
( pp(member(C,B4,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,B4)) = B4 ) )
=> ( ! [B4: C] :
( pp(member(C,B4,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,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5))) )
=> ( ! [A6: B] :
( pp(member(B,A6,S5))
=> ( aa(B,A,G,A6) = one_one(A) ) )
=> ( ! [B4: C] :
( pp(member(C,B4,T4))
=> ( aa(C,A,H2,B4) = one_one(A) ) )
=> ( ! [A6: B] :
( pp(member(B,A6,S3))
=> ( aa(C,A,H2,aa(B,C,J,A6)) = aa(B,A,G,A6) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H2),T5) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
tff(fact_4104_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I6: set(A),I: A,F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),I6))
=> ( pp(member(A,I,I6))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I)))
=> ( ! [I4: A] :
( pp(member(A,I4,I6))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,I4))) )
=> 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),I6))) ) ) ) ) ) ).
% less_1_prod2
tff(fact_4105_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [I6: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),I6))
=> ( ( I6 != bot_bot(set(A)) )
=> ( ! [I4: A] :
( pp(member(A,I4,I6))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I4))) )
=> 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),I6))) ) ) ) ) ).
% less_1_prod
tff(fact_4106_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [B5: set(B),A5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = 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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B5)) ) ) ) ) ).
% prod.subset_diff
tff(fact_4107_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S3: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( 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),H2),S3) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
tff(fact_4108_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S3: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,H2,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),T5) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
tff(fact_4109_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = 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),S3) ) ) ) ) ) ).
% prod.mono_neutral_right
tff(fact_4110_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T5) ) ) ) ) ) ).
% prod.mono_neutral_left
tff(fact_4111_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C6: set(B),A5: set(B),B5: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C6))
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
=> ( aa(B,A,G,A6) = one_one(A) ) )
=> ( ! [B4: B] :
( pp(member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B5)))
=> ( aa(B,A,H2,B4) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),C6) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),B5) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
tff(fact_4112_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C6: set(B),A5: set(B),B5: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C6))
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
=> ( aa(B,A,G,A6) = one_one(A) ) )
=> ( ! [B4: B] :
( pp(member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B5)))
=> ( aa(B,A,H2,B4) = one_one(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),B5) )
<=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),C6) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
tff(fact_4113_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),B5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B5))),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)),inf_inf(set(B)),A5),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),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B5)) ) ) ) ) ).
% prod.union_inter
tff(fact_4114_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A),B5: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = 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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5))),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)),A5),B5))) ) ) ) ).
% prod.Int_Diff
tff(fact_4115_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [T5: set(B),S3: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ! [I4: B] :
( pp(member(B,I4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,H2,I4) = one_one(A) ) )
=> ( ! [I4: B] :
( pp(member(B,I4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),T5)))
=> ( aa(B,A,G,I4) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),T5)))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),T5) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
tff(fact_4116_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( linordered_semidom(A)
=> ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ! [I4: B] :
( pp(member(B,I4,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I4)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I4)),aa(B,A,G,I4))) ) )
=> ( ( A5 != bot_bot(set(B)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5))) ) ) ) ) ).
% prod_mono_strict
tff(fact_4117_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A),X: B] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),insert(B,X,A5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),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)),A5),insert(B,X,bot_bot(set(B)))))) ) ) ) ).
% prod.insert_remove
tff(fact_4118_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),X: B,G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(member(B,X,A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,G,X)),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)),A5),insert(B,X,bot_bot(set(B)))))) ) ) ) ) ).
% prod.remove
tff(fact_4119_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),B5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5)))
=> ( aa(B,A,G,X2) = 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)),A5),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),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B5)) ) ) ) ) ) ).
% prod.union_inter_neutral
tff(fact_4120_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),B5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5) = bot_bot(set(B)) )
=> ( 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)),A5),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),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B5)) ) ) ) ) ) ).
% prod.union_disjoint
tff(fact_4121_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),B5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( 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)),A5),B5)) = aa(A,A,aa(A,fun(A,A),times_times(A),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),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B5))),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)),B5),A5)))),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)),inf_inf(set(B)),A5),B5))) ) ) ) ) ).
% prod.union_diff2
tff(fact_4122_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( linordered_idom(B)
=> ! [B5: set(A),A5: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ! [B4: A] :
( pp(member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,B4))) )
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A6))) )
=> 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),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B5))) ) ) ) ) ) ).
% prod_mono2
tff(fact_4123_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( linordered_semidom(A)
=> ! [A5: set(B),F: fun(B,A),N: A,K: nat] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I4)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),N)) ) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),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),A5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).
% prod_le_power
tff(fact_4124_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( semidom_divide(A)
=> ! [A5: set(B),F: fun(B,A),A3: B] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( aa(B,A,F,A3) != zero_zero(A) )
=> ( ( pp(member(B,A3,A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),insert(B,A3,bot_bot(set(B))))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5)),aa(B,A,F,A3)) ) )
& ( ~ pp(member(B,A3,A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),insert(B,A3,bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5) ) ) ) ) ) ) ).
% prod_diff1
tff(fact_4125_prod__Un,axiom,
! [A: $tType,B: $tType] :
( field(A)
=> ! [A5: set(B),B5: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5)))
=> ( aa(B,A,F,X2) != zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B5)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),B5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5))) ) ) ) ) ) ).
% prod_Un
tff(fact_4126_translation__Compl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,T2: set(A)] : image(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)),image(A,A,aa(A,fun(A,A),plus_plus(A),A3),T2)) ) ).
% translation_Compl
tff(fact_4127_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list(A),Ys2: list(A)] :
( ( remdups_adj(A,Xs) = Ys2 )
<=> ? [F4: fun(nat,nat)] :
( pp(aa(fun(nat,nat),bool,order_mono(nat,nat),F4))
& ( image(nat,nat,F4,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)),Ys2)) )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth(A,Xs),I5) = aa(nat,A,nth(A,Ys2),aa(nat,nat,F4,I5)) ) )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I5),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth(A,Xs),I5) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I5),one_one(nat))) )
<=> ( aa(nat,nat,F4,I5) = aa(nat,nat,F4,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I5),one_one(nat))) ) ) ) ) ) ).
% remdups_adj_altdef
tff(fact_4128_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),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_aa(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),bit0(one2))),N)),one_one(nat)))) ) ).
% pochhammer_times_pochhammer_half
tff(fact_4129_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_4130_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ab(B,A)),A5) = one_one(A) ) ).
% prod.neutral_const
tff(fact_4131_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( comm_ring_1(A)
=> ! [F: fun(B,int),A5: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ac(fun(B,int),fun(B,A),F)),A5) ) ).
% of_int_prod
tff(fact_4132_finite__Collect__less__nat,axiom,
! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),K)))) ).
% finite_Collect_less_nat
tff(fact_4133_finite__Collect__le__nat,axiom,
! [K: nat] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ae(nat,fun(nat,bool)),K)))) ).
% finite_Collect_le_nat
tff(fact_4134_card__Collect__less__nat,axiom,
! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N))) = N ).
% card_Collect_less_nat
tff(fact_4135_finite__interval__int1,axiom,
! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_af(int,fun(int,fun(int,bool)),A3),B2)))) ).
% finite_interval_int1
tff(fact_4136_finite__interval__int4,axiom,
! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_ag(int,fun(int,fun(int,bool)),A3),B2)))) ).
% finite_interval_int4
tff(fact_4137_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),A3: B,B2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( pp(member(B,A3,S3))
=> ( 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_ah(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
& ( ~ pp(member(B,A3,S3))
=> ( 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_ah(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = one_one(A) ) ) ) ) ) ).
% prod.delta
tff(fact_4138_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),A3: B,B2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( pp(member(B,A3,S3))
=> ( 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_ai(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
& ( ~ pp(member(B,A3,S3))
=> ( 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_ai(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = one_one(A) ) ) ) ) ) ).
% prod.delta'
tff(fact_4139_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aTP_Lamp_aj(B,A),Xs)) = zero_zero(A) ) ).
% sum_list_0
tff(fact_4140_card__Collect__le__nat,axiom,
! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ae(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).
% card_Collect_le_nat
tff(fact_4141_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [F: fun(B,nat),A5: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ak(fun(B,nat),fun(B,A),F)),A5) ) ).
% of_nat_prod
tff(fact_4142_finite__interval__int2,axiom,
! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_al(int,fun(int,fun(int,bool)),A3),B2)))) ).
% finite_interval_int2
tff(fact_4143_finite__interval__int3,axiom,
! [A3: int,B2: int] : pp(aa(set(int),bool,finite_finite2(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_am(int,fun(int,fun(int,bool)),A3),B2)))) ).
% finite_interval_int3
tff(fact_4144_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_4145_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
=> ( image(A,A,aTP_Lamp_an(A,fun(A,A),D2),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).
% image_divide_atLeastAtMost
tff(fact_4146_Gcd__int__eq,axiom,
! [N6: set(nat)] : gcd_Gcd(int,image(nat,int,semiring_1_of_nat(int),N6)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N6)) ).
% Gcd_int_eq
tff(fact_4147_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I6: set(B),X: fun(B,A),Y: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),X))))
=> ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),Y))))
=> pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ap(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I6),X),Y)))) ) ) ) ).
% prod.finite_Collect_op
tff(fact_4148_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_aq(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.shift_bounds_Suc_ivl
tff(fact_4149_nat__seg__image__imp__finite,axiom,
! [A: $tType,A5: set(A),F: fun(nat,A),N: nat] :
( ( A5 = image(nat,A,F,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N))) )
=> pp(aa(set(A),bool,finite_finite2(A),A5)) ) ).
% nat_seg_image_imp_finite
tff(fact_4150_finite__conv__nat__seg__image,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
<=> ? [N5: nat,F4: fun(nat,A)] : A5 = image(nat,A,F4,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N5))) ) ).
% finite_conv_nat_seg_image
tff(fact_4151_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I6: set(B),X: fun(B,A),Y: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),X))))
=> ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),Y))))
=> pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_as(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I6),X),Y)))) ) ) ) ).
% sum.finite_Collect_op
tff(fact_4152_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_at(set(B),fun(fun(B,bool),fun(B,bool)),A5),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_au(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A5) ) ) ) ).
% prod.inter_filter
tff(fact_4153_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),H2: fun(B,A),A5: 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),aTP_Lamp_av(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),A5) = 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),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),A5)) ) ).
% prod.distrib
tff(fact_4154_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_aq(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.shift_bounds_cl_Suc_ivl
tff(fact_4155_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [F: fun(B,A),A3: A,A5: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aTP_Lamp_aw(fun(B,A),fun(A,fun(B,A)),F),A3)),A5)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5)),A3) ) ).
% mod_prod_eq
tff(fact_4156_int__prod,axiom,
! [B: $tType,F: fun(B,nat),A5: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7121269368397514597t_prod(B,nat),F),A5)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),aTP_Lamp_ax(fun(B,nat),fun(B,int),F)),A5) ).
% int_prod
tff(fact_4157_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_ay(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_4158_translation__subtract__Compl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,T2: set(A)] : image(A,A,aTP_Lamp_az(A,fun(A,A),A3),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),image(A,A,aTP_Lamp_az(A,fun(A,A),A3),T2)) ) ).
% translation_subtract_Compl
tff(fact_4159_Collect__imp__eq,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ba(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P))),collect(A,Q)) ).
% Collect_imp_eq
tff(fact_4160_nat__less__as__int,axiom,
! [X3: nat,Xa: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),Xa))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).
% nat_less_as_int
tff(fact_4161_len__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_len(A,A3) = heap_Time_tap(nat,aTP_Lamp_bb(array(A),fun(heap_ext(product_unit),nat),A3)) ) ).
% len_def
tff(fact_4162_lambda__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ( aTP_Lamp_bc(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).
% lambda_one
tff(fact_4163_lambda__zero,axiom,
! [A: $tType] :
( mult_zero(A)
=> ( aTP_Lamp_bd(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).
% lambda_zero
tff(fact_4164_Compl__eq,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = collect(A,aTP_Lamp_be(set(A),fun(A,bool),A5)) ).
% Compl_eq
tff(fact_4165_Collect__neg__eq,axiom,
! [A: $tType,P: fun(A,bool)] : collect(A,aTP_Lamp_bf(fun(A,bool),fun(A,bool),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).
% Collect_neg_eq
tff(fact_4166_uminus__set__def,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = collect(A,aa(fun(A,bool),fun(A,bool),uminus_uminus(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A5))) ).
% uminus_set_def
tff(fact_4167_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),M: A,N: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,M)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),M),N)) ) ) ) ).
% min_of_mono
tff(fact_4168_mono__Suc,axiom,
pp(aa(fun(nat,nat),bool,order_mono(nat,nat),suc)) ).
% mono_Suc
tff(fact_4169_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),collect(A,aTP_Lamp_bg(A,fun(A,bool),A3))),collect(A,aTP_Lamp_bg(A,fun(A,bool),B2))))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).
% strict_subset_divisors_dvd
tff(fact_4170_mono__add,axiom,
! [A: $tType] :
( ordere6658533253407199908up_add(A)
=> ! [A3: A] : pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A3))) ) ).
% mono_add
tff(fact_4171_less__set__def,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
<=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A5)),aTP_Lamp_a(set(A),fun(A,bool),B5))) ) ).
% less_set_def
tff(fact_4172_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).
% mono_strict_invE
tff(fact_4173_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),M: A,N: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,M)),aa(A,B,F,N)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),M),N)) ) ) ) ).
% max_of_mono
tff(fact_4174_numeral__code_I2_J,axiom,
! [A: $tType] :
( numeral(A)
=> ! [N: num] : aa(num,A,numeral_numeral(A),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_4175_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
<=> ! [X5: A,Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X5)),aa(A,B,F,Y5))) ) ) ) ).
% mono_def
tff(fact_4176_monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y4))) )
=> pp(aa(fun(A,B),bool,order_mono(A,B),F)) ) ) ).
% monoI
tff(fact_4177_monoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y))) ) ) ) ).
% monoE
tff(fact_4178_monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y))) ) ) ) ).
% monoD
tff(fact_4179_min__def__raw,axiom,
! [A: $tType] :
( ord(A)
=> ! [X3: A,Xa: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa) = X3 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa) = Xa ) ) ) ) ).
% min_def_raw
tff(fact_4180_subset__divisors__dvd,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A3: A,B2: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_bg(A,fun(A,bool),A3))),collect(A,aTP_Lamp_bg(A,fun(A,bool),B2))))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% subset_divisors_dvd
tff(fact_4181_max__def__raw,axiom,
! [A: $tType] :
( ord(A)
=> ! [X3: A,Xa: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa) = Xa ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa) = X3 ) ) ) ) ).
% max_def_raw
tff(fact_4182_nat__leq__as__int,axiom,
! [X3: nat,Xa: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Xa))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ) ).
% nat_leq_as_int
tff(fact_4183_effect__LetI,axiom,
! [B: $tType,A: $tType,X: A,T2: A,F: fun(A,heap_Time_Heap(B)),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: B,N: nat] :
( ( X = T2 )
=> ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,X),H2,H,R,N)
=> heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,T2),H2,H,R,N) ) ) ).
% effect_LetI
tff(fact_4184_lessThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_bh(A,fun(A,bool),U)) ) ).
% lessThan_def
tff(fact_4185_atMost__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_bi(A,fun(A,bool),U)) ) ).
% atMost_def
tff(fact_4186_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [C2: A,F: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bj(A,fun(fun(B,A),fun(B,A)),C2),F),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))) ) ).
% sum_list_const_mult
tff(fact_4187_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),C2: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_bk(fun(B,A),fun(A,fun(B,A)),F),C2),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs))),C2) ) ).
% sum_list_mult_const
tff(fact_4188_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)))
=> pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_bl(fun(nat,nat),fun(nat,fun(nat,bool)),F),U)))) ) ).
% finite_less_ub
tff(fact_4189_finite__int__segment,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A3: A,B2: A] : pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_bm(A,fun(A,fun(A,bool)),A3),B2)))) ) ).
% finite_int_segment
tff(fact_4190_finite__M__bounded__by__nat,axiom,
! [P: fun(nat,bool),I: nat] : pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_bn(fun(nat,bool),fun(nat,fun(nat,bool)),P),I)))) ).
% finite_M_bounded_by_nat
tff(fact_4191_finite__divisors__int,axiom,
! [I: int] :
( ( I != zero_zero(int) )
=> pp(aa(set(int),bool,finite_finite2(int),collect(int,aTP_Lamp_bo(int,fun(int,bool),I)))) ) ).
% finite_divisors_int
tff(fact_4192_integer__of__num_I2_J,axiom,
! [N: num] : aa(num,code_integer,code_integer_of_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_4193_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_4194_power__numeral__even,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Z2: A,W2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),bit0(W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))) ) ).
% power_numeral_even
tff(fact_4195_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
=> pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aTP_Lamp_bp(nat,fun(nat,nat),K))) ) ).
% mono_ge2_power_minus_self
tff(fact_4196_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A),B5: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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)),inf_inf(set(B)),A5),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_bq(fun(B,A),fun(set(B),fun(B,A)),G),B5)),A5) ) ) ) ).
% prod.inter_restrict
tff(fact_4197_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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)),A5),collect(B,aTP_Lamp_br(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) ) ) ) ).
% prod.setdiff_irrelevant
tff(fact_4198_power__numeral__odd,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Z2: A,W2: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z2),aa(num,nat,numeral_numeral(nat),W2))) ) ).
% power_numeral_odd
tff(fact_4199_card__length__sum__list__rec,axiom,
! [M: nat,N6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
=> ( aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_bs(nat,fun(nat,fun(list(nat),bool)),M),N6))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_bt(nat,fun(nat,fun(list(nat),bool)),M),N6)))),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_bu(nat,fun(nat,fun(list(nat),bool)),M),N6)))) ) ) ).
% card_length_sum_list_rec
tff(fact_4200_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add(B)
& ordere6658533253407199908up_add(B) )
=> ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs)))) ) ) ).
% sum_list_mono
tff(fact_4201_finite__abs__int__segment,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [A3: A] : pp(aa(set(A),bool,finite_finite2(A),collect(A,aTP_Lamp_bv(A,fun(A,bool),A3)))) ) ).
% finite_abs_int_segment
tff(fact_4202_nat__plus__as__int,axiom,
! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).
% nat_plus_as_int
tff(fact_4203_nat__times__as__int,axiom,
! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).
% nat_times_as_int
tff(fact_4204_card__length__sum__list,axiom,
! [M: nat,N6: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_bs(nat,fun(nat,fun(list(nat),bool)),M),N6))) = 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),N6),M)),one_one(nat))),N6) ).
% card_length_sum_list
tff(fact_4205_nat__minus__as__int,axiom,
! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).
% nat_minus_as_int
tff(fact_4206_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_bw(fun(nat,A),fun(nat,fun(nat,A)),G),N)),set_ord_lessThan(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,N)) ) ).
% prod.nat_diff_reindex
tff(fact_4207_finite__divisors__nat,axiom,
! [M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
=> pp(aa(set(nat),bool,finite_finite2(nat),collect(nat,aTP_Lamp_bx(nat,fun(nat,bool),M)))) ) ).
% finite_divisors_nat
tff(fact_4208_card__less,axiom,
! [M7: set(nat),I: nat] :
( pp(member(nat,zero_zero(nat),M7))
=> ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_by(set(nat),fun(nat,fun(nat,bool)),M7),I))) != zero_zero(nat) ) ) ).
% card_less
tff(fact_4209_card__less__Suc,axiom,
! [M7: set(nat),I: nat] :
( pp(member(nat,zero_zero(nat),M7))
=> ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_bz(set(nat),fun(nat,fun(nat,bool)),M7),I)))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_by(set(nat),fun(nat,fun(nat,bool)),M7),I))) ) ) ).
% card_less_Suc
tff(fact_4210_card__less__Suc2,axiom,
! [M7: set(nat),I: nat] :
( ~ pp(member(nat,zero_zero(nat),M7))
=> ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_bz(set(nat),fun(nat,fun(nat,bool)),M7),I))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_by(set(nat),fun(nat,fun(nat,bool)),M7),I))) ) ) ).
% card_less_Suc2
tff(fact_4211_sum__list__Suc,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,aTP_Lamp_ca(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),map(A,nat,F,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).
% sum_list_Suc
tff(fact_4212_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_ay(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_4213_nat__div__as__int,axiom,
! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).
% nat_div_as_int
tff(fact_4214_nat__mod__as__int,axiom,
! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X3),Xa) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).
% nat_mod_as_int
tff(fact_4215_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).
% mono_invE
tff(fact_4216_mono__iff__le__Suc,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A)] :
( pp(aa(fun(nat,A),bool,order_mono(nat,A),F))
<=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N5)),aa(nat,A,F,aa(nat,nat,suc,N5)))) ) ) ).
% mono_iff_le_Suc
tff(fact_4217_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] : image(int,int,aTP_Lamp_cb(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_4218_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf(A)
& semilattice_inf(B) )
=> ! [F: fun(A,B),A5: A,B5: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B5))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F,A5)),aa(A,B,F,B5)))) ) ) ).
% mono_inf
tff(fact_4219_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] : map(nat,A,aTP_Lamp_cc(A,fun(nat,A),X),upt(zero_zero(nat),I)) = replicate(A,I,X) ).
% map_replicate_trivial
tff(fact_4220_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup(A)
& semilattice_sup(B) )
=> ! [F: fun(A,B),A5: A,B5: A] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F,A5)),aa(A,B,F,B5))),aa(A,B,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B5)))) ) ) ).
% mono_sup
tff(fact_4221_semiring__char__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,collect(nat,aTP_Lamp_cd(nat,bool))) ) ).
% semiring_char_def
tff(fact_4222_map__add__upt,axiom,
! [N: nat,M: nat] : map(nat,nat,aTP_Lamp_ce(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_4223_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_4224_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_aq(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% prod.Suc_reindex_ivl
tff(fact_4225_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),P: fun(B,bool),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_cf(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H2),G)),A5) = 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),H2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),collect(B,P)))),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)),inf_inf(set(B)),A5),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).
% prod.If_cases
tff(fact_4226_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( semiring_1(B)
=> ! [R: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),map(C,B,aTP_Lamp_cg(B,fun(C,B),R),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R) ) ).
% sum_list_triv
tff(fact_4227_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),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_aq(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).
% prod.atMost_Suc_shift
tff(fact_4228_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),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_aq(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.lessThan_Suc_shift
tff(fact_4229_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_ch(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).
% prod.atLeastLessThan_rev
tff(fact_4230_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_ci(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_ck(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% prod.nested_swap
tff(fact_4231_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_aq(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).
% prod.atLeast1_atMost_eq
tff(fact_4232_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_cl(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).
% prod.nat_group
tff(fact_4233_map__upt__Suc,axiom,
! [A: $tType,F: fun(nat,A),N: nat] : map(nat,A,F,upt(zero_zero(nat),aa(nat,nat,suc,N))) = cons(A,aa(nat,A,F,zero_zero(nat)),map(nat,A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),F),upt(zero_zero(nat),N))) ).
% map_upt_Suc
tff(fact_4234_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_cn(fun(nat,fun(nat,A)),fun(nat,A),A3)),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_ck(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_ord_lessThan(nat,N)) ) ).
% prod.nested_swap'
tff(fact_4235_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_co(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,one_one(A)) ) ).
% prod_atLeastAtMost_code
tff(fact_4236_map__nth,axiom,
! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).
% map_nth
tff(fact_4237_ratrel__def,axiom,
! [X3: product_prod(int,int),Xa: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X3),Xa))
<=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X3) != zero_zero(int) )
& ( aa(product_prod(int,int),int,product_snd(int,int),Xa) != zero_zero(int) )
& ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X3)),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_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X3)) ) ) ) ).
% ratrel_def
tff(fact_4238_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_cp(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).
% fact_prod
tff(fact_4239_map__decr__upt,axiom,
! [M: nat,N: nat] : map(nat,nat,aTP_Lamp_cq(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).
% map_decr_upt
tff(fact_4240_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( pp(member(B,A3,S3))
=> ( 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(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_cr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),insert(B,A3,bot_bot(set(B)))))) ) )
& ( ~ pp(member(B,A3,S3))
=> ( 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(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_cr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),insert(B,A3,bot_bot(set(B))))) ) ) ) ) ) ).
% prod.delta_remove
tff(fact_4241_diff__nat__eq__if,axiom,
! [Z5: int,Z2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),zero_zero(int)))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) = aa(int,nat,nat2,Z2) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z5),zero_zero(int)))
=> ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z2)),aa(int,nat,nat2,Z5)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z5)),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z2),Z5))) ) ) ) ).
% diff_nat_eq_if
tff(fact_4242_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_cs(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_4243_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),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_aq(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% prod.atMost_shift
tff(fact_4244_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_ct(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% pochhammer_prod
tff(fact_4245_mono__mult,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
=> pp(aa(fun(A,A),bool,order_mono(A,A),aa(A,fun(A,A),times_times(A),A3))) ) ) ).
% mono_mult
tff(fact_4246_mono__times__nat,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(fun(nat,nat),bool,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),N))) ) ).
% mono_times_nat
tff(fact_4247_finite__lists__length__le,axiom,
! [A: $tType,A5: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> pp(aa(set(list(A)),bool,finite_finite2(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_cu(set(A),fun(nat,fun(list(A),bool)),A5),N)))) ) ).
% finite_lists_length_le
tff(fact_4248_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [F: fun(A,B),M: A,N: A,M4: B,N2: B] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( ( image(A,B,F,set_or7035219750837199246ssThan(A,M,N)) = set_or7035219750837199246ssThan(B,M4,N2) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
=> ( aa(A,B,F,M) = M4 ) ) ) ) ) ).
% mono_image_least
tff(fact_4249_fact__prod__rev,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),aa(nat,fun(nat,nat),minus_minus(nat),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% fact_prod_rev
tff(fact_4250_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
=> ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_cp(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).
% fact_eq_fact_times
tff(fact_4251_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A,C2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
=> ( image(A,A,aTP_Lamp_cv(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),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))
=> ( image(A,A,aTP_Lamp_cv(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,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),X),C2)) ) ) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( image(A,A,aTP_Lamp_cv(A,fun(A,A),C2),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
tff(fact_4252_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)) )
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cw(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cw(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cw(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_4253_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)) )
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cx(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cx(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cx(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_4254_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)) )
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cy(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cy(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cy(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_4255_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)) )
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cz(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cz(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))
=> ( image(A,A,aa(A,fun(A,A),aTP_Lamp_cz(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_4256_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),A3: B,B2: fun(B,A),C2: A] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( pp(member(B,A3,S3))
=> ( 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_da(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S3) = 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),aa(set(B),nat,finite_card(B),S3)),one_one(nat)))) ) )
& ( ~ pp(member(B,A3,S3))
=> ( 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_da(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,finite_card(B),S3)) ) ) ) ) ) ).
% prod_gen_delta
tff(fact_4257_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_ct(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).
% pochhammer_Suc_prod
tff(fact_4258_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_db(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).
% pochhammer_prod_rev
tff(fact_4259_set__decode__def,axiom,
! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_dc(nat,fun(nat,bool),X)) ).
% set_decode_def
tff(fact_4260_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
=> ( image(nat,nat,aTP_Lamp_dd(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),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),X),Y))
=> ( image(nat,nat,aTP_Lamp_dd(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,X,Y)) = insert(nat,zero_zero(nat),bot_bot(set(nat))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> ( image(nat,nat,aTP_Lamp_dd(nat,fun(nat,nat),C2),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_4261_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_cp(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_4262_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),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.in_pairs
tff(fact_4263_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),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),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% prod.in_pairs_0
tff(fact_4264_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_df(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_altdef_of_nat
tff(fact_4265_signed__take__bit__code,axiom,
! [A: $tType] :
( bit_ri3973907225187159222ations(A)
=> ! [N: nat,A3: A] : aa(A,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_4266_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_db(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).
% pochhammer_Suc_prod_rev
tff(fact_4267_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
=> ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(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_dg(nat,fun(nat,fun(nat,A)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).
% binomial_altdef_of_nat
tff(fact_4268_gbinomial__mult__fact,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dh(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact
tff(fact_4269_gbinomial__mult__fact_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)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_dh(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).
% gbinomial_mult_fact'
tff(fact_4270_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_di(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).
% gbinomial_prod_rev
tff(fact_4271_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),butlast(A,Xs))))
=> ( aa(nat,A,nth(A,butlast(A,Xs)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).
% nth_butlast
tff(fact_4272_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( take(A,N,butlast(A,Xs)) = take(A,N,Xs) ) ) ).
% take_butlast
tff(fact_4273_prod_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [P3: nat,K: nat,G: fun(nat,A),H2: 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_dj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),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_dk(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),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_4274_gbinomial__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_di(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).
% gbinomial_Suc
tff(fact_4275_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_dl(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_4276_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),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),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),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_4277_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_4278_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list(A),X: 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,X)) = 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,X)) = list_update(A,butlast(A,Xs),K,X) ) ) ) ).
% butlast_list_update
tff(fact_4279_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_dm(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_4280_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A5: set(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(set(A),nat,finite_card(A),A5)))
=> ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(set(A),fun(list(A),bool),aTP_Lamp_dn(nat,fun(set(A),fun(list(A),bool)),K),A5))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_cp(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),aa(set(A),nat,finite_card(A),A5)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ).
% card_lists_distinct_length_eq'
tff(fact_4281_card__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set(A),K: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A5)))
=> ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_do(set(A),fun(nat,fun(list(A),bool)),A5),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_cp(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),aa(set(A),nat,finite_card(A),A5)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A5))) ) ) ) ).
% card_lists_distinct_length_eq
tff(fact_4282_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),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),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool),aTP_Lamp_dp(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),bit0(one2)))) ) ) ) ).
% bit_cut_integer_code
tff(fact_4283_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),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),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),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),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),bit0(one2)))))),one_one(code_integer))) ) ) ) ) ) ).
% integer_of_int_code
tff(fact_4284_swap__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X: A,A3: array(A)] : array_swap(A,I,X,A3) = heap_Time_guard(A,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_dq(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_dr(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))))),I),X),A3)) ) ).
% swap_def
tff(fact_4285_choose__even__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(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ds(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).
% choose_even_sum
tff(fact_4286_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_4287_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ring_1(B)
=> ! [A5: set(A),F: fun(A,B)] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(member(B,aa(A,B,F,X2),ring_1_Ints(B))) )
=> pp(member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A5),ring_1_Ints(B))) ) ) ).
% Ints_sum
tff(fact_4288_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_dt(B,A)),A5) = zero_zero(A) ) ).
% sum.neutral_const
tff(fact_4289_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [F: fun(B,nat),A5: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_du(fun(B,nat),fun(B,A),F)),A5) ) ).
% of_nat_sum
tff(fact_4290_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ring_1(A)
=> ! [F: fun(B,int),A5: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_dv(fun(B,int),fun(B,A),F)),A5) ) ).
% of_int_sum
tff(fact_4291_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty
tff(fact_4292_sum__eq__0__iff,axiom,
! [B: $tType,A: $tType] :
( canoni5634975068530333245id_add(A)
=> ! [F3: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),F3))
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),F3) = zero_zero(A) )
<=> ! [X5: B] :
( pp(member(B,X5,F3))
=> ( aa(B,A,F,X5) = zero_zero(A) ) ) ) ) ) ).
% sum_eq_0_iff
tff(fact_4293_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),G: fun(B,A)] :
( ~ pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = zero_zero(A) ) ) ) ).
% sum.infinite
tff(fact_4294_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(B),A3: B,B2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( pp(member(B,A3,S3))
=> ( 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_dw(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
& ( ~ pp(member(B,A3,S3))
=> ( 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_dw(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = zero_zero(A) ) ) ) ) ) ).
% sum.delta
tff(fact_4295_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(B),A3: B,B2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( pp(member(B,A3,S3))
=> ( 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_dx(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = aa(B,A,B2,A3) ) )
& ( ~ pp(member(B,A3,S3))
=> ( 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_dx(B,fun(fun(B,A),fun(B,A)),A3),B2)),S3) = zero_zero(A) ) ) ) ) ) ).
% sum.delta'
tff(fact_4296_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ordere166539214618696060dd_abs(B)
=> ! [F: fun(A,B),A5: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_dy(fun(A,B),fun(A,B),F)),A5))) ) ).
% sum_abs
tff(fact_4297_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),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),set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).
% sum.atMost_Suc
tff(fact_4298_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),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),set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ).
% sum.lessThan_Suc
tff(fact_4299_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ordere166539214618696060dd_abs(B)
=> ! [F: fun(A,B),A5: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_dy(fun(A,B),fun(A,B),F)),A5))) ) ).
% sum_abs_ge_zero
tff(fact_4300_sum__constant,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Y: A,A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_dz(A,fun(B,A),Y)),A5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),Y) ) ).
% sum_constant
tff(fact_4301_sum__list__upt,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% sum_list_upt
tff(fact_4302_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A5: set(B),P: fun(B,bool),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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_ea(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F)),A5) = 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)),inf_inf(set(B)),A5),collect(B,P))) ) ) ) ).
% sum_of_bool_mult_eq
tff(fact_4303_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A5: set(B),F: fun(B,A),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_eb(fun(B,A),fun(fun(B,bool),fun(B,A)),F),P)),A5) = 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)),inf_inf(set(B)),A5),collect(B,P))) ) ) ) ).
% sum_mult_of_bool_eq
tff(fact_4304_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_4305_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_4306_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [A5: set(B),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ec(fun(B,bool),fun(B,A),P)),A5) = aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),collect(B,P)))) ) ) ) ) ).
% sum_of_bool_eq
tff(fact_4307_sum__zero__power,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A5: set(nat),C2: fun(nat,A)] :
( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
& pp(member(nat,zero_zero(nat),A5)) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ed(fun(nat,A),fun(nat,A),C2)),A5) = aa(nat,A,C2,zero_zero(nat)) ) )
& ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
& pp(member(nat,zero_zero(nat),A5)) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ed(fun(nat,A),fun(nat,A),C2)),A5) = zero_zero(A) ) ) ) ) ).
% sum_zero_power
tff(fact_4308_sum__zero__power_H,axiom,
! [A: $tType] :
( field(A)
=> ! [A5: set(nat),C2: fun(nat,A),D2: fun(nat,A)] :
( ( ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
& pp(member(nat,zero_zero(nat),A5)) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A5) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))) ) )
& ( ~ ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
& pp(member(nat,zero_zero(nat),A5)) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A5) = zero_zero(A) ) ) ) ) ).
% sum_zero_power'
tff(fact_4309_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] : unique8689654367752047608divmod(A,bit0(M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_ef(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).
% divmod_algorithm_code(5)
tff(fact_4310_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_eg(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).
% divmod_algorithm_code(6)
tff(fact_4311_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [K5: set(B),F: fun(B,A),G: fun(B,A)] :
( ! [I4: B] :
( pp(member(B,I4,K5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),aa(B,A,G,I4))) )
=> 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),groups7311177749621191930dd_sum(B,A),F),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),K5))) ) ) ).
% sum_mono
tff(fact_4312_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_eh(fun(B,A),fun(B,A),F)),A5) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)) ) ).
% sum_negf
tff(fact_4313_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [R: A,F: fun(B,A),A5: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)) = 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_bj(A,fun(fun(B,A),fun(B,A)),R),F)),A5) ) ).
% sum_distrib_left
tff(fact_4314_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [F: fun(B,A),A5: set(B),R: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)),R) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_bk(fun(B,A),fun(A,fun(B,A)),F),R)),A5) ) ).
% sum_distrib_right
tff(fact_4315_sum__product,axiom,
! [C: $tType,B: $tType,A: $tType] :
( semiring_0(B)
=> ! [F: fun(A,B),A5: set(A),G: fun(C,B),B5: set(C)] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A5)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_ej(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F),G),B5)),A5) ) ).
% sum_product
tff(fact_4316_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [F: fun(B,A),A3: A,A5: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_aw(fun(B,A),fun(A,fun(B,A)),F),A3)),A5)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)),A3) ) ).
% mod_sum_eq
tff(fact_4317_unset__bit__integer_Oabs__eq,axiom,
! [Xa4: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),Xa4),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Xa4),X)) ).
% unset_bit_integer.abs_eq
tff(fact_4318_bit__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,fun(nat,bool),bit_se5641148757651400278ts_bit(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),X) ).
% bit_integer.abs_eq
tff(fact_4319_or__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa4),X)) ).
% or_integer.abs_eq
tff(fact_4320_and__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa4),X)) ).
% and_integer.abs_eq
tff(fact_4321_euclidean__size__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,nat,euclid6346220572633701492n_size(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,nat,euclid6346220572633701492n_size(int),X) ).
% euclidean_size_integer.abs_eq
tff(fact_4322_not__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,bit_ri4277139882892585799ns_not(int),X)) ).
% not_integer.abs_eq
tff(fact_4323_take__bit__integer_Oabs__eq,axiom,
! [Xa4: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),Xa4),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Xa4),X)) ).
% take_bit_integer.abs_eq
tff(fact_4324_xor__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa4),X)) ).
% xor_integer.abs_eq
tff(fact_4325_times__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),times_times(int),Xa4),X)) ).
% times_integer.abs_eq
tff(fact_4326_push__bit__integer_Oabs__eq,axiom,
! [Xa4: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Xa4),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Xa4),X)) ).
% push_bit_integer.abs_eq
tff(fact_4327_drop__bit__integer_Oabs__eq,axiom,
! [Xa4: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Xa4),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa4),X)) ).
% drop_bit_integer.abs_eq
tff(fact_4328_mask__integer_Oabs__eq,axiom,
! [X: nat] : aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),X) = aa(int,code_integer,code_integer_of_int,aa(nat,int,bit_se2239418461657761734s_mask(int),X)) ).
% mask_integer.abs_eq
tff(fact_4329_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod(A,B),F: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P3: product_prod(A,B)] :
( ! [X2: A,Y4: B] :
( ( aa(B,product_prod(A,B),product_Pair(A,B,X2),Y4) = Q4 )
=> ( aa(B,C,aa(A,fun(B,C),F,X2),Y4) = aa(B,C,aa(A,fun(B,C),G,X2),Y4) ) )
=> ( ( P3 = Q4 )
=> ( aa(product_prod(A,B),C,product_case_prod(A,B,C,F),P3) = aa(product_prod(A,B),C,product_case_prod(A,B,C,G),Q4) ) ) ) ).
% split_cong
tff(fact_4330_set__bit__integer_Oabs__eq,axiom,
! [Xa4: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xa4),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xa4),X)) ).
% set_bit_integer.abs_eq
tff(fact_4331_flip__bit__integer_Oabs__eq,axiom,
! [Xa4: nat,X: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),Xa4),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Xa4),X)) ).
% flip_bit_integer.abs_eq
tff(fact_4332_division__segment__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_integer,euclid7384307370059645450egment(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,euclid7384307370059645450egment(int),X)) ).
% division_segment_integer.abs_eq
tff(fact_4333_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),G: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = zero_zero(A) ) ) ) ).
% sum.neutral
tff(fact_4334_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),A5: set(B)] :
( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) != zero_zero(A) )
=> ~ ! [A6: B] :
( pp(member(B,A6,A5))
=> ( aa(B,A,G,A6) = zero_zero(A) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
tff(fact_4335_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),zero_zero(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),groups7311177749621191930dd_sum(B,A),F),A5)),zero_zero(A))) ) ) ).
% sum_nonpos
tff(fact_4336_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5))) ) ) ).
% sum_nonneg
tff(fact_4337_sum__mono__inv,axiom,
! [A: $tType,I7: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [F: fun(I7,A),I6: set(I7),G: fun(I7,A),I: I7] :
( ( aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),F),I6) = aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),G),I6) )
=> ( ! [I4: I7] :
( pp(member(I7,I4,I6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I7,A,F,I4)),aa(I7,A,G,I4))) )
=> ( pp(member(I7,I,I6))
=> ( pp(aa(set(I7),bool,finite_finite2(I7),I6))
=> ( aa(I7,A,F,I) = aa(I7,A,G,I) ) ) ) ) ) ) ).
% sum_mono_inv
tff(fact_4338_sum__cong__Suc,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(nat),F: fun(nat,A),G: fun(nat,A)] :
( ~ pp(member(nat,zero_zero(nat),A5))
=> ( ! [X2: nat] :
( pp(member(nat,aa(nat,nat,suc,X2),A5))
=> ( aa(nat,A,F,aa(nat,nat,suc,X2)) = aa(nat,A,G,aa(nat,nat,suc,X2)) ) )
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),A5) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A5) ) ) ) ) ).
% sum_cong_Suc
tff(fact_4339_zero__integer__def,axiom,
zero_zero(code_integer) = aa(int,code_integer,code_integer_of_int,zero_zero(int)) ).
% zero_integer_def
tff(fact_4340_less__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa4),X)) ) ).
% less_integer.abs_eq
tff(fact_4341_plus__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa4),X)) ).
% plus_integer.abs_eq
tff(fact_4342_uminus__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),X)) ).
% uminus_integer.abs_eq
tff(fact_4343_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),G: fun(B,A),P: fun(B,bool)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_at(set(B),fun(fun(B,bool),fun(B,bool)),A5),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_ek(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A5) ) ) ) ).
% sum.inter_filter
tff(fact_4344_one__integer__def,axiom,
one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).
% one_integer_def
tff(fact_4345_less__eq__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Xa4),X)) ) ).
% less_eq_integer.abs_eq
tff(fact_4346_minus__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa4),X)) ).
% minus_integer.abs_eq
tff(fact_4347_abs__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_integer,abs_abs(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,abs_abs(int),X)) ).
% abs_integer.abs_eq
tff(fact_4348_divide__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa4),X)) ).
% divide_integer.abs_eq
tff(fact_4349_nat__of__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,nat,code_nat_of_integer,aa(int,code_integer,code_integer_of_int,X)) = aa(int,nat,nat2,X) ).
% nat_of_integer.abs_eq
tff(fact_4350_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_el(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.shift_bounds_cl_Suc_ivl
tff(fact_4351_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_el(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.shift_bounds_Suc_ivl
tff(fact_4352_modulo__integer_Oabs__eq,axiom,
! [Xa4: int,X: 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,Xa4)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa4),X)) ).
% modulo_integer.abs_eq
tff(fact_4353_sgn__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_integer,sgn_sgn(code_integer),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,sgn_sgn(int),X)) ).
% sgn_integer.abs_eq
tff(fact_4354_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))),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_guard(A,P,F),H2,H,R,N)
=> ~ ( ( 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H2))) )
=> ( ( 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,H2))) )
=> ( ( 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,H2)) )
=> ~ pp(aa(heap_ext(product_unit),bool,P,H2)) ) ) ) ) ).
% effect_guardE
tff(fact_4355_effect__guardI,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H2: heap_ext(product_unit),H: 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,H2))
=> ( ( 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H2))) )
=> ( ( 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,H2))) )
=> ( ( 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,H2)) )
=> heap_Time_effect(A,heap_Time_guard(A,P,F),H2,H,R,N) ) ) ) ) ).
% effect_guardI
tff(fact_4356_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [S: set(B),T2: set(C),G: fun(C,A),I: fun(C,B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( pp(aa(set(C),bool,finite_finite2(C),T2))
=> ( ! [X2: C] :
( pp(member(C,X2,T2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G,X2))) )
=> ( ! [X2: B] :
( pp(member(B,X2,S))
=> ? [Xa: C] :
( pp(member(C,Xa,T2))
& ( aa(C,B,I,Xa) = X2 )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(C,A,G,Xa))) ) )
=> 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),groups7311177749621191930dd_sum(B,A),F),S)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T2))) ) ) ) ) ) ).
% sum_le_included
tff(fact_4357_sum__nonneg__eq__0__iff,axiom,
! [B: $tType,A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [A5: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X2))) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5) = zero_zero(A) )
<=> ! [X5: B] :
( pp(member(B,X5,A5))
=> ( aa(B,A,F,X5) = zero_zero(A) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
tff(fact_4358_sum__strict__mono__ex1,axiom,
! [A: $tType,I7: $tType] :
( ordere8940638589300402666id_add(A)
=> ! [A5: set(I7),F: fun(I7,A),G: fun(I7,A)] :
( pp(aa(set(I7),bool,finite_finite2(I7),A5))
=> ( ! [X2: I7] :
( pp(member(I7,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I7,A,F,X2)),aa(I7,A,G,X2))) )
=> ( ? [X3: I7] :
( pp(member(I7,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I7,A,F,X3)),aa(I7,A,G,X3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),F),A5)),aa(set(I7),A,aa(fun(I7,A),fun(set(I7),A),groups7311177749621191930dd_sum(I7,A),G),A5))) ) ) ) ) ).
% sum_strict_mono_ex1
tff(fact_4359_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [R2: fun(A,fun(A,bool)),S3: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),R2,zero_zero(A)),zero_zero(A)))
=> ( ! [X1: A,Y1: A,X23: A,Y22: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),R2,X1),X23))
& pp(aa(A,bool,aa(A,fun(A,bool),R2,Y1),Y22)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y22))) )
=> ( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(B,A,H2,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),R2,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),S3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3))) ) ) ) ) ) ).
% sum.related
tff(fact_4360_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( strict7427464778891057005id_add(A)
=> ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( A5 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5))) ) ) ) ) ).
% sum_strict_mono
tff(fact_4361_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& comm_monoid_add(A) )
=> ! [A3: B,C2: B,B2: B,D2: B,G: fun(B,A),H2: fun(B,A)] :
( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X2))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),D2))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) ) )
=> ( 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),H2),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).
% sum.ivl_cong
tff(fact_4362_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [S5: set(B),T4: set(C),S3: set(B),I: fun(C,B),J: fun(B,C),T5: set(C),G: fun(B,A),H2: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S5))
=> ( pp(aa(set(C),bool,finite_finite2(C),T4))
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5)))
=> ( aa(C,B,I,aa(B,C,J,A6)) = A6 ) )
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5)))
=> pp(member(C,aa(B,C,J,A6),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))) )
=> ( ! [B4: C] :
( pp(member(C,B4,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,B4)) = B4 ) )
=> ( ! [B4: C] :
( pp(member(C,B4,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,B4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5))) )
=> ( ! [A6: B] :
( pp(member(B,A6,S5))
=> ( aa(B,A,G,A6) = zero_zero(A) ) )
=> ( ! [B4: C] :
( pp(member(C,B4,T4))
=> ( aa(C,A,H2,B4) = zero_zero(A) ) )
=> ( ! [A6: B] :
( pp(member(B,A6,S3))
=> ( aa(C,A,H2,aa(B,C,J,A6)) = aa(B,A,G,A6) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),H2),T5) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
tff(fact_4363_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_4364_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_4365_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [S: set(B),F: fun(B,A),I: B] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ! [I4: B] :
( pp(member(B,I4,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I4))) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),S) = zero_zero(A) )
=> ( pp(member(B,I,S))
=> ( aa(B,A,F,I) = zero_zero(A) ) ) ) ) ) ) ).
% sum_nonneg_0
tff(fact_4366_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [S: set(B),F: fun(B,A),B5: A,I: B] :
( pp(aa(set(B),bool,finite_finite2(B),S))
=> ( ! [I4: B] :
( pp(member(B,I4,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I4))) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),S) = B5 )
=> ( pp(member(B,I,S))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I)),B5)) ) ) ) ) ) ).
% sum_nonneg_leq_bound
tff(fact_4367_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),G: fun(B,A),B5: set(B)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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)),inf_inf(set(B)),A5),B5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(B),fun(B,A),aTP_Lamp_em(fun(B,A),fun(set(B),fun(B,A)),G),B5)),A5) ) ) ) ).
% sum.inter_restrict
tff(fact_4368_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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)),A5),collect(B,aTP_Lamp_en(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) ) ) ) ).
% sum.setdiff_irrelevant
tff(fact_4369_sum__power__add,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: A,M: nat,I6: 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_eo(A,fun(nat,fun(nat,A)),X),M)),I6) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),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),X)),I6)) ) ).
% sum_power_add
tff(fact_4370_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_ep(fun(nat,A),fun(nat,fun(nat,A)),G),N)),set_ord_lessThan(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,N)) ) ).
% sum.nat_diff_reindex
tff(fact_4371_sum__choose__upper,axiom,
! [M: 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),M)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).
% sum_choose_upper
tff(fact_4372_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [I6: set(B),I: B,F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I6))
=> ( pp(member(B,I,I6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F,I)))
=> ( ! [I4: B] :
( pp(member(B,I4,I6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),I6))) ) ) ) ) ) ).
% sum_pos2
tff(fact_4373_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [I6: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I6))
=> ( ( I6 != bot_bot(set(B)) )
=> ( ! [I4: B] :
( pp(member(B,I4,I6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F,I4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),I6))) ) ) ) ) ).
% sum_pos
tff(fact_4374_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T5: set(B),S3: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),S3) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
tff(fact_4375_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T5: set(B),S3: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,H2,X2) = zero_zero(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),T5) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
tff(fact_4376_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T5: set(B),S3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) ) ) ) ) ) ).
% sum.mono_neutral_right
tff(fact_4377_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T5: set(B),S3: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T5) ) ) ) ) ) ).
% sum.mono_neutral_left
tff(fact_4378_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [C6: set(B),A5: set(B),B5: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C6))
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
=> ( aa(B,A,G,A6) = zero_zero(A) ) )
=> ( ! [B4: B] :
( pp(member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B5)))
=> ( aa(B,A,H2,B4) = zero_zero(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),C6) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),B5) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
tff(fact_4379_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [C6: set(B),A5: set(B),B5: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),C6))
=> ( ! [A6: B] :
( pp(member(B,A6,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
=> ( aa(B,A,G,A6) = zero_zero(A) ) )
=> ( ! [B4: B] :
( pp(member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B5)))
=> ( aa(B,A,H2,B4) = zero_zero(A) ) )
=> ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),B5) )
<=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),C6) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
tff(fact_4380_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add(A)
& semiring_1(A) )
=> ! [A5: set(B),K5: A,F: fun(B,A)] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),aa(B,A,F,I4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5))) ) ) ).
% sum_bounded_below
tff(fact_4381_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add(A)
& semiring_1(A) )
=> ! [A5: set(B),F: fun(B,A),K5: A] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),K5)) )
=> 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),groups7311177749621191930dd_sum(B,A),F),A5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),K5))) ) ) ).
% sum_bounded_above
tff(fact_4382_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [T5: set(B),S3: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ! [I4: B] :
( pp(member(B,I4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,H2,I4) = zero_zero(A) ) )
=> ( ! [I4: B] :
( pp(member(B,I4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),T5)))
=> ( aa(B,A,G,I4) = zero_zero(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),T5)))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),T5) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
tff(fact_4383_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_4384_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_4385_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_4386_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_4387_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_4388_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_4389_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_4390_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_4391_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_4392_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),P: fun(B,bool),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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_er(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H2),G)),A5) = 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),H2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),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),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).
% sum.If_cases
tff(fact_4393_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_el(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% sum.Suc_reindex_ivl
tff(fact_4394_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_es(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_4395_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_es(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_4396_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),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_el(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).
% sum.atMost_Suc_shift
tff(fact_4397_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_et(fun(nat,A),fun(nat,A),F)),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_4398_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_4399_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),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_el(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% sum.lessThan_Suc_shift
tff(fact_4400_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_et(fun(nat,A),fun(nat,A),F)),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_4401_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_es(fun(nat,A),fun(nat,A),F)),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_4402_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_eu(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).
% sum.atLeastLessThan_rev
tff(fact_4403_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_ev(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_ex(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% sum.nested_swap
tff(fact_4404_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_el(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).
% sum.atLeast1_atMost_eq
tff(fact_4405_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_el(fun(nat,A),fun(nat,A),F)),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_4406_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_ey(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).
% sum.nat_group
tff(fact_4407_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_ez(nat,fun(nat,nat),R)),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_4408_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_fa(fun(nat,fun(nat,A)),fun(nat,A),A3)),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_ex(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_ord_lessThan(nat,N)) ) ).
% sum.nested_swap'
tff(fact_4409_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_fb(nat,fun(nat,nat),N)),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_4410_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_fb(nat,fun(nat,nat),N)),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_4411_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_fc(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,zero_zero(A)) ) ).
% sum_atLeastAtMost_code
tff(fact_4412_card__sum__le__nat__sum,axiom,
! [S3: 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_cp(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S3)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),S3))) ).
% card_sum_le_nat_sum
tff(fact_4413_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ordere6911136660526730532id_add(A)
=> ! [B5: set(B),A5: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B5))
=> ( ! [B4: B] :
( pp(member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),B5),A5)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,B4))) )
=> 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),groups7311177749621191930dd_sum(B,A),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),B5))) ) ) ) ) ).
% sum_mono2
tff(fact_4414_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),B5: set(B),G: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5)))
=> ( aa(B,A,G,X2) = 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)),A5),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),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B5)) ) ) ) ) ) ).
% sum.union_inter_neutral
tff(fact_4415_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_4416_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_4417_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( euclid4440199948858584721cancel(A)
=> ! [A5: set(B),F: fun(B,A),B2: A] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( 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),A5)),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_fd(fun(B,A),fun(A,fun(B,A)),F),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),collect(B,aa(A,fun(B,bool),aTP_Lamp_fe(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),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),collect(B,aa(A,fun(B,bool),aTP_Lamp_ff(fun(B,A),fun(A,fun(B,bool)),F),B2))))),B2)) ) ) ) ).
% sum_div_partition
tff(fact_4418_one__diff__power__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: 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),X),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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_ord_lessThan(nat,N))) ) ).
% one_diff_power_eq
tff(fact_4419_power__diff__1__eq,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: 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),X),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),X),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),X)),set_ord_lessThan(nat,N))) ) ).
% power_diff_1_eq
tff(fact_4420_geometric__sum,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,N: nat] :
( ( X != 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),X)),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),X),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).
% geometric_sum
tff(fact_4421_set__encode__def,axiom,
nat_set_encode = 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),bit0(one2)))) ).
% set_encode_def
tff(fact_4422_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_fg(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_4423_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),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_el(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).
% sum.atMost_shift
tff(fact_4424_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_fh(A,fun(nat,A),A3)),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_4425_sum__list__sum__nth,axiom,
! [B: $tType] :
( comm_monoid_add(B)
=> ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% sum_list_sum_nth
tff(fact_4426_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fi(nat,fun(nat,fun(nat,nat)),M),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).
% sum_choose_diagonal
tff(fact_4427_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_fj(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R)),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_4428_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ordere8940638589300402666id_add(B)
=> ! [B5: set(A),A5: set(A),B2: A,F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( pp(member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F,B2)))
=> ( ! [X2: A] :
( pp(member(A,X2,B5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),B5))) ) ) ) ) ) ) ).
% sum_strict_mono2
tff(fact_4429_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add(B)
& semiring_1(B) )
=> ! [I: C,A5: set(C),F: fun(C,B)] :
( pp(member(C,I,A5))
=> ( ! [X2: C] :
( pp(member(C,X2,aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A5),insert(C,I,bot_bot(set(C))))))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F,X2))) )
=> ( pp(aa(set(C),bool,finite_finite2(C),A5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F,I)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A5))) ) ) ) ) ).
% member_le_sum
tff(fact_4430_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( linordered_field(A)
=> ! [A5: set(B),F: fun(B,A),K5: A] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),aa(A,A,aa(A,fun(A,A),divide_divide(A),K5),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))))) )
=> ( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( A5 != bot_bot(set(B)) )
=> 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),groups7311177749621191930dd_sum(B,A),F),A5)),K5)) ) ) ) ) ).
% sum_bounded_above_divide
tff(fact_4431_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add(A)
& semiring_1(A) )
=> ! [A5: set(B),F: fun(B,A),K5: A] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I4)),K5)) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A5)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A5))),K5))) ) ) ) ).
% sum_bounded_above_strict
tff(fact_4432_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( linordered_idom(B)
=> ! [I6: set(A),X: fun(A,B),A3: fun(A,B),B2: B,Delta: B] :
( ! [I4: A] :
( pp(member(A,I4,I6))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I4))) )
=> ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I6) = one_one(B) )
=> ( ! [I4: A] :
( pp(member(A,I4,I6))
=> 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,I4)),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_fk(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I6)),B2))),Delta)) ) ) ) ) ).
% convex_sum_bound_le
tff(fact_4433_sum__fun__comp,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_1(C)
=> ! [S3: set(A),R2: set(B),G: fun(A,B),F: fun(B,C)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( pp(aa(set(B),bool,finite_finite2(B),R2))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,G,S3)),R2))
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_fl(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_fn(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F)),R2) ) ) ) ) ) ).
% sum_fun_comp
tff(fact_4434_sum__gp__basic,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: 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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),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),X),aa(nat,nat,suc,N))) ) ).
% sum_gp_basic
tff(fact_4435_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_fo(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_fo(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).
% sum_natinterval_diff
tff(fact_4436_sum__gp__strict,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N: nat] :
( ( ( X = 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),X)),set_ord_lessThan(nat,N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
& ( ( X != 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),X)),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),X),N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).
% sum_gp_strict
tff(fact_4437_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: 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),X),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),X),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_fp(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,N)))) ) ).
% diff_power_eq_sum
tff(fact_4438_power__diff__sumr2,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: 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),X),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),X),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_fq(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),set_ord_lessThan(nat,N))) ) ).
% power_diff_sumr2
tff(fact_4439_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_fr(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_4440_sum__power__shift,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M: nat,N: nat,X: 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),X)),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),X),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),X)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ).
% sum_power_shift
tff(fact_4441_take__bit__sum,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fs(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).
% take_bit_sum
tff(fact_4442_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_ft(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N)),set_ord_atMost(nat,N)) ).
% binomial
tff(fact_4443_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),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),bit0(one2)))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N))) ) ).
% mask_eq_sum_exp
tff(fact_4444_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [M: nat,N: nat,X: 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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),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),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).
% sum_gp_multiplied
tff(fact_4445_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),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.in_pairs
tff(fact_4446_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( monoid_mult(A)
& comm_ring(A) )
=> ! [X: 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),X),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)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fv(A,fun(nat,fun(nat,A)),X),N)),set_ord_lessThan(nat,N))) ) ).
% one_diff_power_eq'
tff(fact_4447_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),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),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fu(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% sum.in_pairs_0
tff(fact_4448_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_fw(A,fun(nat,A),A3)),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_4449_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_fx(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% binomial_ring
tff(fact_4450_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_fy(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),set_ord_atMost(nat,N)) ) ).
% pochhammer_binomial_sum
tff(fact_4451_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_fz(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum
tff(fact_4452_choose__square__sum,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ga(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N) ).
% choose_square_sum
tff(fact_4453_sum_Ozero__middle,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [P3: nat,K: nat,G: fun(nat,A),H2: 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_gb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),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_gc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),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_4454_mask__eq__sum__exp__nat,axiom,
! [N: nat] : 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),bit0(one2))),N)),aa(nat,nat,suc,zero_zero(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),bit0(one2)))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N))) ).
% mask_eq_sum_exp_nat
tff(fact_4455_gauss__sum__nat,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(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),bit0(one2))) ).
% gauss_sum_nat
tff(fact_4456_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_gd(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_4457_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,A3: A,X: 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_ge(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),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_gf(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),set_ord_atMost(nat,M)) ) ).
% gbinomial_partial_sum_poly
tff(fact_4458_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),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),bit0(one2))),K)),one_one(nat)) ).
% sum_power2
tff(fact_4459_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_cp(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),bit0(one2))) ).
% Sum_Ico_nat
tff(fact_4460_card__lists__length__le,axiom,
! [A: $tType,A5: set(A),N: nat] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_cu(set(A),fun(nat,fun(list(A),bool)),A5),N))) = 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(set(A),nat,finite_card(A),A5))),set_ord_atMost(nat,N)) ) ) ).
% card_lists_length_le
tff(fact_4461_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),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_4462_double__arith__series,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A3: A,D2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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_gg(A,fun(A,fun(nat,A)),A3),D2)),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),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D2))) ) ).
% double_arith_series
tff(fact_4463_sum__gp0,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,N: nat] :
( ( ( X = 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),X)),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))) ) )
& ( ( X != 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),X)),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),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).
% sum_gp0
tff(fact_4464_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_gh(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_linear_sum
tff(fact_4465_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [M: nat,A3: A,X: 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_ge(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),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_gi(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y)),set_ord_atMost(nat,M)) ) ).
% gbinomial_partial_sum_poly_xpos
tff(fact_4466_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_gj(nat,fun(nat,A),M)),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) ) ).
% gbinomial_sum_nat_pow2
tff(fact_4467_arith__series__nat,axiom,
! [A3: nat,D2: 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_gk(nat,fun(nat,fun(nat,nat)),A3),D2)),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),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D2)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% arith_series_nat
tff(fact_4468_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_cp(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),bit0(one2))) ).
% Sum_Icc_nat
tff(fact_4469_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),bit0(one2))),M)),one_one(nat)))),set_ord_atMost(nat,M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).
% binomial_r_part_sum
tff(fact_4470_choose__linear__sum,axiom,
! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_gl(nat,fun(nat,nat),N)),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),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).
% choose_linear_sum
tff(fact_4471_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),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_4472_arith__series,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [A3: A,D2: 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_gm(A,fun(A,fun(nat,A)),A3),D2)),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),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).
% arith_series
tff(fact_4473_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),bit0(one2))) ) ).
% gauss_sum
tff(fact_4474_sum__gp__offset,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X = 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),X)),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)) ) )
& ( ( X != 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),X)),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),X),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),X),aa(nat,nat,suc,N))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).
% sum_gp_offset
tff(fact_4475_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_gn(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).
% choose_alternating_sum
tff(fact_4476_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),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_go(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_4477_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),bit0(one2))) ) ).
% gauss_sum_from_Suc_0
tff(fact_4478_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [R: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gp(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(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(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,M))) ) ).
% gchoose_row_sum_weighted
tff(fact_4479_sum__gp,axiom,
! [A: $tType] :
( ( division_ring(A)
& comm_ring(A) )
=> ! [N: nat,M: nat,X: 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),X)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( ( ( X = 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),X)),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)) ) )
& ( ( X != 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),X)),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),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).
% sum_gp
tff(fact_4480_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),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).
% gbinomial_r_part_sum
tff(fact_4481_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_gp(A,fun(nat,A),A3)),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),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_4482_choose__odd__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(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gq(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).
% choose_odd_sum
tff(fact_4483_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_dq(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_gr(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_4484_Array__Time_Onth__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat] : array_nth(A,A3,I) = heap_Time_guard(A,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_gs(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_gt(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_4485_upd__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X: A,A3: array(A)] : array_upd(A,I,X,A3) = heap_Time_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_dq(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_gu(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))))),I),X),A3)) ) ).
% upd_def
tff(fact_4486_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,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_gv(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).
% nat_of_integer_code
tff(fact_4487_int__sum,axiom,
! [B: $tType,F: fun(B,nat),A5: set(B)] : aa(nat,int,semiring_1_of_nat(int),aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),F),A5)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),aTP_Lamp_ax(fun(B,nat),fun(B,int),F)),A5) ).
% int_sum
tff(fact_4488_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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_gw(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_gy(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% sum.triangle_reindex_eq
tff(fact_4489_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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_gz(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_gy(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).
% sum.triangle_reindex
tff(fact_4490_sum__subtractf__nat,axiom,
! [A: $tType,A5: set(A),G: fun(A,nat),F: fun(A,nat)] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G,X2)),aa(A,nat,F,X2))) )
=> ( 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_ha(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F)),A5) = 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),F),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A5)) ) ) ).
% sum_subtractf_nat
tff(fact_4491_card__eq__sum,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_hb(A,nat)),A5) ).
% card_eq_sum
tff(fact_4492_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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_gw(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_hd(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).
% prod.triangle_reindex_eq
tff(fact_4493_sum__eq__Suc0__iff,axiom,
! [A: $tType,A5: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A5) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ? [X5: A] :
( pp(member(A,X5,A5))
& ( aa(A,nat,F,X5) = aa(nat,nat,suc,zero_zero(nat)) )
& ! [Xa2: A] :
( pp(member(A,Xa2,A5))
=> ( ( X5 != Xa2 )
=> ( aa(A,nat,F,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
tff(fact_4494_sum__SucD,axiom,
! [A: $tType,F: fun(A,nat),A5: set(A),N: nat] :
( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A5) = aa(nat,nat,suc,N) )
=> ? [X2: A] :
( pp(member(A,X2,A5))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F,X2))) ) ) ).
% sum_SucD
tff(fact_4495_sum__eq__1__iff,axiom,
! [A: $tType,A5: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A5) = one_one(nat) )
<=> ? [X5: A] :
( pp(member(A,X5,A5))
& ( aa(A,nat,F,X5) = one_one(nat) )
& ! [Xa2: A] :
( pp(member(A,Xa2,A5))
=> ( ( X5 != Xa2 )
=> ( aa(A,nat,F,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).
% sum_eq_1_iff
tff(fact_4496_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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_gz(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_hd(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).
% prod.triangle_reindex
tff(fact_4497_sum__Suc,axiom,
! [A: $tType,F: fun(A,nat),A5: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ca(fun(A,nat),fun(A,nat),F)),A5) = 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),A5)),aa(set(A),nat,finite_card(A),A5)) ).
% sum_Suc
tff(fact_4498_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set(A),T5: set(B),R2: fun(A,fun(B,bool)),K: nat] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( pp(aa(set(B),bool,finite_finite2(B),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,T5))
=> ( aa(set(A),nat,finite_card(A),collect(A,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_he(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S3),R2),X2))) = K ) )
=> ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_hg(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T5),R2)),S3) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T5)) ) ) ) ) ).
% sum_multicount
tff(fact_4499_sum__diff__distrib,axiom,
! [A: $tType] :
( ord(A)
=> ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
( ! [X2: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X2)),aa(A,nat,P,X2)))
=> ( 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),set_ord_lessThan(A,N))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),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_hh(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,N)) ) ) ) ).
% sum_diff_distrib
tff(fact_4500_prod__encode__def,axiom,
nat_prod_encode = product_case_prod(nat,nat,nat,aTP_Lamp_hi(nat,fun(nat,nat))) ).
% prod_encode_def
tff(fact_4501_rat__less__code,axiom,
! [P3: rat,Q4: rat] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P3),Q4))
<=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_hk(rat,fun(int,fun(int,bool)),Q4)),quotient_of(P3))) ) ).
% rat_less_code
tff(fact_4502_rat__less__eq__code,axiom,
! [P3: rat,Q4: rat] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P3),Q4))
<=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aTP_Lamp_hm(rat,fun(int,fun(int,bool)),Q4)),quotient_of(P3))) ) ).
% rat_less_eq_code
tff(fact_4503_rat__uminus__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_hn(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).
% rat_uminus_code
tff(fact_4504_lenlex__conv,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_ho(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lenlex_conv
tff(fact_4505_Divides_Oadjust__div__def,axiom,
! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,product_case_prod(int,int,int,aTP_Lamp_hp(int,fun(int,int))),Qr) ).
% Divides.adjust_div_def
tff(fact_4506_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_hq(fun(A,nat),fun(list(A),fun(A,nat)),F),Xs)),set2(A,Xs)) ).
% sum_list_map_eq_sum_count
tff(fact_4507_rat__divide__code,axiom,
! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_hs(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).
% rat_divide_code
tff(fact_4508_rat__times__code,axiom,
! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_hu(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).
% rat_times_code
tff(fact_4509_rat__plus__code,axiom,
! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_hw(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).
% rat_plus_code
tff(fact_4510_rat__minus__code,axiom,
! [P3: rat,Q4: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P3),Q4)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_hy(rat,fun(int,fun(int,product_prod(int,int))),Q4)),quotient_of(P3)) ).
% rat_minus_code
tff(fact_4511_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list(A),X6: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set2(A,Xs)),X6))
=> ( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs)) = 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_hz(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F)),X6) ) ) ) ).
% sum_list_map_eq_sum_count2
tff(fact_4512_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ia(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).
% divmod_step_nat_def
tff(fact_4513_rat__inverse__code,axiom,
! [P3: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P3)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ib(int,fun(int,product_prod(int,int)))),quotient_of(P3)) ).
% rat_inverse_code
tff(fact_4514_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),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ic(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).
% divmod_step_int_def
tff(fact_4515_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_ay(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),bit0(one2))) ) ) ).
% Sum_Icc_int
tff(fact_4516_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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_id(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).
% divmod_step_integer_def
tff(fact_4517_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,product_case_prod(code_integer,code_integer,num,aTP_Lamp_ie(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).
% num_of_integer_code
tff(fact_4518_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_ig(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_4519_new__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A] : array_new(A,N,X) = 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_ih(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),X)) ) ).
% new_def
tff(fact_4520_of__list__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_ij(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).
% of_list_def
tff(fact_4521_num__of__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,num,code_num_of_integer,aa(int,code_integer,code_integer_of_int,X)) = aa(nat,num,num_of_nat,aa(int,nat,nat2,X)) ).
% num_of_integer.abs_eq
tff(fact_4522_return__def,axiom,
! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),X) = heap_Time_heap(A,aTP_Lamp_ik(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% return_def
tff(fact_4523_ureturn__def,axiom,
! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X) = heap_Time_heap(A,aTP_Lamp_il(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% ureturn_def
tff(fact_4524_effect__heapE,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,heap_Time_heap(A,F),H2,H,R,N)
=> ~ ( ( 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H2))) )
=> ( ( 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,H2))) )
=> ( 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,H2)) ) ) ) ) ).
% effect_heapE
tff(fact_4525_effect__heapI,axiom,
! [A: $tType,N: nat,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H2: heap_ext(product_unit),H: 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,H2))) )
=> ( ( 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H2))) )
=> ( ( 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,H2)) )
=> heap_Time_effect(A,heap_Time_heap(A,F),H2,H,R,N) ) ) ) ).
% effect_heapI
tff(fact_4526_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),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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_im(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_4527_int__ge__less__than__def,axiom,
! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_in(int,fun(int,fun(int,bool)),D2))) ).
% int_ge_less_than_def
tff(fact_4528_int__ge__less__than2__def,axiom,
! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_io(int,fun(int,fun(int,bool)),D2))) ).
% int_ge_less_than2_def
tff(fact_4529_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),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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_ip(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),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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_iq(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_4530_freeze__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : array_freeze(A,A3) = heap_Time_heap(list(A),aTP_Lamp_ir(array(A),fun(heap_ext(product_unit),product_prod(list(A),product_prod(heap_ext(product_unit),nat))),A3)) ) ).
% freeze_def
tff(fact_4531_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,product_case_prod(code_integer,code_integer,int,aTP_Lamp_is(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ) ) ).
% int_of_integer_code
tff(fact_4532_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_it(A,A),N,zero_zero(A)) ) ).
% of_nat_code
tff(fact_4533_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_iu(product_prod(int,int),product_prod(int,int))) ).
% inverse_rat_def
tff(fact_4534_int__of__integer__inverse,axiom,
! [X: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,X)) = X ).
% int_of_integer_inverse
tff(fact_4535_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_4536_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_4537_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_4538_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_min(code_integer),K),L)) = aa(int,int,aa(int,fun(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_4539_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_4540_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_4541_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_4542_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_4543_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% plus_integer.rep_eq
tff(fact_4544_uminus__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),X)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,X)) ).
% uminus_integer.rep_eq
tff(fact_4545_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% times_integer.rep_eq
tff(fact_4546_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_4547_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% minus_integer.rep_eq
tff(fact_4548_abs__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,abs_abs(code_integer),X)) = aa(int,int,abs_abs(int),aa(code_integer,int,code_int_of_integer,X)) ).
% abs_integer.rep_eq
tff(fact_4549_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% divide_integer.rep_eq
tff(fact_4550_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% modulo_integer.rep_eq
tff(fact_4551_sgn__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,sgn_sgn(code_integer),X)) = aa(int,int,sgn_sgn(int),aa(code_integer,int,code_int_of_integer,X)) ).
% sgn_integer.rep_eq
tff(fact_4552_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_4553_division__segment__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,euclid7384307370059645450egment(code_integer),X)) = aa(int,int,euclid7384307370059645450egment(int),aa(code_integer,int,code_int_of_integer,X)) ).
% division_segment_integer.rep_eq
tff(fact_4554_euclidean__size__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,nat,euclid6346220572633701492n_size(code_integer),X) = aa(int,nat,euclid6346220572633701492n_size(int),aa(code_integer,int,code_int_of_integer,X)) ).
% euclidean_size_integer.rep_eq
tff(fact_4555_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),K),L)) = aa(int,int,aa(int,fun(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_4556_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_4557_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_4558_int__of__integer__inject,axiom,
! [X: code_integer,Y: code_integer] :
( ( aa(code_integer,int,code_int_of_integer,X) = aa(code_integer,int,code_int_of_integer,Y) )
<=> ( X = Y ) ) ).
% int_of_integer_inject
tff(fact_4559_less__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Xa4))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4))) ) ).
% less_integer.rep_eq
tff(fact_4560_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_4561_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_4562_less__eq__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: code_integer] :
( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),X),Xa4))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4))) ) ).
% less_eq_integer.rep_eq
tff(fact_4563_nat__of__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,nat,code_nat_of_integer,X) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).
% nat_of_integer.rep_eq
tff(fact_4564_take__bit__integer_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% take_bit_integer.rep_eq
tff(fact_4565_not__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),X)) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(code_integer,int,code_int_of_integer,X)) ).
% not_integer.rep_eq
tff(fact_4566_and__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% and_integer.rep_eq
tff(fact_4567_bit__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,fun(nat,bool),bit_se5641148757651400278ts_bit(code_integer),X) = aa(int,fun(nat,bool),bit_se5641148757651400278ts_bit(int),aa(code_integer,int,code_int_of_integer,X)) ).
% bit_integer.rep_eq
tff(fact_4568_or__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% or_integer.rep_eq
tff(fact_4569_xor__integer_Orep__eq,axiom,
! [X: code_integer,Xa4: 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),X),Xa4)) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% xor_integer.rep_eq
tff(fact_4570_push__bit__integer_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% push_bit_integer.rep_eq
tff(fact_4571_drop__bit__integer_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% drop_bit_integer.rep_eq
tff(fact_4572_mask__integer_Orep__eq,axiom,
! [X: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),X)) = aa(nat,int,bit_se2239418461657761734s_mask(int),X) ).
% mask_integer.rep_eq
tff(fact_4573_unset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% unset_bit_integer.rep_eq
tff(fact_4574_set__bit__integer_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% set_bit_integer.rep_eq
tff(fact_4575_flip__bit__integer_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),X),aa(code_integer,int,code_int_of_integer,Xa4)) ).
% flip_bit_integer.rep_eq
tff(fact_4576_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Inc: fun(A,A),N: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).
% of_nat_aux.simps(2)
tff(fact_4577_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Inc: fun(A,A),I: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I) = I ) ).
% of_nat_aux.simps(1)
tff(fact_4578_num__of__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,num,code_num_of_integer,X) = aa(nat,num,num_of_nat,aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X))) ).
% num_of_integer.rep_eq
tff(fact_4579_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_iv(product_prod(int,int),product_prod(int,int))) ).
% uminus_rat_def
tff(fact_4580_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_iw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% plus_rat_def
tff(fact_4581_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_ix(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).
% times_rat_def
tff(fact_4582_transpose__rectangle,axiom,
! [A: $tType,Xs: list(list(A)),N: nat] :
( ( ( Xs = nil(list(A)) )
=> ( N = zero_zero(nat) ) )
=> ( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
=> ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I4)) = N ) )
=> ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_iz(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),N)) ) ) ) ).
% transpose_rectangle
tff(fact_4583_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ordere6911136660526730532id_add(B)
=> ! [F: fun(nat,B),Ns: list(nat)] :
( ! [X2: nat,Y4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F,X2)),aa(nat,B,F,Y4))) )
=> ( sorted_wrt(nat,ord_less(nat),Ns)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(B),B,groups8242544230860333062m_list(B),map(nat,B,F,Ns)))) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4584_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).
% strict_sorted_imp_sorted
tff(fact_4585_sorted2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A,Zs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),cons(A,X,cons(A,Y,Zs)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
& sorted_wrt(A,ord_less_eq(A),cons(A,Y,Zs)) ) ) ) ).
% sorted2
tff(fact_4586_sorted0,axiom,
! [A: $tType] :
( linorder(A)
=> sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).
% sorted0
tff(fact_4587_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> sorted_wrt(A,ord_less(A),nil(A)) ) ).
% strict_sorted_simps(1)
tff(fact_4588_strict__sorted__equal,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A)] :
( sorted_wrt(A,ord_less(A),Xs)
=> ( sorted_wrt(A,ord_less(A),Ys2)
=> ( ( set2(A,Ys2) = set2(A,Xs) )
=> ( Ys2 = Xs ) ) ) ) ) ).
% strict_sorted_equal
tff(fact_4589_sorted__take,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),N: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),take(A,N,Xs)) ) ) ).
% sorted_take
tff(fact_4590_sorted__replicate,axiom,
! [A: $tType] :
( linorder(A)
=> ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,N,X)) ) ).
% sorted_replicate
tff(fact_4591_sorted__drop,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),N: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),drop(A,N,Xs)) ) ) ).
% sorted_drop
tff(fact_4592_sorted__remdups__adj,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).
% sorted_remdups_adj
tff(fact_4593_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M,N)) ).
% sorted_wrt_upt
tff(fact_4594_sorted__upt,axiom,
! [M: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,N)) ).
% sorted_upt
tff(fact_4595_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A5)) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_4596_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A5)) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_4597_sorted__map,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] :
( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
<=> sorted_wrt(B,aTP_Lamp_ja(fun(B,A),fun(B,fun(B,bool)),F),Xs) ) ) ).
% sorted_map
tff(fact_4598_sorted1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : sorted_wrt(A,ord_less_eq(A),cons(A,X,nil(A))) ) ).
% sorted1
tff(fact_4599_sorted__simps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Ys2: list(A)] :
( sorted_wrt(A,ord_less_eq(A),cons(A,X,Ys2))
<=> ( ! [X5: A] :
( pp(member(A,X5,set2(A,Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) )
& sorted_wrt(A,ord_less_eq(A),Ys2) ) ) ) ).
% sorted_simps(2)
tff(fact_4600_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Ys2: list(A)] :
( sorted_wrt(A,ord_less(A),cons(A,X,Ys2))
<=> ( ! [X5: A] :
( pp(member(A,X5,set2(A,Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X5)) )
& sorted_wrt(A,ord_less(A),Ys2) ) ) ) ).
% strict_sorted_simps(2)
tff(fact_4601_sorted__append,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A)] :
( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys2))
<=> ( sorted_wrt(A,ord_less_eq(A),Xs)
& sorted_wrt(A,ord_less_eq(A),Ys2)
& ! [X5: A] :
( pp(member(A,X5,set2(A,Xs)))
=> ! [Xa2: A] :
( pp(member(A,Xa2,set2(A,Ys2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Xa2)) ) ) ) ) ) ).
% sorted_append
tff(fact_4602_strict__sorted__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [L: list(A)] :
( sorted_wrt(A,ord_less(A),L)
<=> ( sorted_wrt(A,ord_less_eq(A),L)
& distinct(A,L) ) ) ) ).
% strict_sorted_iff
tff(fact_4603_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( distinct(A,Xs)
=> ( sorted_wrt(A,ord_less_eq(A),Ys2)
=> ( distinct(A,Ys2)
=> ( ( set2(A,Xs) = set2(A,Ys2) )
=> ( Xs = Ys2 ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
tff(fact_4604_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_4605_sorted__wrt__iff__nth__less,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( sorted_wrt(A,P,Xs)
<=> ! [I5: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I5)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ).
% sorted_wrt_iff_nth_less
tff(fact_4606_sorted__wrt__nth__less,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I: nat,J: nat] :
( sorted_wrt(A,P,Xs)
=> ( 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)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).
% sorted_wrt_nth_less
tff(fact_4607_sorted__butlast,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( ( Xs != nil(A) )
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),butlast(A,Xs)) ) ) ) ).
% sorted_butlast
tff(fact_4608_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,N,Xs))) ).
% sorted_enumerate
tff(fact_4609_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_4610_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ? [X2: list(A)] :
( ( set2(A,X2) = A5 )
& sorted_wrt(A,ord_less_eq(A),X2)
& distinct(A,X2)
& ! [Y3: list(A)] :
( ( ( set2(A,Y3) = A5 )
& sorted_wrt(A,ord_less_eq(A),Y3)
& distinct(A,Y3) )
=> ( Y3 = X2 ) ) ) ) ) ).
% finite_sorted_distinct_unique
tff(fact_4611_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I5: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I5)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).
% sorted_iff_nth_mono_less
tff(fact_4612_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( distinct(A,Xs)
=> ( linord4507533701916653071of_set(A,set2(A,Xs)) = Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_4613_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ~ ! [L2: list(A)] :
( sorted_wrt(A,ord_less(A),L2)
=> ( ( set2(A,L2) = A5 )
=> ( aa(list(A),nat,size_size(list(A)),L2) != aa(set(A),nat,finite_card(A),A5) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
tff(fact_4614_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I5)),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I5)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I5)))) ) ) ) ).
% sorted_iff_nth_Suc
tff(fact_4615_sorted__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( 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),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).
% sorted_nth_mono
tff(fact_4616_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
<=> ! [I5: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I5),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I5)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).
% sorted_iff_nth_mono
tff(fact_4617_sorted__wrt__less__idx,axiom,
! [Ns: list(nat),I: nat] :
( sorted_wrt(nat,ord_less(nat),Ns)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I))) ) ) ).
% sorted_wrt_less_idx
tff(fact_4618_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),L: list(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( sorted_wrt(A,ord_less(A),L)
& ( set2(A,L) = A5 )
& ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A5) ) )
<=> ( linord4507533701916653071of_set(A,A5) = L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_4619_length__transpose__sorted,axiom,
! [A: $tType,Xs: list(list(A))] :
( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
=> ( ( ( Xs = nil(list(A)) )
=> ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
& ( ( Xs != nil(list(A)) )
=> ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).
% length_transpose_sorted
tff(fact_4620_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [F3: set(A),I6: set(A),F: fun(A,B),I: A] :
( pp(aa(set(A),bool,finite_finite2(A),F3))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_jb(set(A),fun(fun(A,B),fun(A,bool)),I6),F))),F3))
=> ( ( pp(member(A,I,I6))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),insert(A,I,bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I6)),aa(A,B,F,I)) ) )
& ( ~ pp(member(A,I,I6))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),insert(A,I,bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I6) ) ) ) ) ) ) ).
% sum_diff1'_aux
tff(fact_4621_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A)] :
( pp(aa(set(A),bool,finite_finite2(A),image(nat,A,F,top_top(set(nat)))))
=> ( pp(aa(fun(nat,A),bool,order_mono(nat,A),F))
=> ( ! [N3: nat] :
( ( aa(nat,A,F,N3) = aa(nat,A,F,aa(nat,nat,suc,N3)) )
=> ( aa(nat,A,F,aa(nat,nat,suc,N3)) = aa(nat,A,F,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) )
=> ? [N7: nat] :
( ! [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N4),N7))
=> ! [M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N7))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,M2)),aa(nat,A,F,N4))) ) ) )
& ! [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N4))
=> ( aa(nat,A,F,N7) = aa(nat,A,F,N4) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
tff(fact_4622_top__apply,axiom,
! [D: $tType,C: $tType] :
( top(C)
=> ! [X: D] : aa(D,C,top_top(fun(D,C)),X) = top_top(C) ) ).
% top_apply
tff(fact_4623_inf__top__left,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),X) = X ) ).
% inf_top_left
tff(fact_4624_inf__top__right,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).
% inf_top_right
tff(fact_4625_inf__eq__top__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = top_top(A) )
<=> ( ( X = top_top(A) )
& ( Y = top_top(A) ) ) ) ) ).
% inf_eq_top_iff
tff(fact_4626_top__eq__inf__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [X: A,Y: A] :
( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) )
<=> ( ( X = top_top(A) )
& ( Y = top_top(A) ) ) ) ) ).
% top_eq_inf_iff
tff(fact_4627_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = top_top(A) )
<=> ( ( A3 = top_top(A) )
& ( B2 = top_top(A) ) ) ) ) ).
% inf_top.eq_neutr_iff
tff(fact_4628_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A3) = A3 ) ).
% inf_top.left_neutral
tff(fact_4629_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A,B2: A] :
( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) )
<=> ( ( A3 = top_top(A) )
& ( B2 = top_top(A) ) ) ) ) ).
% inf_top.neutr_eq_iff
tff(fact_4630_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),top_top(A)) = A3 ) ).
% inf_top.right_neutral
tff(fact_4631_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).
% boolean_algebra.disj_one_right
tff(fact_4632_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).
% boolean_algebra.disj_one_left
tff(fact_4633_sup__top__right,axiom,
! [A: $tType] :
( bounded_lattice_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),top_top(A)) = top_top(A) ) ).
% sup_top_right
tff(fact_4634_sup__top__left,axiom,
! [A: $tType] :
( bounded_lattice_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),X) = top_top(A) ) ).
% sup_top_left
tff(fact_4635_max__top,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),X) = top_top(A) ) ).
% max_top
tff(fact_4636_max__top2,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),X),top_top(A)) = top_top(A) ) ).
% max_top2
tff(fact_4637_min__top2,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),X),top_top(A)) = X ) ).
% min_top2
tff(fact_4638_min__top,axiom,
! [A: $tType] :
( order_top(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),X) = X ) ).
% min_top
tff(fact_4639_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).
% boolean_algebra.compl_zero
tff(fact_4640_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).
% boolean_algebra.compl_one
tff(fact_4641_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,uminus_uminus(A),X)) = top_top(A) ) ).
% boolean_algebra.disj_cancel_right
tff(fact_4642_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),X) = top_top(A) ) ).
% boolean_algebra.disj_cancel_left
tff(fact_4643_sup__compl__top__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),Y)) = top_top(A) ) ).
% sup_compl_top_left2
tff(fact_4644_sup__compl__top__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = top_top(A) ) ).
% sup_compl_top_left1
tff(fact_4645_finite__compl,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A5)))
<=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).
% finite_compl
tff(fact_4646_Gcd__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).
% Gcd_UNIV
tff(fact_4647_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [P3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),bot_bot(set(B))) = zero_zero(A) ) ).
% sum.empty'
tff(fact_4648_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I6: set(B),P3: fun(B,A),I: B] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))))
=> ( ( pp(member(B,I,I6))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),insert(B,I,I6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),I6) ) )
& ( ~ pp(member(B,I,I6))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),insert(B,I,I6)) = 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),I6)) ) ) ) ) ) ).
% sum.insert'
tff(fact_4649_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X),top_top(A)) = X ) ).
% boolean_algebra.conj_one_right
tff(fact_4650_top__greatest,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),top_top(A))) ) ).
% top_greatest
tff(fact_4651_top_Oextremum__unique,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
<=> ( A3 = top_top(A) ) ) ) ).
% top.extremum_unique
tff(fact_4652_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A3))
=> ( A3 = top_top(A) ) ) ) ).
% top.extremum_uniqueI
tff(fact_4653_top_Oextremum__strict,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A3)) ) ).
% top.extremum_strict
tff(fact_4654_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( order_top(A)
=> ! [A3: A] :
( ( A3 != top_top(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),top_top(A))) ) ) ).
% top.not_eq_extremum
tff(fact_4655_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),I6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),collect(B,aa(set(B),fun(B,bool),aTP_Lamp_jc(fun(B,A),fun(set(B),fun(B,bool)),G),I6))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I6) ) ).
% sum.non_neutral'
tff(fact_4656_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),top_top(set(fun(A,B)))))
=> ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).
% finite_fun_UNIVD1
tff(fact_4657_sorted__upto,axiom,
! [M: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M,N)) ).
% sorted_upto
tff(fact_4658_sorted__wrt__upto,axiom,
! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).
% sorted_wrt_upto
tff(fact_4659_sup__cancel__left2,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),B2)) = top_top(A) ) ).
% sup_cancel_left2
tff(fact_4660_sup__cancel__left1,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),X)),B2)) = top_top(A) ) ).
% sup_cancel_left1
tff(fact_4661_Compl__empty__eq,axiom,
! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).
% Compl_empty_eq
tff(fact_4662_Compl__UNIV__eq,axiom,
! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).
% Compl_UNIV_eq
tff(fact_4663_Compl__partition2,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),A5) = top_top(set(A)) ).
% Compl_partition2
tff(fact_4664_Compl__partition,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = top_top(set(A)) ).
% Compl_partition
tff(fact_4665_Compl__eq__Diff__UNIV,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A5) ).
% Compl_eq_Diff_UNIV
tff(fact_4666_sup__shunt,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).
% sup_shunt
tff(fact_4667_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [A3: A,X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),X) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),X) = top_top(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Y) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Y) = top_top(A) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
tff(fact_4668_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A5: set(B)] :
( ( image(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)),image(B,A,F,A5))),image(B,A,F,aa(set(B),set(B),uminus_uminus(set(B)),A5)))) ) ).
% surj_Compl_image_subset
tff(fact_4669_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S3: 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)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T5) ) ) ) ) ).
% sum.mono_neutral_left'
tff(fact_4670_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S3: 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)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S3) ) ) ) ) ).
% sum.mono_neutral_right'
tff(fact_4671_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(B),T5: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [I4: B] :
( pp(member(B,I4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,H2,I4) = zero_zero(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H2),T5) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
tff(fact_4672_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(B),T5: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = zero_zero(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H2),S3) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
tff(fact_4673_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ! [X: A,Y: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
=> ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
=> ( aa(A,A,uminus_uminus(A),X) = Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_4674_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I6: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),G))))
=> ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),H2))))
=> ( 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_jd(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),I6) = 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),I6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H2),I6)) ) ) ) ) ).
% sum.distrib'
tff(fact_4675_sum_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [I6: set(B),P3: fun(B,A)] :
( ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))) ) )
& ( ~ pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ar(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P3),I6) = zero_zero(A) ) ) ) ) ).
% sum.G_def
tff(fact_4676_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).
% finite_UNIV_card_ge_0
tff(fact_4677_sorted__transpose,axiom,
! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),transpose(A,Xs)))) ).
% sorted_transpose
tff(fact_4678_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( 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,rev(A,Xs)),N) = 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)),aa(nat,nat,suc,N))) ) ) ).
% rev_nth
tff(fact_4679_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_4680_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).
% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_4681_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F: fun(B,A)] :
( pp(aa(set(A),bool,finite_finite2(A),image(B,A,F,top_top(set(B)))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),image(B,A,F,top_top(set(B)))))) ) ).
% card_range_greater_zero
tff(fact_4682_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
<=> ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I5)),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I5))),aa(nat,A,nth(A,Xs),I5))) ) ) ) ).
% sorted_rev_iff_nth_Suc
tff(fact_4683_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
<=> ! [I5: nat,J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I5),J3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I5))) ) ) ) ) ).
% sorted_rev_iff_nth_mono
tff(fact_4684_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I: nat,J: nat] :
( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
=> ( 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),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I))) ) ) ) ) ).
% sorted_rev_nth_mono
tff(fact_4685_range__mod,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( image(nat,nat,aTP_Lamp_je(nat,fun(nat,nat),N),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).
% range_mod
tff(fact_4686_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ab_group_add(B)
=> ! [I6: set(A),F: fun(A,B),I: A] :
( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_jb(set(A),fun(fun(A,B),fun(A,bool)),I6),F))))
=> ( ( pp(member(A,I,I6))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),insert(A,I,bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I6)),aa(A,B,F,I)) ) )
& ( ~ pp(member(A,I,I6))
=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I6),insert(A,I,bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I6) ) ) ) ) ) ).
% sum_diff1'
tff(fact_4687_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),filter(list(A),aTP_Lamp_jf(nat,fun(list(A),bool),I),Xs))))
=> ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).
% nth_nth_transpose_sorted
tff(fact_4688_transpose__column,axiom,
! [A: $tType,Xs: list(list(A)),I: nat] :
( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
=> ( map(list(A),A,aTP_Lamp_jg(nat,fun(list(A),A),I),filter(list(A),aTP_Lamp_jf(nat,fun(list(A),bool),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).
% transpose_column
tff(fact_4689_transpose__column__length,axiom,
! [A: $tType,Xs: list(list(A)),I: nat] :
( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
=> ( aa(list(list(A)),nat,size_size(list(list(A))),filter(list(A),aTP_Lamp_jf(nat,fun(list(A),bool),I),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).
% transpose_column_length
tff(fact_4690_card__UNIV__unit,axiom,
aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).
% card_UNIV_unit
tff(fact_4691_infinite__UNIV__int,axiom,
~ pp(aa(set(int),bool,finite_finite2(int),top_top(set(int)))) ).
% infinite_UNIV_int
tff(fact_4692_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y4: int] :
( ( X = aa(int,code_integer,code_integer_of_int,Y4) )
=> ~ pp(member(int,Y4,top_top(set(int)))) ) ).
% integer_of_int_cases
tff(fact_4693_integer__of__int__induct,axiom,
! [P: fun(code_integer,bool),X: 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,X)) ) ).
% integer_of_int_induct
tff(fact_4694_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( pp(member(int,X,top_top(set(int))))
=> ( pp(member(int,Y,top_top(set(int))))
=> ( ( aa(int,code_integer,code_integer_of_int,X) = aa(int,code_integer,code_integer_of_int,Y) )
<=> ( X = Y ) ) ) ) ).
% integer_of_int_inject
tff(fact_4695_int__of__integer,axiom,
! [X: code_integer] : pp(member(int,aa(code_integer,int,code_int_of_integer,X),top_top(set(int)))) ).
% int_of_integer
tff(fact_4696_int__of__integer__cases,axiom,
! [Y: int] :
( pp(member(int,Y,top_top(set(int))))
=> ~ ! [X2: code_integer] : Y != aa(code_integer,int,code_int_of_integer,X2) ) ).
% int_of_integer_cases
tff(fact_4697_int__of__integer__induct,axiom,
! [Y: int,P: fun(int,bool)] :
( pp(member(int,Y,top_top(set(int))))
=> ( ! [X2: code_integer] : pp(aa(int,bool,P,aa(code_integer,int,code_int_of_integer,X2)))
=> pp(aa(int,bool,P,Y)) ) ) ).
% int_of_integer_induct
tff(fact_4698_length__filter__le,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),filter(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_filter_le
tff(fact_4699_sorted__same,axiom,
! [A: $tType] :
( linorder(A)
=> ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter(A,aa(list(A),fun(A,bool),aTP_Lamp_jh(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs),Xs)) ) ).
% sorted_same
tff(fact_4700_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_4701_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
( pp(member(A,X,set2(A,Xs)))
=> ( ~ pp(aa(A,bool,P,X))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),filter(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).
% length_filter_less
tff(fact_4702_surj__list__encode,axiom,
image(list(nat),nat,nat_list_encode,top_top(set(list(nat)))) = top_top(set(nat)) ).
% surj_list_encode
tff(fact_4703_surj__prod__encode,axiom,
image(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).
% surj_prod_encode
tff(fact_4704_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),P: fun(B,bool)] :
( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
=> sorted_wrt(A,ord_less_eq(A),map(B,A,F,filter(B,P,Xs))) ) ) ).
% sorted_filter
tff(fact_4705_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F,filter(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_ji(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F),G),Xs),Xs))) ) ).
% sorted_map_same
tff(fact_4706_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( monoid_add(A)
=> ! [F: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,filter(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_jj(fun(B,A),fun(fun(B,bool),fun(B,A)),F),P),Xs)) ) ).
% sum_list_map_filter'
tff(fact_4707_sum__list__filter__le__nat,axiom,
! [A: $tType,F: fun(A,nat),P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,filter(A,P,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F,Xs)))) ).
% sum_list_filter_le_nat
tff(fact_4708_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( monoid_add(A)
=> ! [Xs: list(B),P: fun(B,bool),F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,set2(B,Xs)))
=> ( ~ pp(aa(B,bool,P,X2))
=> ( aa(B,A,F,X2) = zero_zero(A) ) ) )
=> ( aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,filter(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,F,Xs)) ) ) ) ).
% sum_list_map_filter
tff(fact_4709_Ints__def,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_Ints(A) = image(int,A,ring_1_of_int(A),top_top(set(int))) ) ) ).
% Ints_def
tff(fact_4710_int__in__range__abs,axiom,
! [N: nat] : pp(member(int,aa(nat,int,semiring_1_of_nat(int),N),image(int,int,abs_abs(int),top_top(set(int))))) ).
% int_in_range_abs
tff(fact_4711_length__filter__conv__card,axiom,
! [A: $tType,P3: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter(A,P3,Xs)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_jk(fun(A,bool),fun(list(A),fun(nat,bool)),P3),Xs))) ).
% length_filter_conv_card
tff(fact_4712_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list(list(A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
=> ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = map(list(A),A,aTP_Lamp_jg(nat,fun(list(A),A),I),filter(list(A),aTP_Lamp_jf(nat,fun(list(A),bool),I),Xs)) ) ) ).
% nth_transpose
tff(fact_4713_transpose__max__length,axiom,
! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_jl(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter(list(A),aTP_Lamp_jm(list(A),bool),Xs)) ).
% transpose_max_length
tff(fact_4714_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_jn(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_jo(list(B),fun(nat,nat)),filter(list(B),aTP_Lamp_jp(list(B),bool),Xss),zero_zero(nat)))) ).
% transpose_aux_max
tff(fact_4715_transpose__transpose,axiom,
! [A: $tType,Xs: list(list(A))] :
( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
=> ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_jm(list(A),bool),Xs) ) ) ).
% transpose_transpose
tff(fact_4716_length__takeWhile__le,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_takeWhile_le
tff(fact_4717_sorted__takeWhile,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).
% sorted_takeWhile
tff(fact_4718_takeWhile__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)),takeWhile(A,P,Xs))))
=> ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).
% takeWhile_nth
tff(fact_4719_nth__length__takeWhile,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))))) ) ).
% nth_length_takeWhile
tff(fact_4720_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = foldr(A,A,plus_plus(A),Xs,zero_zero(A)) ) ).
% sum_list.eq_foldr
tff(fact_4721_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) )
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).
% length_takeWhile_less_P_nth
tff(fact_4722_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) ) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) )
=> ( takeWhile(A,P,Xs) = take(A,N,Xs) ) ) ) ).
% takeWhile_eq_take_P_nth
tff(fact_4723_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: fun(nat,bool),Xs: list(A),Is: list(nat)] : map(product_prod(A,nat),A,product_fst(A,nat),filter(product_prod(A,nat),aTP_Lamp_jq(fun(nat,bool),fun(product_prod(A,nat),bool),P),zip(A,nat,Xs,Is))) = map(product_prod(A,nat),A,product_fst(A,nat),filter(product_prod(A,nat),aTP_Lamp_jr(fun(nat,bool),fun(product_prod(A,nat),bool),P),zip(A,nat,Xs,map(nat,nat,suc,Is)))) ).
% nths_shift_lemma_Suc
tff(fact_4724_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) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_js(fun(B,A),fun(A,fun(B,fun(A,A))),F),A3),Xs,zero_zero(A)) ) ).
% horner_sum_foldr
tff(fact_4725_nths__shift__lemma,axiom,
! [A: $tType,A5: set(nat),Xs: list(A),I: nat] : map(product_prod(A,nat),A,product_fst(A,nat),filter(product_prod(A,nat),aTP_Lamp_jt(set(nat),fun(product_prod(A,nat),bool),A5),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)))))) = 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_ju(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A5),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).
% nths_shift_lemma
tff(fact_4726_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),T2: A] :
( sorted_wrt(A,ord_less_eq(A),rev(A,map(B,A,F,Xs)))
=> ( filter(B,aa(A,fun(B,bool),aTP_Lamp_jv(fun(B,A),fun(A,fun(B,bool)),F),T2),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_jv(fun(B,A),fun(A,fun(B,bool)),F),T2),Xs) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
tff(fact_4727_length__transpose,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_jl(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).
% length_transpose
tff(fact_4728_foldr__max__sorted,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Y: A] :
( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
=> ( ( ( Xs = nil(A) )
=> ( foldr(A,A,ord_max(A),Xs,Y) = Y ) )
& ( ( Xs != nil(A) )
=> ( foldr(A,A,ord_max(A),Xs,Y) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y) ) ) ) ) ) ).
% foldr_max_sorted
tff(fact_4729_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num,N: nat] : aa(nat,A,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)),aa(nat,A,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_4730_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_4731_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),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),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))),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))),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),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_jw(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_4732_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: fun(nat,fun(T,T)),Nat: nat] : aa(nat,T,rec_nat(T,F1,F22),aa(nat,nat,suc,Nat)) = aa(T,T,aa(nat,fun(T,T),F22,Nat),aa(nat,T,rec_nat(T,F1,F22),Nat)) ).
% old.nat.simps(7)
tff(fact_4733_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: fun(nat,fun(T,T))] : aa(nat,T,rec_nat(T,F1,F22),zero_zero(nat)) = F1 ).
% old.nat.simps(6)
tff(fact_4734_case__nat__numeral,axiom,
! [A: $tType,A3: A,F: fun(nat,A),V: num] : case_nat(A,A3,F,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F,pred_numeral(V)) ).
% case_nat_numeral
tff(fact_4735_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num] : aa(nat,A,rec_nat(A,A3,F),aa(num,nat,numeral_numeral(nat),V)) = aa(A,A,aa(nat,fun(A,A),F,pred_numeral(V)),aa(nat,A,rec_nat(A,A3,F),pred_numeral(V))) ).
% rec_nat_numeral
tff(fact_4736_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E4: $tType,F: fun(B,A),G: fun(C,B),X: C,F5: fun(D,A),G3: fun(E4,D),X4: E4] :
( ( aa(B,A,F,aa(C,B,G,X)) = aa(D,A,F5,aa(E4,D,G3,X4)) )
=> ( aa(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F),G),X) = aa(E4,A,aa(fun(E4,D),fun(E4,A),comp(D,A,E4,F5),G3),X4) ) ) ).
% comp_cong
tff(fact_4737_nat_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H2: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H2,case_nat(A,F1,F22,Nat)) = case_nat(B,aa(A,B,H2,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_jx(fun(A,B),fun(fun(nat,A),fun(nat,B)),H2),F22),Nat) ).
% nat.case_distrib
tff(fact_4738_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: fun(nat,A),X22: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X22)) = aa(nat,A,F22,X22) ).
% old.nat.simps(5)
tff(fact_4739_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).
% old.nat.simps(4)
tff(fact_4740_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero(nat) )
<=> pp(case_nat(bool,fFalse,aTP_Lamp_jy(nat,bool),Nat)) ) ).
% nat.disc_eq_case(2)
tff(fact_4741_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero(nat) )
<=> pp(case_nat(bool,fTrue,aTP_Lamp_jz(nat,bool),Nat)) ) ).
% nat.disc_eq_case(1)
tff(fact_4742_rec__nat__Suc__imp,axiom,
! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N: nat] :
( ( F = rec_nat(A,F1,F22) )
=> ( aa(nat,A,F,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F22,N),aa(nat,A,F,N)) ) ) ).
% rec_nat_Suc_imp
tff(fact_4743_rec__nat__0__imp,axiom,
! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
( ( F = rec_nat(A,F1,F22) )
=> ( aa(nat,A,F,zero_zero(nat)) = F1 ) ) ).
% rec_nat_0_imp
tff(fact_4744_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add(B)
& comm_monoid_add(A) )
=> ! [H2: fun(B,A),G: fun(C,B),A5: set(C)] :
( ( aa(B,A,H2,zero_zero(B)) = zero_zero(A) )
=> ( ! [X2: B,Y4: B] : aa(B,A,H2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H2,X2)),aa(B,A,H2,Y4))
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H2),G)),A5) = aa(B,A,H2,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A5)) ) ) ) ) ).
% sum_comp_morphism
tff(fact_4745_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),map(B,A,F,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),map(B,A,aa(fun(B,A),fun(B,A),comp(A,A,B,uminus_uminus(A)),F),Xs)) ) ).
% uminus_sum_list_map
tff(fact_4746_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
<=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).
% less_eq_nat.simps(2)
tff(fact_4747_max__Suc1,axiom,
! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_ka(nat,fun(nat,nat),N),M) ).
% max_Suc1
tff(fact_4748_max__Suc2,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_kb(nat,fun(nat,nat),N),M) ).
% max_Suc2
tff(fact_4749_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A5: set(B),H2: fun(B,C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ! [X2: B,Y4: B] :
( pp(member(B,X2,A5))
=> ( pp(member(B,Y4,A5))
=> ( ( X2 != Y4 )
=> ( ( aa(B,C,H2,X2) = aa(B,C,H2,Y4) )
=> ( aa(C,A,G,aa(B,C,H2,X2)) = zero_zero(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),image(B,C,H2,A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H2)),A5) ) ) ) ) ).
% sum.reindex_nontrivial
tff(fact_4750_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),H2: fun(B,C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ! [X2: B,Y4: B] :
( pp(member(B,X2,A5))
=> ( pp(member(B,Y4,A5))
=> ( ( X2 != Y4 )
=> ( ( aa(B,C,H2,X2) = aa(B,C,H2,Y4) )
=> ( aa(C,A,G,aa(B,C,H2,X2)) = one_one(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),image(B,C,H2,A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H2)),A5) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_4751_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ordere6911136660526730532id_add(B)
=> ! [I6: set(C),G: fun(A,B),F: fun(C,A)] :
( pp(aa(set(C),bool,finite_finite2(C),I6))
=> ( ! [I4: C] :
( pp(member(C,I4,I6))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F,I4)))) )
=> 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),image(C,A,F,I6))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),comp(A,B,C,G),F)),I6))) ) ) ) ).
% sum_image_le
tff(fact_4752_diff__Suc,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_cp(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ).
% diff_Suc
tff(fact_4753_min__Suc1,axiom,
! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_kc(nat,fun(nat,nat),N),M) ).
% min_Suc1
tff(fact_4754_min__Suc2,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_kd(nat,fun(nat,nat),N),M) ).
% min_Suc2
tff(fact_4755_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType,N: nat,X: A,F: fun(nat,A)] :
( ( ( N = zero_zero(nat) )
=> ( case_nat(A,X,F,N) = X ) )
& ( ( N != zero_zero(nat) )
=> ( case_nat(A,X,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_4756_nths__def,axiom,
! [A: $tType,Xs: list(A),A5: set(nat)] : nths(A,Xs,A5) = map(product_prod(A,nat),A,product_fst(A,nat),filter(product_prod(A,nat),aTP_Lamp_jt(set(nat),fun(product_prod(A,nat),bool),A5),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).
% nths_def
tff(fact_4757_length__product__lists,axiom,
! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = foldr(nat,nat,times_times(nat),map(list(B),nat,size_size(list(B)),Xss),one_one(nat)) ).
% length_product_lists
tff(fact_4758_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
<=> ~ ( ( ( Nat = zero_zero(nat) )
& ~ pp(aa(A,bool,P,F1)) )
| ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
& ~ pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).
% nat.split_sels(2)
tff(fact_4759_nths__singleton,axiom,
! [A: $tType,A5: set(nat),X: A] :
( ( pp(member(nat,zero_zero(nat),A5))
=> ( nths(A,cons(A,X,nil(A)),A5) = cons(A,X,nil(A)) ) )
& ( ~ pp(member(nat,zero_zero(nat),A5))
=> ( nths(A,cons(A,X,nil(A)),A5) = nil(A) ) ) ) ).
% nths_singleton
tff(fact_4760_card_Ocomp__fun__commute__on,axiom,
aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) = aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) ).
% card.comp_fun_commute_on
tff(fact_4761_nths__all,axiom,
! [A: $tType,Xs: list(A),I6: set(nat)] :
( ! [I4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(member(nat,I4,I6)) )
=> ( nths(A,Xs,I6) = Xs ) ) ).
% nths_all
tff(fact_4762_sorted__nths,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),I6: set(nat)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I6)) ) ) ).
% sorted_nths
tff(fact_4763_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),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_4764_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),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_4765_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),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_4766_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),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_4767_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)),comp(B,heap_Time_Heap(B),A,heap_Time_return(B)),F) ).
% lift_def
tff(fact_4768_drop__eq__nths,axiom,
! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,collect(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).
% drop_eq_nths
tff(fact_4769_euclidean__size__int__def,axiom,
euclid6346220572633701492n_size(int) = aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)) ).
% euclidean_size_int_def
tff(fact_4770_pred__def,axiom,
! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_cp(nat,nat),Nat) ).
% pred_def
tff(fact_4771_length__nths,axiom,
! [A: $tType,Xs: list(A),I6: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I6)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_ke(list(A),fun(set(nat),fun(nat,bool)),Xs),I6))) ).
% length_nths
tff(fact_4772_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),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).
% sum.atLeast0_atMost_Suc_shift
tff(fact_4773_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),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% sum.atLeast0_lessThan_Suc_shift
tff(fact_4774_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),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).
% prod.atLeast0_atMost_Suc_shift
tff(fact_4775_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),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).
% prod.atLeast0_lessThan_Suc_shift
tff(fact_4776_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),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_4777_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),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_4778_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),comp(nat,A,nat,G),aTP_Lamp_cq(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_4779_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),comp(nat,A,nat,G),aTP_Lamp_cq(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_4780_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),comp(nat,A,nat,G),aTP_Lamp_cq(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_4781_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),comp(nat,A,nat,G),aTP_Lamp_cq(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_4782_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),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% sum.atLeast_int_atMost_int_shift
tff(fact_4783_filter__eq__nths,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : filter(A,P,Xs) = nths(A,Xs,collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_jk(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).
% filter_eq_nths
tff(fact_4784_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),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).
% prod.atLeast_int_atMost_int_shift
tff(fact_4785_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),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% sum.atLeast_int_lessThan_int_shift
tff(fact_4786_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),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_4787_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),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_4788_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),comp(int,A,nat,G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).
% prod.atLeast_int_lessThan_int_shift
tff(fact_4789_Gcd__int__def,axiom,
! [K5: set(int)] : gcd_Gcd(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),K5))) ).
% Gcd_int_def
tff(fact_4790_nths__Cons,axiom,
! [A: $tType,X: A,L: list(A),A5: set(nat)] : nths(A,cons(A,X,L),A5) = append(A,if(list(A),member(nat,zero_zero(nat),A5),cons(A,X,nil(A)),nil(A)),nths(A,L,collect(nat,aTP_Lamp_kf(set(nat),fun(nat,bool),A5)))) ).
% nths_Cons
tff(fact_4791_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: fun(A,bool),F1: A,F22: fun(nat,A),Nat: nat] :
( pp(aa(A,bool,P,case_nat(A,F1,F22,Nat)))
<=> ( ( ( Nat = zero_zero(nat) )
=> pp(aa(A,bool,P,F1)) )
& ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
=> pp(aa(A,bool,P,aa(nat,A,F22,pred(Nat)))) ) ) ) ).
% nat.split_sels(1)
tff(fact_4792_Code__Numeral_Onegative__def,axiom,
code_negative = aa(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_4793_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F: fun(nat,set(A)),S3: set(A)] :
( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F,I4)),S3))
=> ( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ? [N8: nat] :
( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N8))
=> ! [M5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N8))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N3))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F,M5)),aa(nat,set(A),F,N3))) ) ) )
& ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N3))
=> ( aa(nat,set(A),F,N8) = aa(nat,set(A),F,N3) ) ) )
=> ( aa(nat,set(A),F,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),F,top_top(set(nat)))) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_4794_range__abs__Nats,axiom,
image(int,int,abs_abs(int),top_top(set(int))) = semiring_1_Nats(int) ).
% range_abs_Nats
tff(fact_4795_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).
% Sup_atLeastAtMost
tff(fact_4796_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).
% cSup_atLeastAtMost
tff(fact_4797_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& dense_linorder(A) )
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,X)) = X ) ) ) ).
% cSup_atLeastLessThan
tff(fact_4798_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).
% Sup_atLeastLessThan
tff(fact_4799_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,X)) = X ) ) ) ).
% cSup_greaterThanAtMost
tff(fact_4800_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).
% Sup_greaterThanAtMost
tff(fact_4801_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).
% Sup_greaterThanLessThan
tff(fact_4802_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& dense_linorder(A) )
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,X)) = X ) ) ) ).
% cSup_greaterThanLessThan
tff(fact_4803_Nats__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> pp(member(A,zero_zero(A),semiring_1_Nats(A))) ) ).
% Nats_0
tff(fact_4804_cSup__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice(A)
& no_bot(A) )
=> ! [X6: set(A),A3: A] :
( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3)) )
=> ( ! [Y4: A] :
( ! [X3: A] :
( pp(member(A,X3,X6))
=> 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),A3),Y4)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ).
% cSup_eq
tff(fact_4805_cSup__eq__maximum,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Z2: A,X6: set(A)] :
( pp(member(A,Z2,X6))
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z2 ) ) ) ) ).
% cSup_eq_maximum
tff(fact_4806_Nats__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> pp(member(A,one_one(A),semiring_1_Nats(A))) ) ).
% Nats_1
tff(fact_4807_Nats__add,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [A3: A,B2: A] :
( pp(member(A,A3,semiring_1_Nats(A)))
=> ( pp(member(A,B2,semiring_1_Nats(A)))
=> pp(member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),semiring_1_Nats(A))) ) ) ) ).
% Nats_add
tff(fact_4808_Nats__mult,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [A3: A,B2: A] :
( pp(member(A,A3,semiring_1_Nats(A)))
=> ( pp(member(A,B2,semiring_1_Nats(A)))
=> pp(member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),semiring_1_Nats(A))) ) ) ) ).
% Nats_mult
tff(fact_4809_Nats__numeral,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [W2: num] : pp(member(A,aa(num,A,numeral_numeral(A),W2),semiring_1_Nats(A))) ) ).
% Nats_numeral
tff(fact_4810_of__nat__in__Nats,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : pp(member(A,aa(nat,A,semiring_1_of_nat(A),N),semiring_1_Nats(A))) ) ).
% of_nat_in_Nats
tff(fact_4811_Nats__induct,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [X: A,P: fun(A,bool)] :
( pp(member(A,X,semiring_1_Nats(A)))
=> ( ! [N3: nat] : pp(aa(A,bool,P,aa(nat,A,semiring_1_of_nat(A),N3)))
=> pp(aa(A,bool,P,X)) ) ) ) ).
% Nats_induct
tff(fact_4812_Nats__cases,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [X: A] :
( pp(member(A,X,semiring_1_Nats(A)))
=> ~ ! [N3: nat] : X != aa(nat,A,semiring_1_of_nat(A),N3) ) ) ).
% Nats_cases
tff(fact_4813_fact__in__Nats,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N: nat] : pp(member(A,semiring_char_0_fact(A,N),semiring_1_Nats(A))) ) ).
% fact_in_Nats
tff(fact_4814_cSup__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),A3: A] :
( ( X6 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3)) )
=> ( ! [Y4: A] :
( ! [X3: A] :
( pp(member(A,X3,X6))
=> 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),A3),Y4)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),X6) = A3 ) ) ) ) ) ).
% cSup_eq_non_empty
tff(fact_4815_cSup__least,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),Z2: A] :
( ( X6 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z2)) ) ) ) ).
% cSup_least
tff(fact_4816_le__cSup__finite,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( pp(member(A,X,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).
% le_cSup_finite
tff(fact_4817_less__cSupD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),Z2: A] :
( ( X6 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),aa(set(A),A,complete_Sup_Sup(A),X6)))
=> ? [X2: A] :
( pp(member(A,X2,X6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2)) ) ) ) ) ).
% less_cSupD
tff(fact_4818_less__cSupE,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [Y: A,X6: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
=> ( ( X6 != bot_bot(set(A)) )
=> ~ ! [X2: A] :
( pp(member(A,X2,X6))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X2)) ) ) ) ) ).
% less_cSupE
tff(fact_4819_finite__imp__Sup__less,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),X: A,A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( pp(member(A,X,X6))
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> 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),aa(set(A),A,complete_Sup_Sup(A),X6)),A3)) ) ) ) ) ).
% finite_imp_Sup_less
tff(fact_4820_inf__Sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,B5: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B5)) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,aa(A,fun(A,A),inf_inf(A),A3),B5)) ) ).
% inf_Sup
tff(fact_4821_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B5: set(A),A3: A] :
( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A3) = bot_bot(A) )
<=> ! [X5: A] :
( pp(member(A,X5,B5))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X5),A3) = bot_bot(A) ) ) ) ) ).
% Sup_inf_eq_bot_iff
tff(fact_4822_finite__subset__Union,axiom,
! [A: $tType,A5: set(A),B12: set(set(A))] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)))
=> ~ ! [F6: set(set(A))] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),F6))
=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F6),B12))
=> ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6))) ) ) ) ) ).
% finite_subset_Union
tff(fact_4823_card__Union__le__sum__card,axiom,
! [A: $tType,U3: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U3))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U3))) ).
% card_Union_le_sum_card
tff(fact_4824_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),B5: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,B5))),A3) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_kg(fun(B,A),fun(A,fun(B,A)),F),A3),B5)) ) ).
% SUP_inf
tff(fact_4825_Sup__inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B5: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A3) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,aTP_Lamp_kh(A,fun(A,A),A3),B5)) ) ).
% Sup_inf
tff(fact_4826_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F: fun(B,A),B5: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,B5))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ki(A,fun(fun(B,A),fun(B,A)),A3),F),B5)) ) ).
% inf_SUP
tff(fact_4827_SUP__inf__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),A5: set(B),G: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(C,A,G,B5))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_kk(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B5),A5)) ) ).
% SUP_inf_distrib2
tff(fact_4828_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),F: fun(B,A),M7: A] :
( ( A5 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),M7)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),M7)) ) ) ) ).
% cSUP_least
tff(fact_4829_finite__Sup__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( ( X6 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A3))
<=> ! [X5: A] :
( pp(member(A,X5,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),A3)) ) ) ) ) ) ).
% finite_Sup_less_iff
tff(fact_4830_cSup__abs__le,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S3: set(A),A3: A] :
( ( S3 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S3))),A3)) ) ) ) ).
% cSup_abs_le
tff(fact_4831_Nats__diff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A,B2: A] :
( pp(member(A,A3,semiring_1_Nats(A)))
=> ( pp(member(A,B2,semiring_1_Nats(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
=> pp(member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2),semiring_1_Nats(A))) ) ) ) ) ).
% Nats_diff
tff(fact_4832_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [B5: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member(set(B),X2,B5))
=> pp(aa(set(B),bool,finite_finite2(B),X2)) )
=> ( ! [A13: set(B)] :
( pp(member(set(B),A13,B5))
=> ! [A24: set(B)] :
( pp(member(set(B),A24,B5))
=> ( ( A13 != A24 )
=> ! [X2: B] :
( pp(member(B,X2,A13))
=> ( pp(member(B,X2,A24))
=> ( aa(B,A,G,X2) = zero_zero(A) ) ) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(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)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G),B5) ) ) ) ) ).
% sum.Union_comp
tff(fact_4833_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [B5: set(set(B)),G: fun(B,A)] :
( ! [X2: set(B)] :
( pp(member(set(B),X2,B5))
=> pp(aa(set(B),bool,finite_finite2(B),X2)) )
=> ( ! [A13: set(B)] :
( pp(member(set(B),A13,B5))
=> ! [A24: set(B)] :
( pp(member(set(B),A24,B5))
=> ( ( A13 != A24 )
=> ! [X2: B] :
( pp(member(B,X2,A13))
=> ( pp(member(B,X2,A24))
=> ( aa(B,A,G,X2) = 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)),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_4834_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set(set(A))] :
( ! [X2: set(A)] :
( pp(member(set(A),X2,U3))
=> pp(aa(set(A),bool,finite_finite2(A),X2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U3))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U3))) ) ).
% card_Union_le_sum_card_weak
tff(fact_4835_Nats__subset__Ints,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A))) ) ).
% Nats_subset_Ints
tff(fact_4836_UN__UN__finite__eq,axiom,
! [A: $tType,A5: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),aTP_Lamp_kl(fun(nat,set(A)),fun(nat,set(A)),A5),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A5,top_top(set(nat)))) ).
% UN_UN_finite_eq
tff(fact_4837_cSup__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S3: set(A),L: A,E: A] :
( ( S3 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,S3))
=> 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),L))),E)) )
=> 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(set(A),A,complete_Sup_Sup(A),S3)),L))),E)) ) ) ) ).
% cSup_asclose
tff(fact_4838_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set(A),A5: fun(A,set(B))] :
( pp(aa(set(A),bool,finite_finite2(A),I6))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),image(A,set(B),A5,I6)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_km(fun(A,set(B)),fun(A,nat),A5)),I6))) ) ).
% card_UN_le
tff(fact_4839_UN__finite__subset,axiom,
! [A: $tType,A5: fun(nat,set(A)),C6: 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)),image(nat,set(A),A5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),C6))
=> 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)),image(nat,set(A),A5,top_top(set(nat))))),C6)) ) ).
% UN_finite_subset
tff(fact_4840_UN__finite2__eq,axiom,
! [A: $tType,A5: fun(nat,set(A)),B5: fun(nat,set(A)),K: nat] :
( ! [N3: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),A5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N3))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(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)),image(nat,set(A),A5,top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,top_top(set(nat)))) ) ) ).
% UN_finite2_eq
tff(fact_4841_Nats__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_Nats(A) = image(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ) ).
% Nats_def
tff(fact_4842_card__partition,axiom,
! [A: $tType,C6: set(set(A)),K: nat] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),C6))
=> ( pp(aa(set(A),bool,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)))
=> ( ! [C3: set(A)] :
( pp(member(set(A),C3,C6))
=> ( aa(set(A),nat,finite_card(A),C3) = K ) )
=> ( ! [C1: set(A),C22: set(A)] :
( pp(member(set(A),C1,C6))
=> ( pp(member(set(A),C22,C6))
=> ( ( C1 != C22 )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C6)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)) ) ) ) ) ) ).
% card_partition
tff(fact_4843_Nats__altdef2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( semiring_1_Nats(A) = collect(A,aTP_Lamp_kn(A,bool)) ) ) ).
% Nats_altdef2
tff(fact_4844_UN__finite2__subset,axiom,
! [A: $tType,A5: 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)),image(nat,set(A),A5,set_or7035219750837199246ssThan(nat,zero_zero(nat),N3)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(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)),image(nat,set(A),A5,top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),B5,top_top(set(nat)))))) ) ).
% UN_finite2_subset
tff(fact_4845_UN__le__eq__Un0,axiom,
! [A: $tType,M7: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),M7,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)),image(nat,set(A),M7,set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M7,zero_zero(nat))) ).
% UN_le_eq_Un0
tff(fact_4846_SUP__eq__top__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A5: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)) = top_top(A) )
<=> ! [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),top_top(A)))
=> ? [Xa2: B] :
( pp(member(B,Xa2,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),aa(B,A,F,Xa2))) ) ) ) ) ).
% SUP_eq_top_iff
tff(fact_4847_Sup__eq__top__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A5: set(A)] :
( ( aa(set(A),A,complete_Sup_Sup(A),A5) = top_top(A) )
<=> ! [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),top_top(A)))
=> ? [Xa2: A] :
( pp(member(A,Xa2,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Xa2)) ) ) ) ) ).
% Sup_eq_top_iff
tff(fact_4848_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [F: fun(A,B),A5: fun(C,A),I6: set(C)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ko(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5),I6))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),image(C,A,A5,I6))))) ) ) ).
% mono_SUP
tff(fact_4849_Sup__nat__empty,axiom,
aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).
% Sup_nat_empty
tff(fact_4850_Sup__upper2,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [U: A,A5: set(A),V: A] :
( pp(member(A,U,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),U))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).
% Sup_upper2
tff(fact_4851_Sup__le__iff,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B2))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),B2)) ) ) ) ).
% Sup_le_iff
tff(fact_4852_Sup__upper,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A,A5: set(A)] :
( pp(member(A,X,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ).
% Sup_upper
tff(fact_4853_Sup__least,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),Z2: A] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z2)) ) ) ).
% Sup_least
tff(fact_4854_Sup__mono,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),B5: set(A)] :
( ! [A6: A] :
( pp(member(A,A6,A5))
=> ? [X3: A] :
( pp(member(A,X3,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X3)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ).
% Sup_mono
tff(fact_4855_Sup__eqI,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),X: A] :
( ! [Y4: A] :
( pp(member(A,Y4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
=> ( ! [Y4: A] :
( ! [Z4: A] :
( pp(member(A,Z4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z4),Y4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),A5) = X ) ) ) ) ).
% Sup_eqI
tff(fact_4856_less__Sup__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A3: A,S3: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S3)))
<=> ? [X5: A] :
( pp(member(A,X5,S3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X5)) ) ) ) ).
% less_Sup_iff
tff(fact_4857_le__Sup__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [X: A,A5: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A5)))
<=> ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
=> ? [X5: A] :
( pp(member(A,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X5)) ) ) ) ) ).
% le_Sup_iff
tff(fact_4858_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),B5: set(C),F: fun(B,A),G: fun(C,A)] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> ? [X3: C] :
( pp(member(C,X3,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),aa(C,A,G,X3))) ) )
=> ( ! [J2: C] :
( pp(member(C,J2,B5))
=> ? [X3: B] :
( pp(member(B,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,J2)),aa(B,A,F,X3))) ) )
=> ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,G,B5)) ) ) ) ) ).
% SUP_eq
tff(fact_4859_less__eq__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),U: A] :
( ! [V4: A] :
( pp(member(A,V4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V4)) )
=> ( ( A5 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).
% less_eq_Sup
tff(fact_4860_Sup__subset__mono,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ).
% Sup_subset_mono
tff(fact_4861_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A5: set(B),U: A,F: fun(B,A)] :
( pp(member(B,I,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,I)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)))) ) ) ) ).
% SUP_upper2
tff(fact_4862_SUP__le__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A5: set(B),U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),U))
<=> ! [X5: B] :
( pp(member(B,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X5)),U)) ) ) ) ).
% SUP_le_iff
tff(fact_4863_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A5: set(B),F: fun(B,A)] :
( pp(member(B,I,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I)),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)))) ) ) ).
% SUP_upper
tff(fact_4864_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),G: fun(B,A),A5: set(B)] :
( ! [X2: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,G,X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(B,A,G,A5)))) ) ) ).
% SUP_mono'
tff(fact_4865_SUP__least,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),F: fun(B,A),U: A] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),U)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),U)) ) ) ).
% SUP_least
tff(fact_4866_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),B5: set(C),F: fun(B,A),G: fun(C,A)] :
( ! [N3: B] :
( pp(member(B,N3,A5))
=> ? [X3: C] :
( pp(member(C,X3,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,N3)),aa(C,A,G,X3))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(C,A,G,B5)))) ) ) ).
% SUP_mono
tff(fact_4867_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),F: fun(B,A),X: A] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),X)) )
=> ( ! [Y4: A] :
( ! [I3: B] :
( pp(member(B,I3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I3)),Y4)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)) = X ) ) ) ) ).
% SUP_eqI
tff(fact_4868_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A5: set(B),Y: A,I: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),Y))
=> ( pp(member(B,I,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I)),Y)) ) ) ) ).
% SUP_lessD
tff(fact_4869_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [A3: A,F: fun(B,A),A5: set(B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))))
<=> ? [X5: B] :
( pp(member(B,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X5))) ) ) ) ).
% less_SUP_iff
tff(fact_4870_le__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [X: A,F: fun(B,A),A5: set(B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))))
<=> ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
=> ? [X5: B] :
( pp(member(B,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),aa(B,A,F,X5))) ) ) ) ) ).
% le_SUP_iff
tff(fact_4871_SUP__eq__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I6: set(B),C2: A,F: fun(B,A)] :
( ( I6 != bot_bot(set(B)) )
=> ( ! [I4: B] :
( pp(member(B,I4,I6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(B,A,F,I4))) )
=> ( ( aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,I6)) = C2 )
<=> ! [X5: B] :
( pp(member(B,X5,I6))
=> ( aa(B,A,F,X5) = C2 ) ) ) ) ) ) ).
% SUP_eq_iff
tff(fact_4872_Sup__inter__less__eq,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),B5: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)))) ) ).
% Sup_inter_less_eq
tff(fact_4873_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),B5: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B5))
=> ( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(B,A,G,B5)))) ) ) ) ).
% SUP_subset_mono
tff(fact_4874_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F,A5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A5)))) ) ) ).
% mono_Sup
tff(fact_4875_card__UNION,axiom,
! [A: $tType,A5: set(set(A))] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A5))
=> ( ! [X2: set(A)] :
( pp(member(set(A),X2,A5))
=> pp(aa(set(A),bool,finite_finite2(A),X2)) )
=> ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) = 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_kp(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_kq(set(set(A)),fun(set(set(A)),bool),A5)))) ) ) ) ).
% card_UNION
tff(fact_4876_surj__int__encode,axiom,
image(int,nat,nat_int_encode,top_top(set(int))) = top_top(set(nat)) ).
% surj_int_encode
tff(fact_4877_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $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(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_kr(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).
% horner_sum_eq_sum_funpow
tff(fact_4878_Suc__funpow,axiom,
! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).
% Suc_funpow
tff(fact_4879_funpow__0,axiom,
! [A: $tType,F: fun(A,A),X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F),X) = X ).
% funpow_0
tff(fact_4880_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A5: set(A)] :
( ( aa(set(A),A,complete_Inf_Inf(A),A5) = bot_bot(A) )
<=> ! [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X5))
=> ? [Xa2: A] :
( pp(member(A,Xa2,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa2),X5)) ) ) ) ) ).
% Inf_eq_bot_iff
tff(fact_4881_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).
% Inf_atLeastAtMost
tff(fact_4882_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).
% cInf_atLeastAtMost
tff(fact_4883_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,X)) = Y ) ) ) ).
% cInf_atLeastLessThan
tff(fact_4884_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).
% Inf_atLeastLessThan
tff(fact_4885_surj__fn,axiom,
! [A: $tType,F: fun(A,A),N: nat] :
( ( image(A,A,F,top_top(set(A))) = top_top(set(A)) )
=> ( image(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),top_top(set(A))) = top_top(set(A)) ) ) ).
% surj_fn
tff(fact_4886_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& dense_linorder(A) )
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,X)) = Y ) ) ) ).
% cInf_greaterThanAtMost
tff(fact_4887_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).
% Inf_greaterThanAtMost
tff(fact_4888_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice(A)
& dense_linorder(A) )
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).
% Inf_greaterThanLessThan
tff(fact_4889_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& dense_linorder(A) )
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,X)) = Y ) ) ) ).
% cInf_greaterThanLessThan
tff(fact_4890_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A5: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5)) = bot_bot(A) )
<=> ! [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X5))
=> ? [Xa2: B] :
( pp(member(B,Xa2,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,Xa2)),X5)) ) ) ) ) ).
% INF_eq_bot_iff
tff(fact_4891_Compl__UN,axiom,
! [A: $tType,B: $tType,B5: fun(B,set(A)),A5: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),B5,A5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),aTP_Lamp_ks(fun(B,set(A)),fun(B,set(A)),B5),A5)) ).
% Compl_UN
tff(fact_4892_Compl__INT,axiom,
! [A: $tType,B: $tType,B5: fun(B,set(A)),A5: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),image(B,set(A),B5,A5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),aTP_Lamp_ks(fun(B,set(A)),fun(B,set(A)),B5),A5)) ).
% Compl_INT
tff(fact_4893_Inf__atMostLessThan,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
=> ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,X)) = bot_bot(A) ) ) ) ).
% Inf_atMostLessThan
tff(fact_4894_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),comp(A,A,B,F)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ).
% comp_funpow
tff(fact_4895_Inf__greatest,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),Z2: A] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),A5))) ) ) ).
% Inf_greatest
tff(fact_4896_le__Inf__iff,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B2: A,A5: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X5)) ) ) ) ).
% le_Inf_iff
tff(fact_4897_Inf__lower2,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [U: A,A5: set(A),V: A] :
( pp(member(A,U,A5))
=> ( 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),aa(set(A),A,complete_Inf_Inf(A),A5)),V)) ) ) ) ).
% Inf_lower2
tff(fact_4898_Inf__lower,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A,A5: set(A)] :
( pp(member(A,X,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),X)) ) ) ).
% Inf_lower
tff(fact_4899_Inf__mono,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B5: set(A),A5: set(A)] :
( ! [B4: A] :
( pp(member(A,B4,B5))
=> ? [X3: A] :
( pp(member(A,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B4)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))) ) ) ).
% Inf_mono
tff(fact_4900_Inf__eqI,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),X: A] :
( ! [I4: A] :
( pp(member(A,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I4)) )
=> ( ! [Y4: A] :
( ! [I3: A] :
( pp(member(A,I3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),I3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),A5) = X ) ) ) ) ).
% Inf_eqI
tff(fact_4901_cInf__eq__minimum,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Z2: A,X6: set(A)] :
( pp(member(A,Z2,X6))
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X2)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z2 ) ) ) ) ).
% cInf_eq_minimum
tff(fact_4902_cInf__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice(A)
& no_top(A) )
=> ! [X6: set(A),A3: A] :
( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X2)) )
=> ( ! [Y4: A] :
( ! [X3: A] :
( pp(member(A,X3,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),A3)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ).
% cInf_eq
tff(fact_4903_Inf__less__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [S3: set(A),A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A3))
<=> ? [X5: A] :
( pp(member(A,X5,S3))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),A3)) ) ) ) ).
% Inf_less_iff
tff(fact_4904_mono__pow,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),N: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> pp(aa(fun(A,A),bool,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F))) ) ) ).
% mono_pow
tff(fact_4905_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),F) ).
% funpow_mult
tff(fact_4906_int__encode__eq,axiom,
! [X: int,Y: int] :
( ( aa(int,nat,nat_int_encode,X) = aa(int,nat,nat_int_encode,Y) )
<=> ( X = Y ) ) ).
% int_encode_eq
tff(fact_4907_funpow__swap1,axiom,
! [A: $tType,F: fun(A,A),N: nat,X: A] : aa(A,A,F,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),aa(A,A,F,X)) ).
% funpow_swap1
tff(fact_4908_funpow__mod__eq,axiom,
! [A: $tType,N: nat,F: fun(A,A),X: A,M: nat] :
( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),X) = X )
=> ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)),F),X) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F),X) ) ) ).
% funpow_mod_eq
tff(fact_4909_Inf__le__iff,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),X))
<=> ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
=> ? [X5: A] :
( pp(member(A,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5)) ) ) ) ) ).
% Inf_le_iff
tff(fact_4910_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),B5: set(C),G: fun(C,A),F: fun(B,A)] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> ? [X3: C] :
( pp(member(C,X3,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,X3)),aa(B,A,F,I4))) ) )
=> ( ! [J2: C] :
( pp(member(C,J2,B5))
=> ? [X3: B] :
( pp(member(B,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X3)),aa(C,A,G,J2))) ) )
=> ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5)) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,G,B5)) ) ) ) ) ).
% INF_eq
tff(fact_4911_cInf__eq__non__empty,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),A3: A] :
( ( X6 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X2)) )
=> ( ! [Y4: A] :
( ! [X3: A] :
( pp(member(A,X3,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),A3)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),X6) = A3 ) ) ) ) ) ).
% cInf_eq_non_empty
tff(fact_4912_cInf__greatest,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),Z2: A] :
( ( X6 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ).
% cInf_greatest
tff(fact_4913_Inf__less__eq,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),U: A] :
( ! [V4: A] :
( pp(member(A,V4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V4),U)) )
=> ( ( A5 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),U)) ) ) ) ).
% Inf_less_eq
tff(fact_4914_cInf__le__finite,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X6: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( pp(member(A,X,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).
% cInf_le_finite
tff(fact_4915_cInf__lessD,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),Z2: A] :
( ( X6 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z2))
=> ? [X2: A] :
( pp(member(A,X2,X6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2)) ) ) ) ) ).
% cInf_lessD
tff(fact_4916_Inf__superset__mono,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B5: set(A),A5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))) ) ) ).
% Inf_superset_mono
tff(fact_4917_finite__imp__less__Inf,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),X: A,A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( pp(member(A,X,X6))
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ) ).
% finite_imp_less_Inf
tff(fact_4918_funpow__times__power,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [F: fun(A,nat),X: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F,X)),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(A,nat,F,X))) ) ).
% funpow_times_power
tff(fact_4919_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),F) ).
% funpow_Suc_right
tff(fact_4920_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F) = aa(fun(A,A),fun(A,A),comp(A,A,A,F),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ).
% funpow.simps(2)
tff(fact_4921_sup__Inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,B5: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B5)) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,aa(A,fun(A,A),sup_sup(A),A3),B5)) ) ).
% sup_Inf
tff(fact_4922_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B5: set(A),A3: A] :
( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B5)),A3) = top_top(A) )
<=> ! [X5: A] :
( pp(member(A,X5,B5))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X5),A3) = top_top(A) ) ) ) ) ).
% Inf_sup_eq_top_iff
tff(fact_4923_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),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),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ).
% funpow_add
tff(fact_4924_funpow__mono,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(A,A),A5: A,B5: A,N: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),A5)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),B5))) ) ) ) ).
% funpow_mono
tff(fact_4925_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),U: A,F: fun(B,A)] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,I4))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5)))) ) ) ).
% INF_greatest
tff(fact_4926_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [U: A,F: fun(B,A),A5: set(B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))))
<=> ! [X5: B] :
( pp(member(B,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X5))) ) ) ) ).
% le_INF_iff
tff(fact_4927_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A5: set(B),F: fun(B,A),U: A] :
( pp(member(B,I,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I)),U))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),U)) ) ) ) ).
% INF_lower2
tff(fact_4928_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),G: fun(B,A),A5: set(B)] :
( ! [X2: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,G,X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),aa(set(A),A,complete_Inf_Inf(A),image(B,A,G,A5)))) ) ) ).
% INF_mono'
tff(fact_4929_INF__lower,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A5: set(B),F: fun(B,A)] :
( pp(member(B,I,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),aa(B,A,F,I))) ) ) ).
% INF_lower
tff(fact_4930_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [B5: set(B),A5: set(C),F: fun(C,A),G: fun(B,A)] :
( ! [M5: B] :
( pp(member(B,M5,B5))
=> ? [X3: C] :
( pp(member(C,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X3)),aa(B,A,G,M5))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(C,A,F,A5))),aa(set(A),A,complete_Inf_Inf(A),image(B,A,G,B5)))) ) ) ).
% INF_mono
tff(fact_4931_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),X: A,F: fun(B,A)] :
( ! [I4: B] :
( pp(member(B,I4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F,I4))) )
=> ( ! [Y4: A] :
( ! [I3: B] :
( pp(member(B,I3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),aa(B,A,F,I3))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5)) = X ) ) ) ) ).
% INF_eqI
tff(fact_4932_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Y: A,F: fun(B,A),A5: set(B),I: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))))
=> ( pp(member(B,I,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F,I))) ) ) ) ).
% less_INF_D
tff(fact_4933_INF__less__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A5: set(B),A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),A3))
<=> ? [X5: B] :
( pp(member(B,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X5)),A3)) ) ) ) ).
% INF_less_iff
tff(fact_4934_INF__sup,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),B5: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,B5))),A3) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_kt(fun(B,A),fun(A,fun(B,A)),F),A3),B5)) ) ).
% INF_sup
tff(fact_4935_Inf__sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B5: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B5)),A3) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,aTP_Lamp_ku(A,fun(A,A),A3),B5)) ) ).
% Inf_sup
tff(fact_4936_sup__INF,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F: fun(B,A),B5: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,B5))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kv(A,fun(fun(B,A),fun(B,A)),A3),F),B5)) ) ).
% sup_INF
tff(fact_4937_INF__sup__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),A5: set(B),G: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),aa(set(A),A,complete_Inf_Inf(A),image(C,A,G,B5))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_kx(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B5),A5)) ) ).
% INF_sup_distrib2
tff(fact_4938_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A5: set(B),X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),X))
<=> ! [Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
=> ? [X5: B] :
( pp(member(B,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X5)),Y5)) ) ) ) ) ).
% INF_le_iff
tff(fact_4939_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),M: A,F: fun(B,A)] :
( ( A5 != bot_bot(set(B)) )
=> ( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5)))) ) ) ) ).
% cINF_greatest
tff(fact_4940_INF__eq__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I6: set(B),F: fun(B,A),C2: A] :
( ( I6 != bot_bot(set(B)) )
=> ( ! [I4: B] :
( pp(member(B,I4,I6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I4)),C2)) )
=> ( ( aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,I6)) = C2 )
<=> ! [X5: B] :
( pp(member(B,X5,I6))
=> ( aa(B,A,F,X5) = C2 ) ) ) ) ) ) ).
% INF_eq_iff
tff(fact_4941_finite__less__Inf__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),X6))
=> ( ( X6 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X6)))
<=> ! [X5: A] :
( pp(member(A,X5,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),X5)) ) ) ) ) ) ).
% finite_less_Inf_iff
tff(fact_4942_Inf__le__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A)] :
( ( A5 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ).
% Inf_le_Sup
tff(fact_4943_cInf__abs__ge,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S3: set(A),A3: A] :
( ( S3 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X2)),A3)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S3))),A3)) ) ) ) ).
% cInf_abs_ge
tff(fact_4944_uminus__Sup,axiom,
! [A: $tType] :
( comple489889107523837845lgebra(A)
=> ! [A5: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),A5)) = aa(set(A),A,complete_Inf_Inf(A),image(A,A,uminus_uminus(A),A5)) ) ).
% uminus_Sup
tff(fact_4945_uminus__Inf,axiom,
! [A: $tType] :
( comple489889107523837845lgebra(A)
=> ! [A5: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),A5)) = aa(set(A),A,complete_Sup_Sup(A),image(A,A,uminus_uminus(A),A5)) ) ).
% uminus_Inf
tff(fact_4946_less__eq__Inf__inter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A),B5: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)))) ) ).
% less_eq_Inf_inter
tff(fact_4947_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [F: fun(A,A),P3: A,K: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,P3)),P3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),bot_bot(A))),P3)) ) ) ) ).
% Kleene_iter_lpfp
tff(fact_4948_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( order_top(A)
=> ! [F: fun(A,A),P3: A,K: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,F,P3)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P3),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),top_top(A)))) ) ) ) ).
% Kleene_iter_gpfp
tff(fact_4949_funpow__mono2,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(A,A),I: nat,J: nat,X: A,Y: A] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( 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),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F,X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I),F),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F),Y))) ) ) ) ) ) ).
% funpow_mono2
tff(fact_4950_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [B5: set(B),A5: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B5),A5))
=> ( ! [X2: B] :
( pp(member(B,X2,B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),aa(set(A),A,complete_Inf_Inf(A),image(B,A,G,B5)))) ) ) ) ).
% INF_superset_mono
tff(fact_4951_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( comple489889107523837845lgebra(A)
=> ! [B5: fun(B,A),A5: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,B5,A5))) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aTP_Lamp_ky(fun(B,A),fun(B,A),B5),A5)) ) ).
% uminus_INF
tff(fact_4952_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( comple489889107523837845lgebra(A)
=> ! [B5: fun(B,A),A5: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,B5,A5))) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aTP_Lamp_ky(fun(B,A),fun(B,A),B5),A5)) ) ).
% uminus_SUP
tff(fact_4953_mono__INF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice(B)
& comple6319245703460814977attice(A) )
=> ! [F: fun(A,B),A5: fun(C,A),I6: set(C)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),image(C,A,A5,I6)))),aa(set(B),B,complete_Inf_Inf(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ko(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5),I6)))) ) ) ).
% mono_INF
tff(fact_4954_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F,A5)))) ) ) ).
% mono_Inf
tff(fact_4955_of__nat__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(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_4956_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,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(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_4957_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(B),F: fun(B,A)] :
( ( A5 != bot_bot(set(B)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)))) ) ) ).
% INF_le_SUP
tff(fact_4958_mono__funpow,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Q: fun(A,A)] :
( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
=> pp(aa(fun(nat,A),bool,order_mono(nat,A),aTP_Lamp_kz(fun(A,A),fun(nat,A),Q))) ) ) ).
% mono_funpow
tff(fact_4959_cInf__asclose,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder(A)
& linordered_idom(A) )
=> ! [S3: set(A),L: A,E: A] :
( ( S3 != bot_bot(set(A)) )
=> ( ! [X2: A] :
( pp(member(A,X2,S3))
=> 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),L))),E)) )
=> 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(set(A),A,complete_Inf_Inf(A),S3)),L))),E)) ) ) ) ).
% cInf_asclose
tff(fact_4960_funpow__decreasing,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [M: nat,N: nat,F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),bot_bot(A)))) ) ) ) ).
% funpow_decreasing
tff(fact_4961_funpow__increasing,axiom,
! [A: $tType] :
( ( lattice(A)
& order_top(A) )
=> ! [M: nat,N: nat,F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
=> ( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),top_top(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F),top_top(A)))) ) ) ) ).
% funpow_increasing
tff(fact_4962_numeral__unfold__funpow,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(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_4963_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),image(B,A,aTP_Lamp_lb(fun(C,fun(B,A)),fun(B,A),P),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),image(fun(B,C),A,aTP_Lamp_ld(fun(C,fun(B,A)),fun(fun(B,C),A),P),top_top(set(fun(B,C))))) ) ).
% SUP_INF
tff(fact_4964_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [P: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),image(B,A,aTP_Lamp_le(fun(C,fun(B,A)),fun(B,A),P),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),image(fun(B,C),A,aTP_Lamp_lf(fun(C,fun(B,A)),fun(fun(B,C),A),P),top_top(set(fun(B,C))))) ) ).
% INF_SUP
tff(fact_4965_relpowp__fun__conv,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y))
<=> ? [F4: fun(nat,A)] :
( ( aa(nat,A,F4,zero_zero(nat)) = X )
& ( aa(nat,A,F4,N) = Y )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,F4,I5)),aa(nat,A,F4,aa(nat,nat,suc,I5)))) ) ) ) ).
% relpowp_fun_conv
tff(fact_4966_relpowp__1,axiom,
! [A: $tType,P: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),one_one(nat)),P) = P ).
% relpowp_1
tff(fact_4967_Nat_Ofunpow__code__def,axiom,
! [A: $tType] : funpow(A) = compow(fun(A,A)) ).
% Nat.funpow_code_def
tff(fact_4968_Inf__int__def,axiom,
! [X6: set(int)] : aa(set(int),int,complete_Inf_Inf(int),X6) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),image(int,int,uminus_uminus(int),X6))) ).
% Inf_int_def
tff(fact_4969_relpowp__Suc__I2,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A,N: nat,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z2)) ) ) ).
% relpowp_Suc_I2
tff(fact_4970_relpowp__Suc__E2,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z2))
=> ~ ! [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y4),Z2)) ) ) ).
% relpowp_Suc_E2
tff(fact_4971_relpowp__Suc__D2,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z2))
=> ? [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
& pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y4),Z2)) ) ) ).
% relpowp_Suc_D2
tff(fact_4972_relpowp__Suc__I,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z2)) ) ) ).
% relpowp_Suc_I
tff(fact_4973_relpowp__Suc__E,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z2))
=> ~ ! [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z2)) ) ) ).
% relpowp_Suc_E
tff(fact_4974_relpowp__0__I,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A] : pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),X)) ).
% relpowp_0_I
tff(fact_4975_relpowp__0__E,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),Y))
=> ( X = Y ) ) ).
% relpowp_0_E
tff(fact_4976_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R2: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),R2) = fequal(A) ).
% relpowp.simps(1)
tff(fact_4977_relpowp__E,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z2))
=> ( ( ( N = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y4: A,M5: nat] :
( ( N = aa(nat,nat,suc,M5) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M5),P),X),Y4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z2)) ) ) ) ) ).
% relpowp_E
tff(fact_4978_relpowp__E2,axiom,
! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z2))
=> ( ( ( N = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y4: A,M5: nat] :
( ( N = aa(nat,nat,suc,M5) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M5),P),Y4),Z2)) ) ) ) ) ).
% relpowp_E2
tff(fact_4979_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).
% relpowp_bot
tff(fact_4980_set__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : set2(product_prod(A,B),zip(A,B,Xs,Ys2)) = collect(product_prod(A,B),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_lg(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys2)) ).
% set_zip
tff(fact_4981_measure__function__int,axiom,
fun_is_measure(int,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))) ).
% measure_function_int
tff(fact_4982_surj__int__decode,axiom,
image(nat,int,nat_int_decode,top_top(set(nat))) = top_top(set(int)) ).
% surj_int_decode
tff(fact_4983_int__decode__inverse,axiom,
! [N: nat] : aa(int,nat,nat_int_encode,aa(nat,int,nat_int_decode,N)) = N ).
% int_decode_inverse
tff(fact_4984_int__encode__inverse,axiom,
! [X: int] : aa(nat,int,nat_int_decode,aa(int,nat,nat_int_encode,X)) = X ).
% int_encode_inverse
tff(fact_4985_measure__snd,axiom,
! [B: $tType,A: $tType,F: fun(A,nat)] :
( fun_is_measure(A,F)
=> fun_is_measure(product_prod(B,A),aTP_Lamp_lh(fun(A,nat),fun(product_prod(B,A),nat),F)) ) ).
% measure_snd
tff(fact_4986_measure__fst,axiom,
! [B: $tType,A: $tType,F: fun(A,nat)] :
( fun_is_measure(A,F)
=> fun_is_measure(product_prod(A,B),aTP_Lamp_li(fun(A,nat),fun(product_prod(A,B),nat),F)) ) ).
% measure_fst
tff(fact_4987_lexordp_Omono,axiom,
! [A: $tType] :
( ord(A)
=> pp(aa(fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),bool,order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),aTP_Lamp_lj(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))))) ) ).
% lexordp.mono
tff(fact_4988_is__measure_Osimps,axiom,
! [A: $tType,A3: fun(A,nat)] :
( fun_is_measure(A,A3)
<=> ? [X_13: fun(A,nat)] : A3 = X_13 ) ).
% is_measure.simps
tff(fact_4989_is__measure__trivial,axiom,
! [A: $tType,F: fun(A,nat)] : fun_is_measure(A,F) ).
% is_measure_trivial
tff(fact_4990_int__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( aa(nat,int,nat_int_decode,X) = aa(nat,int,nat_int_decode,Y) )
<=> ( X = Y ) ) ).
% int_decode_eq
tff(fact_4991_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X),Y)) ).
% bot2E
tff(fact_4992_measure__size,axiom,
! [A: $tType] :
( size(A)
=> fun_is_measure(A,size_size(A)) ) ).
% measure_size
tff(fact_4993_set__conv__nth,axiom,
! [A: $tType,Xs: list(A)] : set2(A,Xs) = collect(A,aTP_Lamp_lk(list(A),fun(A,bool),Xs)) ).
% set_conv_nth
tff(fact_4994_set__nths,axiom,
! [A: $tType,Xs: list(A),I6: set(nat)] : set2(A,nths(A,Xs,I6)) = collect(A,aa(set(nat),fun(A,bool),aTP_Lamp_ll(list(A),fun(set(nat),fun(A,bool)),Xs),I6)) ).
% set_nths
tff(fact_4995_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_lm(A,fun(A,bool)),aTP_Lamp_ln(A,fun(A,bool))) ) ).
% sup_bot.semilattice_neutr_order_axioms
tff(fact_4996_max__nat_Osemilattice__neutr__order__axioms,axiom,
semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ae(nat,fun(nat,bool)),aTP_Lamp_ad(nat,fun(nat,bool))) ).
% max_nat.semilattice_neutr_order_axioms
tff(fact_4997_times__int_Oabs__eq,axiom,
! [Xa4: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa4)),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa4),X)) ).
% times_int.abs_eq
tff(fact_4998_int_Oabs__induct,axiom,
! [P: fun(int,bool),X: int] :
( ! [Y4: product_prod(nat,nat)] : pp(aa(int,bool,P,aa(product_prod(nat,nat),int,abs_Integ,Y4)))
=> pp(aa(int,bool,P,X)) ) ).
% int.abs_induct
tff(fact_4999_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( ( Z2 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
tff(fact_5000_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z2 )
<=> ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
tff(fact_5001_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X2: nat,Y4: nat] : Z2 != aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X2),Y4)) ).
% eq_Abs_Integ
tff(fact_5002_mono__compose,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Q: fun(A,fun(B,C)),F: fun(D,B)] :
( pp(aa(fun(A,fun(B,C)),bool,order_mono(A,fun(B,C)),Q))
=> pp(aa(fun(A,fun(D,C)),bool,order_mono(A,fun(D,C)),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_lq(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F))) ) ) ).
% mono_compose
tff(fact_5003_zero__int__def,axiom,
zero_zero(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))) ).
% zero_int_def
tff(fact_5004_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),product_Pair(nat,nat,N),zero_zero(nat))) ).
% int_def
tff(fact_5005_nat_Oabs__eq,axiom,
! [X: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),nat,product_case_prod(nat,nat,nat,minus_minus(nat)),X) ).
% nat.abs_eq
tff(fact_5006_Nats__altdef1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( semiring_1_Nats(A) = collect(A,aTP_Lamp_lr(A,bool)) ) ) ).
% Nats_altdef1
tff(fact_5007_uminus__int_Oabs__eq,axiom,
! [X: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ls(nat,fun(nat,product_prod(nat,nat)))),X)) ).
% uminus_int.abs_eq
tff(fact_5008_one__int__def,axiom,
one_one(int) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).
% one_int_def
tff(fact_5009_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_lt(nat,fun(nat,A))),X) ) ).
% of_int.abs_eq
tff(fact_5010_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_lu(nat,fun(nat,bool))) ).
% gcd_nat.semilattice_neutr_order_axioms
tff(fact_5011_less__int_Oabs__eq,axiom,
! [Xa4: product_prod(nat,nat),X: product_prod(nat,nat)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa4)),aa(product_prod(nat,nat),int,abs_Integ,X)))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lw(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa4),X)) ) ).
% less_int.abs_eq
tff(fact_5012_less__eq__int_Oabs__eq,axiom,
! [Xa4: product_prod(nat,nat),X: 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,Xa4)),aa(product_prod(nat,nat),int,abs_Integ,X)))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ly(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa4),X)) ) ).
% less_eq_int.abs_eq
tff(fact_5013_plus__int_Oabs__eq,axiom,
! [Xa4: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa4)),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_ma(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa4),X)) ).
% plus_int.abs_eq
tff(fact_5014_minus__int_Oabs__eq,axiom,
! [Xa4: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa4)),aa(product_prod(nat,nat),int,abs_Integ,X)) = 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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_mc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa4),X)) ).
% minus_int.abs_eq
tff(fact_5015_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).
% inf_top.semilattice_neutr_order_axioms
tff(fact_5016_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))) = case_option(int,num,zero_zero(int),aTP_Lamp_md(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_5017_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I6: set(B),P3: fun(B,A),I: B] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))))
=> ( ( pp(member(B,I,I6))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),insert(B,I,I6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I6) ) )
& ( ~ pp(member(B,I,I6))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),insert(B,I,I6)) = 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),I6)) ) ) ) ) ) ).
% prod.insert'
tff(fact_5018_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),A3: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),A3)) )
=> ( linorder_insort_key(A,A,aTP_Lamp_me(A,A),A3,Xs) = append(A,Xs,cons(A,A3,nil(A))) ) ) ) ) ).
% sorted_insort_is_snoc
tff(fact_5019_length__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),linorder_insort_key(B,A,F,X,Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).
% length_insort
tff(fact_5020_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_5021_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)) = 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_5022_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),I6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),collect(B,aa(set(B),fun(B,bool),aTP_Lamp_mf(fun(B,A),fun(set(B),fun(B,bool)),G),I6))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I6) ) ).
% prod.non_neutral'
tff(fact_5023_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Y: B,Ys2: list(B)] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y)))
=> ( linorder_insort_key(B,A,F,X,cons(B,Y,Ys2)) = cons(B,X,cons(B,Y,Ys2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,F,Y)))
=> ( linorder_insort_key(B,A,F,X,cons(B,Y,Ys2)) = cons(B,Y,linorder_insort_key(B,A,F,X,Ys2)) ) ) ) ) ).
% insort_key.simps(2)
tff(fact_5024_sorted__insort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),linorder_insort_key(A,A,aTP_Lamp_me(A,A),X,Xs))
<=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).
% sorted_insort
tff(fact_5025_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I6: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),I6))
=> ( 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_av(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),I6) = 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),I6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H2),I6)) ) ) ) ).
% prod.distrib_triv'
tff(fact_5026_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Xs: list(B),F: fun(B,A),A3: B] :
( ! [X2: B] :
( pp(member(B,X2,set2(B,Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,A3)),aa(B,A,F,X2))) )
=> ( linorder_insort_key(B,A,F,A3,Xs) = cons(B,A3,Xs) ) ) ) ).
% insort_is_Cons
tff(fact_5027_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X: B,Xs: list(B)] :
( sorted_wrt(A,ord_less_eq(A),map(B,A,F,linorder_insort_key(B,A,F,X,Xs)))
<=> sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs)) ) ) ).
% sorted_insort_key
tff(fact_5028_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),T5: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( 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),H2),S3) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
tff(fact_5029_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),T5: set(B),H2: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S3),T5))
=> ( ! [I4: B] :
( pp(member(B,I4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,H2,I4) = one_one(A) ) )
=> ( ! [X2: B] :
( pp(member(B,X2,S3))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H2),T5) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
tff(fact_5030_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: 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)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = 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),S3) ) ) ) ) ).
% prod.mono_neutral_right'
tff(fact_5031_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: 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)),S3),T5))
=> ( ! [X2: B] :
( pp(member(B,X2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T5),S3)))
=> ( aa(B,A,G,X2) = one_one(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T5) ) ) ) ) ).
% prod.mono_neutral_left'
tff(fact_5032_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I6: set(B),G: fun(B,A),H2: fun(B,A)] :
( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),G))))
=> ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),H2))))
=> ( 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_av(fun(B,A),fun(fun(B,A),fun(B,A)),G),H2)),I6) = 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),I6)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H2),I6)) ) ) ) ) ).
% prod.distrib'
tff(fact_5033_prod_OG__def,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [I6: set(B),P3: fun(B,A)] :
( ( pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))) ) )
& ( ~ pp(aa(set(B),bool,finite_finite2(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ao(set(B),fun(fun(B,A),fun(B,bool)),I6),P3))))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P3),I6) = one_one(A) ) ) ) ) ).
% prod.G_def
tff(fact_5034_filter__insort,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),P: fun(B,bool),X: B] :
( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
=> ( pp(aa(B,bool,P,X))
=> ( filter(B,P,linorder_insort_key(B,A,F,X,Xs)) = linorder_insort_key(B,A,F,X,filter(B,P,Xs)) ) ) ) ) ).
% filter_insort
tff(fact_5035_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),bit0(N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).
% and_minus_numerals(3)
tff(fact_5036_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),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).
% and_minus_numerals(7)
tff(fact_5037_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)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(N))) ).
% and_minus_numerals(8)
tff(fact_5038_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)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(N))) ).
% and_minus_numerals(4)
tff(fact_5039_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)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).
% int_numeral_not_and_num
tff(fact_5040_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))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).
% int_numeral_and_not_num
tff(fact_5041_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [A3: B,Xs: list(B),F: fun(B,A)] :
( pp(member(B,A3,set2(B,Xs)))
=> ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
=> ( ( hd(B,filter(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_mg(B,fun(fun(B,A),fun(B,bool)),A3),F),Xs)) = A3 )
=> ( linorder_insort_key(B,A,F,A3,remove1(B,A3,Xs)) = Xs ) ) ) ) ) ).
% insort_key_remove1
tff(fact_5042_less__eq__int_Orep__eq,axiom,
! [X: int,Xa4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa4))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ly(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa4))) ) ).
% less_eq_int.rep_eq
tff(fact_5043_less__int_Orep__eq,axiom,
! [X: int,Xa4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa4))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lw(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X)),aa(int,product_prod(nat,nat),rep_Integ,Xa4))) ) ).
% less_int.rep_eq
tff(fact_5044_sorted__remove1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),A3: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),remove1(A,A3,Xs)) ) ) ).
% sorted_remove1
tff(fact_5045_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),X: B] :
( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
=> sorted_wrt(A,ord_less_eq(A),map(B,A,F,remove1(B,X,Xs))) ) ) ).
% sorted_map_remove1
tff(fact_5046_length__remove1,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( ( pp(member(A,X,set2(A,Xs)))
=> ( aa(list(A),nat,size_size(list(A)),remove1(A,X,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,X,set2(A,Xs)))
=> ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).
% length_remove1
tff(fact_5047_nat_Orep__eq,axiom,
! [X: int] : aa(int,nat,nat2,X) = aa(product_prod(nat,nat),nat,product_case_prod(nat,nat,nat,minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,X)) ).
% nat.rep_eq
tff(fact_5048_insort__remove1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A3: A,Xs: list(A)] :
( pp(member(A,A3,set2(A,Xs)))
=> ( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( linorder_insort_key(A,A,aTP_Lamp_me(A,A),A3,remove1(A,A3,Xs)) = Xs ) ) ) ) ).
% insort_remove1
tff(fact_5049_of__int_Orep__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: int] : aa(int,A,ring_1_of_int(A),X) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_lt(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).
% of_int.rep_eq
tff(fact_5050_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ls(nat,fun(nat,product_prod(nat,nat))))) ).
% uminus_int_def
tff(fact_5051_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)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(M,N)) ) ).
% numeral_xor_num
tff(fact_5052_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)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(M,N)) ) ).
% numeral_and_num
tff(fact_5053_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int_def
tff(fact_5054_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_mc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% minus_int_def
tff(fact_5055_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_ma(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% plus_int_def
tff(fact_5056_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] : fun_rp_inv_image(A,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_mh(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_5057_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),product_case_prod(nat,nat,A,aTP_Lamp_lt(nat,fun(nat,A)))) ) ) ).
% ring_1_class.of_int_def
tff(fact_5058_card__Min__le__sum,axiom,
! [A: $tType,A5: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),image(A,nat,F,A5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A5))) ) ).
% card_Min_le_sum
tff(fact_5059_of__nat__eq__id,axiom,
semiring_1_of_nat(nat) = id(nat) ).
% of_nat_eq_id
tff(fact_5060_id__funpow,axiom,
! [A: $tType,N: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),id(A)) = id(A) ).
% id_funpow
tff(fact_5061_rotate0,axiom,
! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).
% rotate0
tff(fact_5062_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ( aa(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_5063_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(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_5064_Min__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,X,bot_bot(set(A)))) = X ) ).
% Min_singleton
tff(fact_5065_push__bit__0__id,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ( aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),zero_zero(nat)) = id(A) ) ) ).
% push_bit_0_id
tff(fact_5066_drop__bit__0,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ( aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),zero_zero(nat)) = id(A) ) ) ).
% drop_bit_0
tff(fact_5067_Min_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) ) ) ) ) ) ).
% Min.bounded_iff
tff(fact_5068_Min__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X5)) ) ) ) ) ) ).
% Min_gr_iff
tff(fact_5069_Min__const,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [A5: set(B),C2: A] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( A5 != bot_bot(set(B)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),image(B,A,aTP_Lamp_mi(A,fun(B,A),C2),A5)) = C2 ) ) ) ) ).
% Min_const
tff(fact_5070_Min__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).
% Min_insert
tff(fact_5071_funpow__simps__right_I1_J,axiom,
! [A: $tType,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F) = id(A) ).
% funpow_simps_right(1)
tff(fact_5072_Min_OcoboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A3)) ) ) ) ).
% Min.coboundedI
tff(fact_5073_Min__eqI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ! [Y4: A] :
( pp(member(A,Y4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
=> ( pp(member(A,X,A5))
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = X ) ) ) ) ) ).
% Min_eqI
tff(fact_5074_Min__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),X)) ) ) ) ).
% Min_le
tff(fact_5075_Min__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> pp(member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5),A5)) ) ) ) ).
% Min_in
tff(fact_5076_Min_Oin__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ).
% Min.in_idem
tff(fact_5077_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))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lw(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).
% less_int_def
tff(fact_5078_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))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ly(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).
% less_eq_int_def
tff(fact_5079_nat__def,axiom,
nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))) ).
% nat_def
tff(fact_5080_Min_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A6)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5))) ) ) ) ) ).
% Min.boundedI
tff(fact_5081_Min_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)))
=> ! [A9: A] :
( pp(member(A,A9,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A9)) ) ) ) ) ) ).
% Min.boundedE
tff(fact_5082_eq__Min__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),M: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A5) )
<=> ( pp(member(A,M,A5))
& ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X5)) ) ) ) ) ) ) ).
% eq_Min_iff
tff(fact_5083_Min__le__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),X))
<=> ? [X5: A] :
( pp(member(A,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),X)) ) ) ) ) ) ).
% Min_le_iff
tff(fact_5084_Min__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),M: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = M )
<=> ( pp(member(A,M,A5))
& ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X5)) ) ) ) ) ) ) ).
% Min_eq_iff
tff(fact_5085_Min__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),X))
<=> ? [X5: A] :
( pp(member(A,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),X)) ) ) ) ) ) ).
% Min_less_iff
tff(fact_5086_Min__insert2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ! [B4: A] :
( pp(member(A,B4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B4)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,A3,A5)) = A3 ) ) ) ) ).
% Min_insert2
tff(fact_5087_Min__Inf,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).
% Min_Inf
tff(fact_5088_Min_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5))) ) ) ) ) ).
% Min.subset_imp
tff(fact_5089_Min__antimono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M7: set(A),N6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M7),N6))
=> ( ( M7 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N6)),aa(set(A),A,lattic643756798350308766er_Min(A),M7))) ) ) ) ) ).
% Min_antimono
tff(fact_5090_hom__Min__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H2: fun(A,A),N6: set(A)] :
( ! [X2: A,Y4: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Y4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H2,X2)),aa(A,A,H2,Y4))
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> ( ( N6 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic643756798350308766er_Min(A),N6)) = aa(set(A),A,lattic643756798350308766er_Min(A),image(A,A,H2,N6)) ) ) ) ) ) ).
% hom_Min_commute
tff(fact_5091_Min_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ) ).
% Min.subset
tff(fact_5092_Min_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [X2: A,Y4: A] : pp(member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Y4),insert(A,X2,insert(A,Y4,bot_bot(set(A))))))
=> pp(member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5),A5)) ) ) ) ) ).
% Min.closed
tff(fact_5093_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ~ pp(member(A,X,A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ) ).
% Min.insert_not_elem
tff(fact_5094_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(A,B,F,aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(B),B,lattic643756798350308766er_Min(B),image(A,B,F,A5)) ) ) ) ) ) ).
% mono_Min_commute
tff(fact_5095_Min_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),aa(set(A),A,lattic643756798350308766er_Min(A),B5)) ) ) ) ) ) ) ).
% Min.union
tff(fact_5096_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S3: set(B),F: fun(B,A),K: A] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( S3 != bot_bot(set(B)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_mj(fun(B,A),fun(A,fun(B,A)),F),K),S3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798350308766er_Min(A),image(B,A,F,S3))),K) ) ) ) ) ).
% Min_add_commute
tff(fact_5097_Min_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,X,A5)) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ).
% Min.insert_remove
tff(fact_5098_Min_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Min.remove
tff(fact_5099_positive__def,axiom,
positive = aa(fun(product_prod(int,int),bool),fun(rat,bool),map_fun(rat,product_prod(int,int),bool,bool,rep_Rat,id(bool)),aTP_Lamp_mk(product_prod(int,int),bool)) ).
% positive_def
tff(fact_5100_dual__Max,axiom,
! [A: $tType] :
( linorder(A)
=> ( lattices_Max(A,aTP_Lamp_ml(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).
% dual_Max
tff(fact_5101_dual__max,axiom,
! [A: $tType] :
( linorder(A)
=> ( max(A,aTP_Lamp_ml(A,fun(A,bool))) = ord_min(A) ) ) ).
% dual_max
tff(fact_5102_euclidean__size__nat__def,axiom,
euclid6346220572633701492n_size(nat) = id(nat) ).
% euclidean_size_nat_def
tff(fact_5103_linorder_OMax_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).
% linorder.Max.cong
tff(fact_5104_ord_Omax__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = B2 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B2) = A3 ) ) ) ).
% ord.max_def
tff(fact_5105_ord_Omax_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] : max(A,Less_eq) = max(A,Less_eq) ).
% ord.max.cong
tff(fact_5106_dual__min,axiom,
! [A: $tType] :
( linorder(A)
=> ( min(A,aTP_Lamp_ml(A,fun(A,bool))) = ord_max(A) ) ) ).
% dual_min
tff(fact_5107_sum__le__card__Max,axiom,
! [A: $tType,A5: set(A),F: fun(A,nat)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),image(A,nat,F,A5))))) ) ).
% sum_le_card_Max
tff(fact_5108_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_5109_Max__singleton,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,X,bot_bot(set(A)))) = X ) ).
% Max_singleton
tff(fact_5110_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aTP_Lamp_bx(nat,fun(nat,bool),N))) = N ) ) ).
% Max_divisors_self_nat
tff(fact_5111_Max_Obounded__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),X)) ) ) ) ) ) ).
% Max.bounded_iff
tff(fact_5112_Max__less__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),X)) ) ) ) ) ) ).
% Max_less_iff
tff(fact_5113_Max__const,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [A5: set(B),C2: A] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( ( A5 != bot_bot(set(B)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),image(B,A,aTP_Lamp_mi(A,fun(B,A),C2),A5)) = C2 ) ) ) ) ).
% Max_const
tff(fact_5114_Max__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ).
% Max_insert
tff(fact_5115_minus__Max__eq__Min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S3)) = aa(set(A),A,lattic643756798350308766er_Min(A),image(A,A,uminus_uminus(A),S3)) ) ) ) ) ).
% minus_Max_eq_Min
tff(fact_5116_minus__Min__eq__Max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S3: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S3)) = aa(set(A),A,lattic643756798349783984er_Max(A),image(A,A,uminus_uminus(A),S3)) ) ) ) ) ).
% minus_Min_eq_Max
tff(fact_5117_ord_Omin_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] : min(A,Less_eq) = min(A,Less_eq) ).
% ord.min.cong
tff(fact_5118_ord_Omin__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = A3 ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B2) = B2 ) ) ) ).
% ord.min_def
tff(fact_5119_Max_OcoboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic643756798349783984er_Max(A),A5))) ) ) ) ).
% Max.coboundedI
tff(fact_5120_Max__eq__if,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ! [X2: A] :
( pp(member(A,X2,A5))
=> ? [Xa: A] :
( pp(member(A,Xa,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa)) ) )
=> ( ! [X2: A] :
( pp(member(A,X2,B5))
=> ? [Xa: A] :
( pp(member(A,Xa,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa)) ) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,lattic643756798349783984er_Max(A),B5) ) ) ) ) ) ) ).
% Max_eq_if
tff(fact_5121_Max__eqI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ! [Y4: A] :
( pp(member(A,Y4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
=> ( pp(member(A,X,A5))
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = X ) ) ) ) ) ).
% Max_eqI
tff(fact_5122_Max__ge,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5))) ) ) ) ).
% Max_ge
tff(fact_5123_Max__in,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> pp(member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5),A5)) ) ) ) ).
% Max_in
tff(fact_5124_Max_Oin__idem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ).
% Max.in_idem
tff(fact_5125_Max_OboundedI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X)) ) ) ) ) ).
% Max.boundedI
tff(fact_5126_Max_OboundedE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),X))
=> ! [A9: A] :
( pp(member(A,A9,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A9),X)) ) ) ) ) ) ).
% Max.boundedE
tff(fact_5127_eq__Max__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),M: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A5) )
<=> ( pp(member(A,M,A5))
& ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M)) ) ) ) ) ) ) ).
% eq_Max_iff
tff(fact_5128_Max__ge__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)))
<=> ? [X5: A] :
( pp(member(A,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) ) ) ) ) ) ).
% Max_ge_iff
tff(fact_5129_Max__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),M: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = M )
<=> ( pp(member(A,M,A5))
& ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M)) ) ) ) ) ) ) ).
% Max_eq_iff
tff(fact_5130_Max__gr__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)))
<=> ? [X5: A] :
( pp(member(A,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X5)) ) ) ) ) ) ).
% Max_gr_iff
tff(fact_5131_Max__insert2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ! [B4: A] :
( pp(member(A,B4,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),A3)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,A3,A5)) = A3 ) ) ) ) ).
% Max_insert2
tff(fact_5132_Max__Sup,axiom,
! [A: $tType] :
( comple5582772986160207858norder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).
% Max_Sup
tff(fact_5133_Max_Osubset__imp,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5))) ) ) ) ) ).
% Max.subset_imp
tff(fact_5134_Max__mono,axiom,
! [A: $tType] :
( linorder(A)
=> ! [M7: set(A),N6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M7),N6))
=> ( ( M7 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M7)),aa(set(A),A,lattic643756798349783984er_Max(A),N6))) ) ) ) ) ).
% Max_mono
tff(fact_5135_hom__Max__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H2: fun(A,A),N6: set(A)] :
( ! [X2: A,Y4: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Y4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H2,X2)),aa(A,A,H2,Y4))
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> ( ( N6 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic643756798349783984er_Max(A),N6)) = aa(set(A),A,lattic643756798349783984er_Max(A),image(A,A,H2,N6)) ) ) ) ) ) ).
% hom_Max_commute
tff(fact_5136_Max_Osubset,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B5)),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ) ).
% Max.subset
tff(fact_5137_Max_Oclosed,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [X2: A,Y4: A] : pp(member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Y4),insert(A,X2,insert(A,Y4,bot_bot(set(A))))))
=> pp(member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5),A5)) ) ) ) ) ).
% Max.closed
tff(fact_5138_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ~ pp(member(A,X,A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ) ).
% Max.insert_not_elem
tff(fact_5139_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(A,B,F,aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(B),B,lattic643756798349783984er_Max(B),image(A,B,F,A5)) ) ) ) ) ) ).
% mono_Max_commute
tff(fact_5140_Max_Ounion,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ) ) ).
% Max.union
tff(fact_5141_Sup__nat__def,axiom,
! [X6: set(nat)] :
( ( ( X6 = bot_bot(set(nat)) )
=> ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = zero_zero(nat) ) )
& ( ( X6 != bot_bot(set(nat)) )
=> ( aa(set(nat),nat,complete_Sup_Sup(nat),X6) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),X6) ) ) ) ).
% Sup_nat_def
tff(fact_5142_card__le__Suc__Max,axiom,
! [S3: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),S3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S3)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S3)))) ) ).
% card_le_Suc_Max
tff(fact_5143_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S3: set(B),F: fun(B,A),K: A] :
( pp(aa(set(B),bool,finite_finite2(B),S3))
=> ( ( S3 != bot_bot(set(B)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_mj(fun(B,A),fun(A,fun(B,A)),F),K),S3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798349783984er_Max(A),image(B,A,F,S3))),K) ) ) ) ) ).
% Max_add_commute
tff(fact_5144_divide__nat__def,axiom,
! [N: nat,M: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_mm(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).
% divide_nat_def
tff(fact_5145_gcd__is__Max__divisors__nat,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),gcd_gcd(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_mn(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ).
% gcd_is_Max_divisors_nat
tff(fact_5146_Max_Oinsert__remove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,X,A5)) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ).
% Max.insert_remove
tff(fact_5147_Max_Oremove,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Max.remove
tff(fact_5148_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_5149_Gcd__eq__Max,axiom,
! [M7: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),M7))
=> ( ( M7 != bot_bot(set(nat)) )
=> ( ~ pp(member(nat,zero_zero(nat),M7))
=> ( gcd_Gcd(nat,M7) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),image(nat,set(nat),aTP_Lamp_mo(nat,set(nat)),M7))) ) ) ) ) ).
% Gcd_eq_Max
tff(fact_5150_dual__Min,axiom,
! [A: $tType] :
( linorder(A)
=> ( lattices_Min(A,aTP_Lamp_ml(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).
% dual_Min
tff(fact_5151_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_5152_positive_Orsp,axiom,
pp(aa(fun(product_prod(int,int),bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),bool),bNF_rel_fun(product_prod(int,int),product_prod(int,int),bool,bool,ratrel,fequal(bool)),aTP_Lamp_mk(product_prod(int,int),bool)),aTP_Lamp_mk(product_prod(int,int),bool))) ).
% positive.rsp
tff(fact_5153_Max__divisors__self__int,axiom,
! [N: int] :
( ( N != zero_zero(int) )
=> ( aa(set(int),int,lattic643756798349783984er_Max(int),collect(int,aTP_Lamp_bo(int,fun(int,bool),N))) = aa(int,int,abs_abs(int),N) ) ) ).
% Max_divisors_self_int
tff(fact_5154_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( power(B)
& power(A) )
=> ! [R2: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R2,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),R2,bNF_rel_fun(A,B,A,B,R2,R2)),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),R2,bNF_rel_fun(nat,nat,A,B,fequal(nat),R2)),power_power(A)),power_power(B))) ) ) ) ).
% power_transfer
tff(fact_5155_linorder_OMin_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).
% linorder.Min.cong
tff(fact_5156_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& semiring_numeral(B)
& monoid_add(A)
& semiring_numeral(A) )
=> ! [R2: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,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),R2,bNF_rel_fun(A,B,A,B,R2,R2)),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),R2),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).
% transfer_rule_numeral
tff(fact_5157_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ring_1(B)
& ring_1(A) )
=> ! [R2: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,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),R2,bNF_rel_fun(A,B,A,B,R2,R2)),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,R2,R2),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),R2),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).
% transfer_rule_of_int
tff(fact_5158_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N != zero_zero(int) )
=> ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N) = aa(set(int),int,lattic643756798349783984er_Max(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_mp(int,fun(int,fun(int,bool)),N),M))) ) ) ).
% gcd_is_Max_divisors_int
tff(fact_5159_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& semiring_1(A) )
=> ! [R2: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,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),R2,bNF_rel_fun(A,B,A,B,R2,R2)),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),R2),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).
% transfer_rule_of_nat
tff(fact_5160_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_5161_antimono__funpow,axiom,
! [A: $tType] :
( ( lattice(A)
& order_top(A) )
=> ! [Q: fun(A,A)] :
( pp(aa(fun(A,A),bool,order_mono(A,A),Q))
=> order_antimono(nat,A,aTP_Lamp_mq(fun(A,A),fun(nat,A),Q)) ) ) ).
% antimono_funpow
tff(fact_5162_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_5163_integer__of__nat_Orep__eq,axiom,
! [X: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,X)) = aa(nat,int,semiring_1_of_nat(int),X) ).
% integer_of_nat.rep_eq
tff(fact_5164_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_5165_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_5166_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_5167_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_mr(int,int)),aTP_Lamp_mr(int,int))) ).
% dup.rsp
tff(fact_5168_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_ms(num,fun(num,int))),aTP_Lamp_ms(num,fun(num,int)))) ).
% sub.rsp
tff(fact_5169_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_5170_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_5171_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_5172_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_5173_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_5174_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_5175_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_5176_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_5177_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_5178_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_5179_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_5180_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_5181_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_5182_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_5183_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_5184_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_5185_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_5186_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_5187_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_5188_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_5189_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_5190_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_5191_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_5192_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_5193_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_5194_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_5195_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_5196_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_5197_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_5198_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_5199_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_5200_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_5201_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_5202_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_5203_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_5204_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_5205_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_5206_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_5207_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_5208_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_5209_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_5210_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_5211_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_5212_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_5213_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_5214_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_mt(int,fun(int,product_prod(int,int)))),aTP_Lamp_mt(int,fun(int,product_prod(int,int))))) ).
% Fract.rsp
tff(fact_5215_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),comp(nat,num,int,num_of_nat),nat2)),aa(fun(int,nat),fun(int,num),comp(nat,num,int,num_of_nat),nat2))) ).
% num_of_integer.rsp
tff(fact_5216_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X))) ) ) ) ).
% antimonoD
tff(fact_5217_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X))) ) ) ) ).
% antimonoE
tff(fact_5218_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y4)),aa(A,B,F,X2))) )
=> order_antimono(A,B,F) ) ) ).
% antimonoI
tff(fact_5219_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_antimono(A,B,F)
<=> ! [X5: A,Y5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Y5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y5)),aa(A,B,F,X5))) ) ) ) ).
% antimono_def
tff(fact_5220_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( zero_neq_one(B)
& zero_neq_one(A) )
=> ! [R2: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R2,zero_zero(A)),zero_zero(B)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,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),R2),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).
% transfer_rule_of_bool
tff(fact_5221_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_ix(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_ix(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).
% times_rat.rsp
tff(fact_5222_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_iw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_iw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).
% plus_rat.rsp
tff(fact_5223_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_5224_antimono__iff__le__Suc,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A)] :
( order_antimono(nat,A,F)
<=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,aa(nat,nat,suc,N5))),aa(nat,A,F,N5))) ) ) ).
% antimono_iff_le_Suc
tff(fact_5225_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_5226_integer__of__nat_Oabs__eq,axiom,
! [X: nat] : aa(nat,code_integer,code_integer_of_nat,X) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).
% integer_of_nat.abs_eq
tff(fact_5227_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_5228_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) ) ) ) ).
% min_of_antimono
tff(fact_5229_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),X: A,Y: A] :
( order_antimono(A,B,F)
=> ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F,X)),aa(A,B,F,Y)) = aa(A,B,F,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) ) ) ) ).
% max_of_antimono
tff(fact_5230_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_5231_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_iv(product_prod(int,int),product_prod(int,int))),aTP_Lamp_iv(product_prod(int,int),product_prod(int,int)))) ).
% uminus_rat.rsp
tff(fact_5232_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_iu(product_prod(int,int),product_prod(int,int))),aTP_Lamp_iu(product_prod(int,int),product_prod(int,int)))) ).
% inverse_rat.rsp
tff(fact_5233_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_iw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).
% plus_rat.transfer
tff(fact_5234_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_iu(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).
% inverse_rat.transfer
tff(fact_5235_positive_Otransfer,axiom,
pp(aa(fun(rat,bool),bool,aa(fun(product_prod(int,int),bool),fun(fun(rat,bool),bool),bNF_rel_fun(product_prod(int,int),rat,bool,bool,pcr_rat,fequal(bool)),aTP_Lamp_mk(product_prod(int,int),bool)),positive)) ).
% positive.transfer
tff(fact_5236_one__rat_Otransfer,axiom,
pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),one_one(int))),one_one(rat))) ).
% one_rat.transfer
tff(fact_5237_zero__rat_Otransfer,axiom,
pp(aa(rat,bool,aa(product_prod(int,int),fun(rat,bool),pcr_rat,aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int))),zero_zero(rat))) ).
% zero_rat.transfer
tff(fact_5238_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_iv(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).
% uminus_rat.transfer
tff(fact_5239_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_ix(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).
% times_rat.transfer
tff(fact_5240_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).
% times_int.transfer
tff(fact_5241_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),bool,aa(fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))),fun(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),bool),bNF_rel_fun(A,B,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),S3,bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(fun(num,fun(A,A)),fun(num,A)),fun(fun(num,fun(B,B)),fun(num,B)),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S3,S3)),bNF_rel_fun(fun(num,fun(A,A)),fun(num,fun(B,B)),fun(num,A),fun(num,B),bNF_rel_fun(num,num,fun(A,A),fun(B,B),fequal(num),bNF_rel_fun(A,B,A,B,S3,S3)),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),rec_num(A)),rec_num(B))) ).
% num.rec_transfer
tff(fact_5242_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),comp(nat,num,int,num_of_nat),nat2)) ).
% num_of_integer_def
tff(fact_5243_verit__eq__simplify_I20_J,axiom,
! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X22: num] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),bit0(X22)) = aa(A,A,aa(num,fun(A,A),F22,X22),aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),X22)) ).
% verit_eq_simplify(20)
tff(fact_5244_verit__eq__simplify_I19_J,axiom,
! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A))] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),one2) = F1 ).
% verit_eq_simplify(19)
tff(fact_5245_verit__eq__simplify_I21_J,axiom,
! [A: $tType,F1: A,F22: fun(num,fun(A,A)),F32: fun(num,fun(A,A)),X32: num] : aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),aa(num,num,bit1,X32)) = aa(A,A,aa(num,fun(A,A),F32,X32),aa(num,A,aa(fun(num,fun(A,A)),fun(num,A),aa(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)),aa(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),rec_num(A),F1),F22),F32),X32)) ).
% verit_eq_simplify(21)
tff(fact_5246_zero__int_Otransfer,axiom,
pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))),zero_zero(int))) ).
% zero_int.transfer
tff(fact_5247_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_mu(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).
% int_transfer
tff(fact_5248_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ls(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).
% uminus_int.transfer
tff(fact_5249_nat_Otransfer,axiom,
pp(aa(fun(int,nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),bool),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))),nat2)) ).
% nat.transfer
tff(fact_5250_one__int_Otransfer,axiom,
pp(aa(int,bool,aa(product_prod(nat,nat),fun(int,bool),pcr_int,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))),one_one(int))) ).
% one_int.transfer
tff(fact_5251_of__int_Otransfer,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(aa(fun(int,A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),bool),bNF_rel_fun(product_prod(nat,nat),int,A,A,pcr_int,fequal(A)),product_case_prod(nat,nat,A,aTP_Lamp_lt(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).
% of_int.transfer
tff(fact_5252_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))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lw(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).
% less_int.transfer
tff(fact_5253_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))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ly(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).
% less_eq_int.transfer
tff(fact_5254_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_ma(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).
% plus_int.transfer
tff(fact_5255_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_mc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).
% minus_int.transfer
tff(fact_5256_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: fun(A,fun(B,bool))] : pp(aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),bool,aa(fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))),fun(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),bool),bNF_rel_fun(A,B,fun(fun(num,A),fun(fun(num,A),fun(num,A))),fun(fun(num,B),fun(fun(num,B),fun(num,B))),S3,bNF_rel_fun(fun(num,A),fun(num,B),fun(fun(num,A),fun(num,A)),fun(fun(num,B),fun(num,B)),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(fun(num,A),fun(num,B),fun(num,A),fun(num,B),bNF_rel_fun(num,num,A,B,fequal(num),S3),bNF_rel_fun(num,num,A,B,fequal(num),S3)))),case_num(A)),case_num(B))) ).
% num.case_transfer
tff(fact_5257_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_lp(nat,fun(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_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% times_int.rsp
tff(fact_5258_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : size_list(A,X,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,X,X21)),size_list(A,X,X222))),aa(nat,nat,suc,zero_zero(nat))) ).
% list.size_gen(2)
tff(fact_5259_intrel__iff,axiom,
! [X: 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),product_Pair(nat,nat,X),Y)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,U),V)))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y) ) ) ).
% intrel_iff
tff(fact_5260_int_Orel__eq__transfer,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))),intrel),fequal(int))) ).
% int.rel_eq_transfer
tff(fact_5261_int_Oabs__eq__iff,axiom,
! [X: product_prod(nat,nat),Y: product_prod(nat,nat)] :
( ( aa(product_prod(nat,nat),int,abs_Integ,X) = aa(product_prod(nat,nat),int,abs_Integ,Y) )
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X),Y)) ) ).
% int.abs_eq_iff
tff(fact_5262_num_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H2: fun(A,B),F1: A,F22: fun(num,A),F32: fun(num,A),Num: num] : aa(A,B,H2,aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),Num)) = aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),aa(A,B,H2,F1)),aa(fun(num,A),fun(num,B),aTP_Lamp_mv(fun(A,B),fun(fun(num,A),fun(num,B)),H2),F22)),aa(fun(num,A),fun(num,B),aTP_Lamp_mv(fun(A,B),fun(fun(num,A),fun(num,B)),H2),F32)),Num) ).
% num.case_distrib
tff(fact_5263_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X22: num] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),bit0(X22)) = aa(num,A,F22,X22) ).
% verit_eq_simplify(17)
tff(fact_5264_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),one2) = F1 ).
% verit_eq_simplify(16)
tff(fact_5265_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),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat)))) ).
% zero_int.rsp
tff(fact_5266_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X32: num] : aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),F1),F22),F32),aa(num,num,bit1,X32)) = aa(num,A,F32,X32) ).
% verit_eq_simplify(18)
tff(fact_5267_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: fun(A,nat)] : size_list(A,X,nil(A)) = zero_zero(nat) ).
% list.size_gen(1)
tff(fact_5268_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
( pp(member(A,X,set2(A,Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(A,nat,F,X)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),size_list(A,F,Xs))) ) ) ).
% size_list_estimation
tff(fact_5269_size__list__pointwise,axiom,
! [A: $tType,Xs: list(A),F: fun(A,nat),G: fun(A,nat)] :
( ! [X2: A] :
( pp(member(A,X2,set2(A,Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,G,X2))) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),size_list(A,F,Xs)),size_list(A,G,Xs))) ) ).
% size_list_pointwise
tff(fact_5270_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
( pp(member(A,X,set2(A,Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F,X)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),size_list(A,F,Xs))) ) ) ).
% size_list_estimation'
tff(fact_5271_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),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ls(nat,fun(nat,product_prod(nat,nat))))),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_ls(nat,fun(nat,product_prod(nat,nat)))))) ).
% uminus_int.rsp
tff(fact_5272_nat_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),product_case_prod(nat,nat,nat,minus_minus(nat))),product_case_prod(nat,nat,nat,minus_minus(nat)))) ).
% nat.rsp
tff(fact_5273_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),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat)))) ).
% one_int.rsp
tff(fact_5274_of__int_Orsp,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(aa(fun(product_prod(nat,nat),A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),A,A,intrel,fequal(A)),product_case_prod(nat,nat,A,aTP_Lamp_lt(nat,fun(nat,A)))),product_case_prod(nat,nat,A,aTP_Lamp_lt(nat,fun(nat,A))))) ) ).
% of_int.rsp
tff(fact_5275_intrel__def,axiom,
intrel = product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_mx(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).
% intrel_def
tff(fact_5276_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))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lw(nat,fun(nat,fun(product_prod(nat,nat),bool))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_lw(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).
% less_int.rsp
tff(fact_5277_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))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ly(nat,fun(nat,fun(product_prod(nat,nat),bool))))),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ly(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).
% less_eq_int.rsp
tff(fact_5278_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_ma(nat,fun(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_ma(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% plus_int.rsp
tff(fact_5279_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)),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aTP_Lamp_mc(nat,fun(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_mc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% minus_int.rsp
tff(fact_5280_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_5281_execute__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A),H2: heap_ext(product_unit)] : aa(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)),H2) = 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)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_if(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,map(nat,A,F,upt(zero_zero(nat),N)),H2))) ) ).
% execute_make
tff(fact_5282_relpow__finite__bounded1,axiom,
! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> 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))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_my(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_mz(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ) ).
% relpow_finite_bounded1
tff(fact_5283_tl__upt,axiom,
! [M: nat,N: nat] : tl(nat,upt(M,N)) = upt(aa(nat,nat,suc,M),N) ).
% tl_upt
tff(fact_5284_relpow__1,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),one_one(nat)),R2) = R2 ).
% relpow_1
tff(fact_5285_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_5286_finite__relpow,axiom,
! [A: $tType,R2: set(product_prod(A,A)),N: nat] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).
% finite_relpow
tff(fact_5287_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] : tl(A,replicate(A,N,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ).
% tl_replicate
tff(fact_5288_execute__heap,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),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)))),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_5289_execute__guard_I2_J,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H2: 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,H2))
=> ( aa(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)),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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H2)) ) ) ).
% execute_guard(2)
tff(fact_5290_Heap__eqI,axiom,
! [A: $tType,F: heap_Time_Heap(A),G: heap_Time_Heap(A)] :
( ! [H4: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H4) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,G),H4)
=> ( F = G ) ) ).
% Heap_eqI
tff(fact_5291_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),R: A,H: heap_ext(product_unit),N: nat] :
( ( aa(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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H),N))) )
=> heap_Time_effect(A,C2,H2,H,R,N) ) ).
% effectI
tff(fact_5292_effect__def,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,C2,H2,H,R,N)
<=> ( aa(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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H),N))) ) ) ).
% effect_def
tff(fact_5293_sorted__tl,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),tl(A,Xs)) ) ) ).
% sorted_tl
tff(fact_5294_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z2),R2)) ) ) ).
% relpow_Suc_E
tff(fact_5295_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N: nat,R2: set(product_prod(A,A)),Z2: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z2),R2))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2))) ) ) ).
% relpow_Suc_I
tff(fact_5296_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
=> ? [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),R2))
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).
% relpow_Suc_D2
tff(fact_5297_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2)))
=> ~ ! [Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),R2))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2))) ) ) ).
% relpow_Suc_E2
tff(fact_5298_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A)),Z2: A,N: nat] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),R2))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2))) ) ) ).
% relpow_Suc_I2
tff(fact_5299_relpow__0__I,axiom,
! [A: $tType,X: A,R2: set(product_prod(A,A))] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),X),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2))) ).
% relpow_0_I
tff(fact_5300_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2)))
=> ( X = Y ) ) ).
% relpow_0_E
tff(fact_5301_take__tl,axiom,
! [A: $tType,N: nat,Xs: list(A)] : take(A,N,tl(A,Xs)) = tl(A,take(A,aa(nat,nat,suc,N),Xs)) ).
% take_tl
tff(fact_5302_drop__Suc,axiom,
! [A: $tType,N: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),Xs) = drop(A,N,tl(A,Xs)) ).
% drop_Suc
tff(fact_5303_execute__assert_I1_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,H2: heap_ext(product_unit)] :
( pp(aa(A,bool,P,X))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P,X)),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),X),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H2),one_one(nat)))) ) ) ).
% execute_assert(1)
tff(fact_5304_execute__tap,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_tap(A,F)),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),A,F,H2)),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H2),one_one(nat)))) ).
% execute_tap
tff(fact_5305_execute__return,axiom,
! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X)) = 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)))),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_ik(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% execute_return
tff(fact_5306_execute__ureturn,axiom,
! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X)) = 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)))),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_il(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).
% execute_ureturn
tff(fact_5307_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
=> ( ( ( N = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y4: A,M5: nat] :
( ( N = aa(nat,nat,suc,M5) )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R2)))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z2),R2)) ) ) ) ) ).
% relpow_E
tff(fact_5308_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
=> ( ( ( N = zero_zero(nat) )
=> ( X != Z2 ) )
=> ~ ! [Y4: A,M5: nat] :
( ( N = aa(nat,nat,suc,M5) )
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),R2))
=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M5),R2))) ) ) ) ) ).
% relpow_E2
tff(fact_5309_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_5310_relpow__empty,axiom,
! [A: $tType,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).
% relpow_empty
tff(fact_5311_relpow__finite__bounded,axiom,
! [A: $tType,R2: set(product_prod(A,A)),K: nat] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
=> 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))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),R2)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_my(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_na(set(product_prod(A,A)),fun(nat,bool),R2)))))) ) ).
% relpow_finite_bounded
tff(fact_5312_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType,Xs: list(A)] :
( ( ( Xs = nil(A) )
=> ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
& ( ( Xs != nil(A) )
=> ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),tl(A,Xs))) ) ) ) ).
% Nitpick.size_list_simp(2)
tff(fact_5313_nth__tl,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),tl(A,Xs))))
=> ( aa(nat,A,nth(A,tl(A,Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).
% nth_tl
tff(fact_5314_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2)))
<=> ? [F4: fun(nat,A)] :
( ( aa(nat,A,F4,zero_zero(nat)) = A3 )
& ( aa(nat,A,F4,N) = B2 )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),N))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F4,I5)),aa(nat,A,F4,aa(nat,nat,suc,I5))),R2)) ) ) ) ).
% relpow_fun_conv
tff(fact_5315_execute__of__list,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit)] : aa(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)),H2) = 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)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ii(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,H2))) ) ).
% execute_of_list
tff(fact_5316_execute__new,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A,H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_new(A,N,X)),H2) = 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)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_if(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,X),H2))) ) ).
% execute_new
tff(fact_5317_take__Suc,axiom,
! [A: $tType,Xs: list(A),N: nat] :
( ( Xs != nil(A) )
=> ( take(A,aa(nat,nat,suc,N),Xs) = cons(A,hd(A,Xs),take(A,N,tl(A,Xs))) ) ) ).
% take_Suc
tff(fact_5318_ntrancl__def,axiom,
! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,N,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_my(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_nb(nat,fun(nat,bool),N)))) ).
% ntrancl_def
tff(fact_5319_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_5320_trancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
=> ( transitive_trancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_my(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_mz(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).
% trancl_finite_eq_relpow
tff(fact_5321_ntrancl__Zero,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R2) = R2 ).
% ntrancl_Zero
tff(fact_5322_take__bit__num__simps_I2_J,axiom,
! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).
% take_bit_num_simps(2)
tff(fact_5323_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),case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).
% take_bit_num_simps(4)
tff(fact_5324_execute__len,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(nat,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(nat,array_len(A,A3)),H2) = 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)),product_Pair(nat,product_prod(heap_ext(product_unit),nat),aa(array(A),nat,array_length(A,H2),A3)),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H2),one_one(nat)))) ) ).
% execute_len
tff(fact_5325_finite__trancl__ntranl,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
=> ( transitive_trancl(A,R2) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R2)),one_one(nat)),R2) ) ) ).
% finite_trancl_ntranl
tff(fact_5326_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,set2(product_prod(A,A),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),set2(product_prod(A,A),Xs))),one_one(nat)),set2(product_prod(A,A),Xs)) ).
% trancl_set_ntrancl
tff(fact_5327_trancl__power,axiom,
! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),P3,transitive_trancl(A,R2)))
<=> ? [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
& pp(member(product_prod(A,A),P3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N5),R2))) ) ) ).
% trancl_power
tff(fact_5328_option_Osize__neq,axiom,
! [A: $tType,X: option(A)] : aa(option(A),nat,size_size(option(A)),X) != zero_zero(nat) ).
% option.size_neq
tff(fact_5329_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: fun(A,nat),X22: A] : size_option(A,X,aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% option.size_gen(2)
tff(fact_5330_sorted__find__Min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( ? [X3: A] :
( pp(member(A,X3,set2(A,Xs)))
& pp(aa(A,bool,P,X3)) )
=> ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_nc(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).
% sorted_find_Min
tff(fact_5331_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R: A,H2: 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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H2),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R),aa(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),N),N2)))) ).
% timeFrame.simps(1)
tff(fact_5332_timeFrame__zero,axiom,
! [A: $tType,H2: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,zero_zero(nat),H2) = H2 ).
% timeFrame_zero
tff(fact_5333_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_5334_find__Some__iff2,axiom,
! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
( ( aa(A,option(A),some(A),X) = find(A,P,Xs) )
<=> ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I5)))
& ( X = aa(nat,A,nth(A,Xs),I5) )
& ! [J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I5))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).
% find_Some_iff2
tff(fact_5335_find__Some__iff,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
<=> ? [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I5)))
& ( X = aa(nat,A,nth(A,Xs),I5) )
& ! [J3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I5))
=> ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).
% find_Some_iff
tff(fact_5336_tap__def,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F) = heap_Time_Heap2(A,aTP_Lamp_nd(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_5337_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),X: A,H: 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))),heap_Time_execute(A,F),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),X),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H),N))) )
=> ( aa(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)),H2) = heap_Time_timeFrame(B,N,aa(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,X)),H)) ) ) ).
% execute_bind(1)
tff(fact_5338_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R2))
=> ( transitive_rtrancl(A,R2) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_my(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R2),collect(nat,aTP_Lamp_na(set(product_prod(A,A)),fun(nat,bool),R2)))) ) ) ).
% rtrancl_finite_eq_relpow
tff(fact_5339_Heap_Oinject,axiom,
! [A: $tType,X: 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))))] :
( ( heap_Time_Heap2(A,X) = heap_Time_Heap2(A,Ya) )
<=> ( X = Ya ) ) ).
% Heap.inject
tff(fact_5340_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_ne(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_5341_Heap__execute,axiom,
! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_Heap2(A,heap_Time_execute(A,F)) = F ).
% Heap_execute
tff(fact_5342_execute_Osimps,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_execute(A,heap_Time_Heap2(A,F)) = F ).
% execute.simps
tff(fact_5343_distrib__if__bind,axiom,
! [A: $tType,B: $tType,B2: bool,C2: heap_Time_Heap(B),D2: 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,D2),F) = heap_Time_bind(B,A,C2,F) ) )
& ( ~ pp(B2)
=> ( heap_Time_bind(B,A,if(heap_Time_Heap(B),B2,C2,D2),F) = heap_Time_bind(B,A,D2,F) ) ) ) ).
% distrib_if_bind
tff(fact_5344_Heap_Oexhaust,axiom,
! [A: $tType,Y: heap_Time_Heap(A)] :
~ ! [X2: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : Y != heap_Time_Heap2(A,X2) ).
% Heap.exhaust
tff(fact_5345_assert__cong,axiom,
! [B: $tType,A: $tType,P: fun(A,bool),P2: fun(A,bool),F: fun(A,heap_Time_Heap(B)),F5: fun(A,heap_Time_Heap(B)),X: A] :
( ( P = P2 )
=> ( ! [X2: A] :
( pp(aa(A,bool,P2,X2))
=> ( aa(A,heap_Time_Heap(B),F,X2) = aa(A,heap_Time_Heap(B),F5,X2) ) )
=> ( heap_Time_bind(A,B,heap_Time_assert(A,P,X),F) = heap_Time_bind(A,B,heap_Time_assert(A,P2,X),F5) ) ) ) ).
% assert_cong
tff(fact_5346_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_5347_ureturn__bind,axiom,
! [A: $tType,B: $tType,X: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X),F) = aa(B,heap_Time_Heap(A),F,X) ).
% ureturn_bind
tff(fact_5348_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H2: heap_ext(product_unit),H3: heap_ext(product_unit),R4: A,N: nat] :
( heap_Time_effect(A,heap_Time_bind(B,A,F,G),H2,H3,R4,N)
=> ~ ! [H5: heap_ext(product_unit),R3: B,N1: nat] :
( heap_Time_effect(B,F,H2,H5,R3,N1)
=> ! [N22: nat] :
( heap_Time_effect(A,aa(B,heap_Time_Heap(A),G,R3),H5,H3,R4,N22)
=> ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),N22) ) ) ) ) ).
% effect_bindE
tff(fact_5349_effect__bindI,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat,G: fun(A,heap_Time_Heap(B)),H3: heap_ext(product_unit),R4: B,N2: nat] :
( heap_Time_effect(A,F,H2,H,R,N)
=> ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),G,R),H,H3,R4,N2)
=> heap_Time_effect(B,heap_Time_bind(A,B,F,G),H2,H3,R4,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)) ) ) ).
% effect_bindI
tff(fact_5350_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_nf(fun(B,A),fun(B,heap_Time_Heap(A)),G)) ).
% bind_lift
tff(fact_5351_Heap_Osize_I2_J,axiom,
! [A: $tType,X: 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)),heap_Time_Heap2(A,X)) = aa(nat,nat,suc,zero_zero(nat)) ).
% Heap.size(2)
tff(fact_5352_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) = 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)))),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_5353_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),X: A,H: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B)),Y: B,H3: heap_ext(product_unit),N2: nat] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),X),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H),N))) )
=> ( ( aa(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,X)),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)),product_Pair(B,product_prod(heap_ext(product_unit),nat),Y),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H3),N2))) )
=> ( aa(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)),H2) = 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)),product_Pair(B,product_prod(heap_ext(product_unit),nat),Y),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N2)))) ) ) ) ).
% execute_bind_eq_SomeI
tff(fact_5354_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_ng(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F)) ).
% bind_return
tff(fact_5355_return__bind,axiom,
! [A: $tType,B: $tType,X: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),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_nh(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X),F)) ).
% return_bind
tff(fact_5356_Heap_Osize__gen,axiom,
! [A: $tType,Xa4: fun(A,nat),X: 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,Xa4),heap_Time_Heap2(A,X)) = aa(nat,nat,suc,zero_zero(nat)) ).
% Heap.size_gen
tff(fact_5357_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),X: A,H: 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))),heap_Time_execute(A,F),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),X),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H),N))) )
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X),H)
=> heap_Time_success(B,heap_Time_bind(A,B,F,G),H2) ) ) ).
% success_bind_executeI
tff(fact_5358_take__bit__num__def,axiom,
! [N: nat,M: num] :
( ( ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
=> ( bit_take_bit_num(N,M) = none(num) ) )
& ( ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
=> ( bit_take_bit_num(N,M) = aa(num,option(num),some(num),aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),N),aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).
% take_bit_num_def
tff(fact_5359_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [M: num,N: num] : power_int(A,aa(num,A,numeral_numeral(A),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(M,N))) ) ).
% power_int_numeral_neg_numeral
tff(fact_5360_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_5361_power__int__1__right,axiom,
! [A: $tType] :
( ( inverse(A)
& monoid_mult(A) )
=> ! [Y: A] : power_int(A,Y,one_one(int)) = Y ) ).
% power_int_1_right
tff(fact_5362_power__int__sgn,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,N: int] : aa(A,A,sgn_sgn(A),power_int(A,A3,N)) = power_int(A,aa(A,A,sgn_sgn(A),A3),N) ) ).
% power_int_sgn
tff(fact_5363_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,W2: num,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W2)),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,aa(num,A,numeral_numeral(A),W2),M)) ) ).
% power_int_mult_distrib_numeral2
tff(fact_5364_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( field(A)
=> ! [W2: num,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W2),M)),power_int(A,Y,M)) ) ).
% power_int_mult_distrib_numeral1
tff(fact_5365_power__int__eq__0__iff,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] :
( ( power_int(A,X,N) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( N != zero_zero(int) ) ) ) ) ).
% power_int_eq_0_iff
tff(fact_5366_power__int__0__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [M: int] :
( ( M != zero_zero(int) )
=> ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ).
% power_int_0_left
tff(fact_5367_power__int__0__right,axiom,
! [B: $tType] :
( ( inverse(B)
& power(B) )
=> ! [X: B] : power_int(B,X,zero_zero(int)) = one_one(B) ) ).
% power_int_0_right
tff(fact_5368_abs__power__int__minus,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A3),N)) = aa(A,A,abs_abs(A),power_int(A,A3,N)) ) ).
% abs_power_int_minus
tff(fact_5369_power__int__of__nat,axiom,
! [A: $tType] :
( ( inverse(A)
& power(A) )
=> ! [X: A,N: nat] : power_int(A,X,aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N) ) ).
% power_int_of_nat
tff(fact_5370_power__int__mult__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: num,N: num] : power_int(A,power_int(A,X,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).
% power_int_mult_numeral
tff(fact_5371_take__bit__num__simps_I1_J,axiom,
! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).
% take_bit_num_simps(1)
tff(fact_5372_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_5373_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_5374_power__int__numeral,axiom,
! [A: $tType] :
( ( inverse(A)
& power(A) )
=> ! [X: A,N: num] : power_int(A,X,aa(num,int,numeral_numeral(int),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),N)) ) ).
% power_int_numeral
tff(fact_5375_power__int__add__numeral2,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,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_5376_power__int__add__numeral,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),power_int(A,X,aa(num,int,numeral_numeral(int),N))) = power_int(A,X,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_5377_power__int__minus1__right,axiom,
! [A: $tType] :
( ( inverse(A)
& monoid_mult(A) )
=> ! [Y: A] : power_int(A,Y,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y) ) ).
% power_int_minus1_right
tff(fact_5378_power__int__mono__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,N: int] :
( 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(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_eq(A),power_int(A,A3,N)),power_int(A,B2,N)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).
% power_int_mono_iff
tff(fact_5379_power__int__minus__left__even,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
=> ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = power_int(A,A3,N) ) ) ) ).
% power_int_minus_left_even
tff(fact_5380_power__int__minus__left__odd,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int,A3: A] :
( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
=> ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,uminus_uminus(A),power_int(A,A3,N)) ) ) ) ).
% power_int_minus_left_odd
tff(fact_5381_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_ni(num,option(num)),bit_take_bit_num(N,M)) ).
% take_bit_num_simps(3)
tff(fact_5382_power__int__not__zero,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] :
( ( ( X != zero_zero(A) )
| ( N = zero_zero(int) ) )
=> ( power_int(A,X,N) != zero_zero(A) ) ) ) ).
% power_int_not_zero
tff(fact_5383_zero__le__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N))) ) ) ).
% zero_le_power_int
tff(fact_5384_success__LetI,axiom,
! [A: $tType,B: $tType,X: A,T2: A,F: fun(A,heap_Time_Heap(B)),H2: heap_ext(product_unit)] :
( ( X = T2 )
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),F,X),H2)
=> heap_Time_success(B,aa(A,heap_Time_Heap(B),F,T2),H2) ) ) ).
% success_LetI
tff(fact_5385_power__int__commutes,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N)) ) ).
% power_int_commutes
tff(fact_5386_power__int__mult__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,Y,M)) ) ).
% power_int_mult_distrib
tff(fact_5387_power__int__mult,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int,N: int] : power_int(A,X,aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = power_int(A,power_int(A,X,M),N) ) ).
% power_int_mult
tff(fact_5388_power__int__inverse,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] : power_int(A,aa(A,A,inverse_inverse(A),X),N) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).
% power_int_inverse
tff(fact_5389_success__ifI,axiom,
! [A: $tType,C2: bool,T2: heap_Time_Heap(A),H2: heap_ext(product_unit),E: heap_Time_Heap(A)] :
( ( pp(C2)
=> heap_Time_success(A,T2,H2) )
=> ( ( ~ pp(C2)
=> heap_Time_success(A,E,H2) )
=> heap_Time_success(A,if(heap_Time_Heap(A),C2,T2,E),H2) ) ) ).
% success_ifI
tff(fact_5390_power__int__divide__distrib,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,Y: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y),M) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M)),power_int(A,Y,M)) ) ).
% power_int_divide_distrib
tff(fact_5391_power__int__abs,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,A3,N)) = power_int(A,aa(A,A,abs_abs(A),A3),N) ) ).
% power_int_abs
tff(fact_5392_effect__success,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,C2,H2,H,R,N)
=> heap_Time_success(A,C2,H2) ) ).
% effect_success
tff(fact_5393_success__effectE,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
( heap_Time_success(A,C2,H2)
=> ~ ! [R3: A,H5: heap_ext(product_unit),N3: nat] : ~ heap_Time_effect(A,C2,H2,H5,R3,N3) ) ).
% success_effectE
tff(fact_5394_zero__less__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N))) ) ) ).
% zero_less_power_int
tff(fact_5395_power__int__one__over,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),X),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,X,N)) ) ).
% power_int_one_over
tff(fact_5396_power__int__minus,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,N: int] : power_int(A,X,aa(int,int,uminus_uminus(int),N)) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).
% power_int_minus
tff(fact_5397_success__returnI,axiom,
! [A: $tType,X: A,H2: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H2) ).
% success_returnI
tff(fact_5398_success__heapI,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H2: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_heap(A,F),H2) ).
% success_heapI
tff(fact_5399_success__guardI,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H2: 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,H2))
=> heap_Time_success(A,heap_Time_guard(A,P,F),H2) ) ).
% success_guardI
tff(fact_5400_success__assertI,axiom,
! [A: $tType,P: fun(A,bool),X: A,H2: heap_ext(product_unit)] :
( pp(aa(A,bool,P,X))
=> heap_Time_success(A,heap_Time_assert(A,P,X),H2) ) ).
% success_assertI
tff(fact_5401_success__ureturnI,axiom,
! [A: $tType,X: A,H2: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H2) ).
% success_ureturnI
tff(fact_5402_success__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,F: fun(nat,A),H2: heap_ext(product_unit)] : heap_Time_success(array(A),array_make(A,N,F),H2) ) ).
% success_makeI
tff(fact_5403_success__of__listI,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit)] : heap_Time_success(array(A),array_of_list(A,Xs),H2) ) ).
% success_of_listI
tff(fact_5404_success__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [N: nat,X: A,H2: heap_ext(product_unit)] : heap_Time_success(array(A),array_new(A,N,X),H2) ) ).
% success_newI
tff(fact_5405_success__tapI,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H2: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_tap(A,F),H2) ).
% success_tapI
tff(fact_5406_success__lenI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit)] : heap_Time_success(nat,array_len(A,A3),H2) ) ).
% success_lenI
tff(fact_5407_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_5408_power__int__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N6: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N6))
=> ( 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,N6))) ) ) ) ).
% power_int_increasing
tff(fact_5409_power__int__strict__increasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N6: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N6))
=> ( 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,N6))) ) ) ) ).
% power_int_strict_increasing
tff(fact_5410_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_5411_power__int__diff,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,M: int,N: int] :
( ( ( X != zero_zero(A) )
| ( M != N ) )
=> ( power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).
% power_int_diff
tff(fact_5412_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_5413_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),X: A,N: nat,G: fun(A,heap_Time_Heap(B))] :
( heap_Time_effect(A,F,H2,H,X,N)
=> ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X),H)
=> heap_Time_success(B,heap_Time_bind(A,B,F,G),H2) ) ) ).
% success_bind_effectI
tff(fact_5414_power__int__strict__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N6: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N6))
=> ( 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,N6)),power_int(A,A3,N))) ) ) ) ) ).
% power_int_strict_decreasing
tff(fact_5415_power__int__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,Y: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( 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),zero_zero(A)),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),power_int(A,Y,N))) ) ) ) ) ).
% power_int_mono
tff(fact_5416_power__int__strict__antimono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,N: int] :
( 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),zero_zero(A)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).
% power_int_strict_antimono
tff(fact_5417_one__le__power__int,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
=> ( 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,X,N))) ) ) ) ).
% one_le_power_int
tff(fact_5418_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_5419_power__int__add,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int,N: int] :
( ( ( X != zero_zero(A) )
| ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
=> ( power_int(A,X,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,X,M)),power_int(A,X,N)) ) ) ) ).
% power_int_add
tff(fact_5420_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: fun(A,nat)] : size_option(A,X,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).
% option.size_gen(1)
tff(fact_5421_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_5422_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( division_ring(A)
& one(B)
& uminus(B) )
=> ! [X: C,A3: A,N: int] :
( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
=> ( 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_5423_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> ! [M: nat,N: num] :
( ( bit_take_bit_num(M,N) = none(num) )
=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).
% take_bit_num_eq_None_imp
tff(fact_5424_and__num__eq__None__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num(M,N) = none(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)) = zero_zero(A) ) ) ) ).
% and_num_eq_None_iff
tff(fact_5425_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [M: num,N: num] :
( ( bit_un2480387367778600638or_num(M,N) = none(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)) = zero_zero(A) ) ) ) ).
% xor_num_eq_None_iff
tff(fact_5426_power__int__antimono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,N: int] :
( 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),zero_zero(A)),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).
% power_int_antimono
tff(fact_5427_power__int__strict__mono,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [A3: A,B2: A,N: int] :
( 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),zero_zero(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),power_int(A,A3,N)),power_int(A,B2,N))) ) ) ) ) ).
% power_int_strict_mono
tff(fact_5428_power__int__le__one,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
=> ( 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),X),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).
% power_int_le_one
tff(fact_5429_power__int__decreasing,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [N: int,N6: int,A3: A] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N6))
=> ( 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) )
| ( N6 != zero_zero(int) )
| ( N = zero_zero(int) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N6)),power_int(A,A3,N))) ) ) ) ) ) ).
% power_int_decreasing
tff(fact_5430_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M: int,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,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_5431_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X: A,M: int,N: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M)),power_int(A,X,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_5432_power__int__minus__mult,axiom,
! [A: $tType] :
( field(A)
=> ! [X: A,N: int] :
( ( ( X != zero_zero(A) )
| ( N != zero_zero(int) ) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(A,X,N) ) ) ) ).
% power_int_minus_mult
tff(fact_5433_power__int__add__1_H,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int] :
( ( ( X != zero_zero(A) )
| ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X,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),X),power_int(A,X,M)) ) ) ) ).
% power_int_add_1'
tff(fact_5434_power__int__add__1,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [X: A,M: int] :
( ( ( X != zero_zero(A) )
| ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
=> ( power_int(A,X,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,X,M)),X) ) ) ) ).
% power_int_add_1
tff(fact_5435_power__int__minus__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [N: int,A3: A] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
=> ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = power_int(A,A3,N) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
=> ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,uminus_uminus(A),power_int(A,A3,N)) ) ) ) ) ).
% power_int_minus_left
tff(fact_5436_successE,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
( heap_Time_success(A,F,H2)
=> ~ ! [R3: 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))),heap_Time_execute(A,F),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),H5)) ) ).
% successE
tff(fact_5437_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( bit_and_not_num(M,N) = none(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))) = zero_zero(int) ) ) ).
% and_not_num_eq_None_iff
tff(fact_5438_power__int__def,axiom,
! [A: $tType] :
( ( inverse(A)
& power(A) )
=> ! [N: int,X: A] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> ( power_int(A,X,N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(int,nat,nat2,N)) ) )
& ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
=> ( power_int(A,X,N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N))) ) ) ) ) ).
% power_int_def
tff(fact_5439_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
( heap_Time_success(A,F,H2)
=> ( aa(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)),H2) = 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)),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))),heap_Time_execute(A,F),H2)))),aa(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)),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))),heap_Time_execute(A,F),H2))))),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)),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))),heap_Time_execute(A,F),H2)))))) ) ) ).
% execute_bind_success
tff(fact_5440_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap(A),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] :
( heap_Time_effect(A,C2,H2,H,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)),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))),heap_Time_execute(A,C2),H2))) )
=> ( ( 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)),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))),heap_Time_execute(A,C2),H2)))) )
=> ( ( 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)),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))),heap_Time_execute(A,C2),H2)))) )
=> ~ heap_Time_success(A,C2,H2) ) ) ) ) ).
% effectE
tff(fact_5441_in__finite__psubset,axiom,
! [A: $tType,A5: set(A),B5: set(A)] :
( pp(member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A5),B5),finite_psubset(A)))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
& pp(aa(set(A),bool,finite_finite2(A),B5)) ) ) ).
% in_finite_psubset
tff(fact_5442_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_5443_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(A),H2: 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))),heap_Time_execute(A,F),H2) = 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))),heap_Time_execute(B,heap_Time_bind(A,B,F,G)),H2) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).
% execute_bind(2)
tff(fact_5444_successI,axiom,
! [A: $tType,F: 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))),heap_Time_execute(A,F),H2) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> heap_Time_success(A,F,H2) ) ).
% successI
tff(fact_5445_success__def,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
( heap_Time_success(A,F,H2)
<=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H2) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% success_def
tff(fact_5446_execute__guard_I1_J,axiom,
! [A: $tType,P: fun(heap_ext(product_unit),bool),H2: 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,H2))
=> ( aa(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)),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% execute_guard(1)
tff(fact_5447_execute__assert_I2_J,axiom,
! [A: $tType,P: fun(A,bool),X: A,H2: heap_ext(product_unit)] :
( ~ pp(aa(A,bool,P,X))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P,X)),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% execute_assert(2)
tff(fact_5448_finite__psubset__def,axiom,
! [A: $tType] : finite_psubset(A) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),bool,aTP_Lamp_nj(set(A),fun(set(A),bool)))) ).
% finite_psubset_def
tff(fact_5449_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
( ! [X2: 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))),heap_Time_execute(A,F),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),X2),H5))
=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).
% Heap_cases
tff(fact_5450_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
( ! [N3: nat,R3: A,H4: heap_ext(product_unit),N9: nat] : X != 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)))),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H4),N9))))
=> ~ ! [N3: nat] : X != 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)))),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_5451_Max_Oinfinite,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = the2(A,none(A)) ) ) ) ).
% Max.infinite
tff(fact_5452_Min_Oinfinite,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = the2(A,none(A)) ) ) ) ).
% Min.infinite
tff(fact_5453_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) = 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_nk(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_5454_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa4: 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,X,Xa4) = Y )
=> ( ! [R3: A,H4: heap_ext(product_unit),N9: nat] :
( ( Xa4 = 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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H4),N9))) )
=> ( 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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),aa(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),X),N9)))) ) )
=> ~ ( ( Xa4 = 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_5455_in__measure,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat)] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),measure(A,F)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y))) ) ).
% in_measure
tff(fact_5456_pred__nat__def,axiom,
pred_nat = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_nl(nat,fun(nat,bool)))) ).
% pred_nat_def
tff(fact_5457_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_nm(A,fun(option(A),option(A))),none(A),A5)) ) ).
% Min.eq_fold'
tff(fact_5458_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G),zero_zero(A),A5) ) ).
% sum.eq_fold
tff(fact_5459_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,times_times(A)),G),one_one(A),A5) ) ).
% prod.eq_fold
tff(fact_5460_Max_Oeq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic643756798349783984er_Max(A),insert(A,X,A5)) = finite_fold(A,A,ord_max(A),X,A5) ) ) ) ).
% Max.eq_fold
tff(fact_5461_Min_Oeq__fold,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic643756798350308766er_Min(A),insert(A,X,A5)) = finite_fold(A,A,ord_min(A),X,A5) ) ) ) ).
% Min.eq_fold
tff(fact_5462_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap(B),G: fun(B,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))),heap_Time_execute(A,heap_Time_bind(B,A,F,G)),H2) = 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))),product_case_prod(B,product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_no(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))),heap_Time_execute(B,F),H2)) ).
% execute_bind_case
tff(fact_5463_less__eq,axiom,
! [M: nat,N: nat] :
( pp(member(product_prod(nat,nat),aa(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_5464_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( pp(member(product_prod(nat,nat),aa(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_5465_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) = 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_nr(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_5466_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_ns(A,fun(option(A),option(A))),none(A),A5)) ) ).
% Max.eq_fold'
tff(fact_5467_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_nt(A,fun(option(A),option(A))),none(A),A5)) ) ).
% Sup_fin.eq_fold'
tff(fact_5468_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_nu(A,fun(option(A),option(A))),none(A),A5)) ) ).
% Inf_fin.eq_fold'
tff(fact_5469_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa4: 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,X,Xa4) = Y )
=> ( 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)))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X),Xa4))
=> ( ! [R3: A,H4: heap_ext(product_unit),N9: nat] :
( ( Xa4 = 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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H4),N9))) )
=> ( ( 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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),aa(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),X),N9)))) )
=> ~ 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)))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X),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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),R3),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,H4),N9))))) ) )
=> ~ ( ( Xa4 = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
=> ~ 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)))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X),none(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ) ) ) ) ) ).
% timeFrame.pelims
tff(fact_5470_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),insert(A,X,bot_bot(set(A)))) = X ) ).
% Inf_fin.singleton
tff(fact_5471_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),insert(A,X,bot_bot(set(A)))) = X ) ).
% Sup_fin.singleton
tff(fact_5472_inf__Sup__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = A3 ) ) ) ) ).
% inf_Sup_absorb
tff(fact_5473_sup__Inf__absorb,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A3) = A3 ) ) ) ) ).
% sup_Inf_absorb
tff(fact_5474_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).
% Inf_fin.insert
tff(fact_5475_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ).
% Sup_fin.insert
tff(fact_5476_Inf__fin__Min,axiom,
! [A: $tType] :
( ( semilattice_inf(A)
& linorder(A) )
=> ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).
% Inf_fin_Min
tff(fact_5477_Sup__fin__Max,axiom,
! [A: $tType] :
( ( semilattice_sup(A)
& linorder(A) )
=> ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).
% Sup_fin_Max
tff(fact_5478_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( lattice(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5))) ) ) ) ).
% Inf_fin_le_Sup_fin
tff(fact_5479_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A5))) ) ) ) ).
% Sup_fin.coboundedI
tff(fact_5480_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),A3: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A3)) ) ) ) ).
% Inf_fin.coboundedI
tff(fact_5481_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ).
% Inf_fin.in_idem
tff(fact_5482_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ).
% Sup_fin.in_idem
tff(fact_5483_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),X))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),X)) ) ) ) ) ) ).
% Sup_fin.bounded_iff
tff(fact_5484_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X5)) ) ) ) ) ) ).
% Inf_fin.bounded_iff
tff(fact_5485_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),X)) ) ) ) ) ).
% Sup_fin.boundedI
tff(fact_5486_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),X))
=> ! [A9: A] :
( pp(member(A,A9,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A9),X)) ) ) ) ) ) ).
% Sup_fin.boundedE
tff(fact_5487_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A6)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))) ) ) ) ) ).
% Inf_fin.boundedI
tff(fact_5488_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)))
=> ! [A9: A] :
( pp(member(A,A9,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A9)) ) ) ) ) ) ).
% Inf_fin.boundedE
tff(fact_5489_Sup__fin__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).
% Sup_fin_Sup
tff(fact_5490_card_Oeq__fold,axiom,
! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = finite_fold(A,nat,aTP_Lamp_nv(A,fun(nat,nat)),zero_zero(nat),A5) ).
% card.eq_fold
tff(fact_5491_Inf__fin__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).
% Inf_fin_Inf
tff(fact_5492_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = the2(A,none(A)) ) ) ) ).
% Inf_fin.infinite
tff(fact_5493_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = the2(A,none(A)) ) ) ) ).
% Sup_fin.infinite
tff(fact_5494_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5))) ) ) ) ) ).
% Sup_fin.subset_imp
tff(fact_5495_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))) ) ) ) ) ).
% Inf_fin.subset_imp
tff(fact_5496_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [H2: fun(A,A),N6: set(A)] :
( ! [X2: A,Y4: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H2,X2)),aa(A,A,H2,Y4))
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> ( ( N6 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic7752659483105999362nf_fin(A),N6)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),image(A,A,H2,N6)) ) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_5497_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [H2: fun(A,A),N6: set(A)] :
( ! [X2: A,Y4: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H2,X2)),aa(A,A,H2,Y4))
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> ( ( N6 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic5882676163264333800up_fin(A),N6)) = aa(set(A),A,lattic5882676163264333800up_fin(A),image(A,A,H2,N6)) ) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_5498_Inf__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ) ).
% Inf_fin.subset
tff(fact_5499_Sup__fin_Osubset,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ) ).
% Sup_fin.subset
tff(fact_5500_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ~ pp(member(A,X,A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
tff(fact_5501_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [X2: A,Y4: A] : pp(member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),Y4),insert(A,X2,insert(A,Y4,bot_bot(set(A))))))
=> pp(member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A5),A5)) ) ) ) ) ).
% Inf_fin.closed
tff(fact_5502_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [X2: A,Y4: A] : pp(member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X2),Y4),insert(A,X2,insert(A,Y4,bot_bot(set(A))))))
=> pp(member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A5),A5)) ) ) ) ) ).
% Sup_fin.closed
tff(fact_5503_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ~ pp(member(A,X,A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
tff(fact_5504_Inf__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) ) ) ) ) ) ) ).
% Inf_fin.union
tff(fact_5505_Sup__fin_Ounion,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) ) ) ) ) ) ) ).
% Sup_fin.union
tff(fact_5506_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),insert(A,X,A5)) = finite_fold(A,A,inf_inf(A),X,A5) ) ) ) ).
% Inf_fin.eq_fold
tff(fact_5507_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),insert(A,X,A5)) = finite_fold(A,A,sup_sup(A),X,A5) ) ) ) ).
% Sup_fin.eq_fold
tff(fact_5508_inf__Sup1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_nw(set(A),fun(A,fun(A,bool)),A5),X))) ) ) ) ) ).
% inf_Sup1_distrib
tff(fact_5509_inf__Sup2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_nx(set(A),fun(set(A),fun(A,bool)),A5),B5))) ) ) ) ) ) ) ).
% inf_Sup2_distrib
tff(fact_5510_sup__Inf2__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_ny(set(A),fun(set(A),fun(A,bool)),A5),B5))) ) ) ) ) ) ) ).
% sup_Inf2_distrib
tff(fact_5511_sup__Inf1__distrib,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_nz(set(A),fun(A,fun(A,bool)),A5),X))) ) ) ) ) ).
% sup_Inf1_distrib
tff(fact_5512_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),insert(A,X,A5)) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ).
% Inf_fin.insert_remove
tff(fact_5513_Inf__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Inf_fin.remove
tff(fact_5514_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),insert(A,X,A5)) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ).
% Sup_fin.insert_remove
tff(fact_5515_Sup__fin_Oremove,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A5: set(A),X: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ) ).
% Sup_fin.remove
tff(fact_5516_raise__def,axiom,
! [A: $tType,S: list(char)] : heap_Time_raise(A,S) = heap_Time_Heap2(A,aTP_Lamp_oa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% raise_def
tff(fact_5517_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A5) ) )
& ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = one_one(A) ) ) ) ) ).
% Gcd_fin.eq_fold
tff(fact_5518_funpow__inj__finite,axiom,
! [A: $tType,P3: fun(A,A),X: A] :
( inj_on(A,A,P3,top_top(set(A)))
=> ( pp(aa(set(A),bool,finite_finite2(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_ob(fun(A,A),fun(A,fun(A,bool)),P3),X))))
=> ~ ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N3))
=> ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N3),P3),X) != X ) ) ) ) ).
% funpow_inj_finite
tff(fact_5519_inj__uminus,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [A5: set(A)] : inj_on(A,A,uminus_uminus(A),A5) ) ).
% inj_uminus
tff(fact_5520_raise__bind,axiom,
! [B: $tType,A: $tType,E: list(char),F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,heap_Time_raise(B,E),F) = heap_Time_raise(A,E) ).
% raise_bind
tff(fact_5521_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).
% Gcd_fin.empty
tff(fact_5522_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = one_one(A) ) ) ) ).
% Gcd_fin.infinite
tff(fact_5523_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)),one_one(A)))
<=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = one_one(A) ) ) ) ).
% is_unit_Gcd_fin_iff
tff(fact_5524_inj__mult__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [A3: A] :
( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
<=> ( A3 != zero_zero(A) ) ) ) ).
% inj_mult_left
tff(fact_5525_inj__divide__right,axiom,
! [A: $tType] :
( field(A)
=> ! [A3: A] :
( inj_on(A,A,aTP_Lamp_oc(A,fun(A,A),A3),top_top(set(A)))
<=> ( A3 != zero_zero(A) ) ) ) ).
% inj_divide_right
tff(fact_5526_inj__fn,axiom,
! [A: $tType,F: fun(A,A),N: nat] :
( inj_on(A,A,F,top_top(set(A)))
=> inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),top_top(set(A))) ) ).
% inj_fn
tff(fact_5527_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set(A),F: fun(B,A),T5: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),image(B,A,F,T5)))
<=> ? [U4: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U4),T5))
& inj_on(B,A,F,U4)
& ( S3 = image(B,A,F,U4) ) ) ) ).
% subset_image_inj
tff(fact_5528_inj__list__encode,axiom,
! [A5: set(list(nat))] : inj_on(list(nat),nat,nat_list_encode,A5) ).
% inj_list_encode
tff(fact_5529_inj__prod__encode,axiom,
! [A5: set(product_prod(nat,nat))] : inj_on(product_prod(nat,nat),nat,nat_prod_encode,A5) ).
% inj_prod_encode
tff(fact_5530_inj__Suc,axiom,
! [N6: set(nat)] : inj_on(nat,nat,suc,N6) ).
% inj_Suc
tff(fact_5531_inj__on__of__nat,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N6: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N6) ) ).
% inj_on_of_nat
tff(fact_5532_inj__int__encode,axiom,
! [A5: set(int)] : inj_on(int,nat,nat_int_encode,A5) ).
% inj_int_encode
tff(fact_5533_inj__int__decode,axiom,
! [A5: set(nat)] : inj_on(nat,int,nat_int_decode,A5) ).
% inj_int_decode
tff(fact_5534_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( order(A)
=> ! [A5: set(A),F: fun(A,B)] :
( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> ( pp(member(A,X2,A5))
=> ( pp(member(A,Y4,A5))
=> ( aa(A,B,F,X2) != aa(A,B,F,Y4) ) ) ) )
=> ( ! [X2: A,Y4: A] :
( pp(member(A,X2,A5))
=> ( pp(member(A,Y4,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X2)) ) ) )
=> inj_on(A,B,F,A5) ) ) ) ).
% linorder_inj_onI
tff(fact_5535_inj__on__mult,axiom,
! [A: $tType] :
( semidom_divide(A)
=> ! [A3: A,A5: set(A)] :
( ( A3 != zero_zero(A) )
=> inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A5) ) ) ).
% inj_on_mult
tff(fact_5536_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [F: fun(A,B)] :
( ! [X2: A,Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> ( aa(A,B,F,X2) != aa(A,B,F,Y4) ) )
=> inj_on(A,B,F,top_top(set(A))) ) ) ).
% linorder_injI
tff(fact_5537_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B5: set(A),A5: set(A)] :
( inj_on(A,B,F,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),image(A,B,F,A5)),image(A,B,F,B5))) ) ) ).
% inj_on_strict_subset
tff(fact_5538_inj__of__nat,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> inj_on(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ).
% inj_of_nat
tff(fact_5539_inj__on__diff__nat,axiom,
! [N6: set(nat),K: nat] :
( ! [N3: nat] :
( pp(member(nat,N3,N6))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N3)) )
=> inj_on(nat,nat,aTP_Lamp_dd(nat,fun(nat,nat),K),N6) ) ).
% inj_on_diff_nat
tff(fact_5540_inj__on__set__encode,axiom,
inj_on(set(nat),nat,nat_set_encode,collect(set(nat),finite_finite2(nat))) ).
% inj_on_set_encode
tff(fact_5541_inj__on__iff__surj,axiom,
! [B: $tType,A: $tType,A5: set(A),A10: set(B)] :
( ( A5 != bot_bot(set(A)) )
=> ( ? [F4: fun(A,B)] :
( inj_on(A,B,F4,A5)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,F4,A5)),A10)) )
<=> ? [G4: fun(B,A)] : image(B,A,G4,A10) = A5 ) ) ).
% inj_on_iff_surj
tff(fact_5542_pigeonhole,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A5: set(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),image(B,A,F,A5))),aa(set(B),nat,finite_card(B),A5)))
=> ~ inj_on(B,A,F,A5) ) ).
% pigeonhole
tff(fact_5543_effect__raiseE,axiom,
! [A: $tType,X: list(char),H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,N: nat] : ~ heap_Time_effect(A,heap_Time_raise(A,X),H2,H,R,N) ).
% effect_raiseE
tff(fact_5544_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ? [N3: nat,F2: fun(nat,A)] :
( ( A5 = image(nat,A,F2,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N3))) )
& inj_on(nat,A,F2,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N3))) ) ) ).
% finite_imp_nat_seg_image_inj_on
tff(fact_5545_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ? [F2: fun(A,nat),N3: nat] :
( ( image(A,nat,F2,A5) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N3)) )
& inj_on(A,nat,F2,A5) ) ) ).
% finite_imp_inj_to_nat_seg
tff(fact_5546_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: 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)),image(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),A5))),aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,F,A5)))) ) ).
% inj_image_Compl_subset
tff(fact_5547_inj__on__nth,axiom,
! [A: $tType,Xs: list(A),I6: set(nat)] :
( distinct(A,Xs)
=> ( ! [X2: nat] :
( pp(member(nat,X2,I6))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),aa(list(A),nat,size_size(list(A)),Xs))) )
=> inj_on(nat,A,nth(A,Xs),I6) ) ) ).
% inj_on_nth
tff(fact_5548_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),S3))
<=> ? [F4: fun(nat,A)] :
( inj_on(nat,A,F4,top_top(set(nat)))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(nat,A,F4,top_top(set(nat)))),S3)) ) ) ).
% infinite_iff_countable_subset
tff(fact_5549_infinite__countable__subset,axiom,
! [A: $tType,S3: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),S3))
=> ? [F2: fun(nat,A)] :
( inj_on(nat,A,F2,top_top(set(nat)))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),image(nat,A,F2,top_top(set(nat)))),S3)) ) ) ).
% infinite_countable_subset
tff(fact_5550_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F: fun(A,A),S: A] :
( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),S) = S )
=> ( ! [M5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
=> ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M5),F),S) != S ) ) )
=> inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_od(fun(A,A),fun(A,fun(nat,A)),F),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).
% inj_on_funpow_least
tff(fact_5551_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( ? [F4: fun(A,B)] :
( inj_on(A,B,F4,A5)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,F4,A5)),B5)) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5))) ) ) ) ).
% inj_on_iff_card_le
tff(fact_5552_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B5: set(B)] :
( inj_on(A,B,F,A5)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,F,A5)),B5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5))) ) ) ) ).
% card_inj_on_le
tff(fact_5553_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(B),bool,finite_finite2(B),B5))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)))
=> ? [F2: fun(A,B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),image(A,B,F2,A5)),B5))
& inj_on(A,B,F2,A5) ) ) ) ) ).
% card_le_inj
tff(fact_5554_execute__raise,axiom,
! [A: $tType,S: list(char),X3: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_raise(A,S)),X3) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).
% execute_raise
tff(fact_5555_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),Ys2: list(B)] :
( inj_on(B,A,F,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),set2(B,Xs)),set2(B,Ys2)))
=> ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
=> ( distinct(A,map(B,A,F,Xs))
=> ( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Ys2))
=> ( distinct(A,map(B,A,F,Ys2))
=> ( ( set2(B,Xs) = set2(B,Ys2) )
=> ( Xs = Ys2 ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
tff(fact_5556_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = zero_zero(A) )
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),insert(A,zero_zero(A),bot_bot(set(A)))))
& pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).
% Gcd_fin_0_iff
tff(fact_5557_greaterThan__Suc,axiom,
! [K: 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)),set_ord_greaterThan(nat,K)),insert(nat,aa(nat,nat,suc,K),bot_bot(set(nat)))) ).
% greaterThan_Suc
tff(fact_5558_last__list__update,axiom,
! [A: $tType,Xs: list(A),K: nat,X: 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,X)) = X ) )
& ( ( 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,X)) = last(A,Xs) ) ) ) ) ).
% last_list_update
tff(fact_5559_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_5560_greaterThan__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,set_ord_greaterThan(A,K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),I)) ) ) ).
% greaterThan_iff
tff(fact_5561_greaterThan__subset__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_greaterThan(A,X)),set_ord_greaterThan(A,Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% greaterThan_subset_iff
tff(fact_5562_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N != zero_zero(nat) )
=> ( last(A,replicate(A,N,X)) = X ) ) ).
% last_replicate
tff(fact_5563_Compl__greaterThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_greaterThan(A,K)) = set_ord_atMost(A,K) ) ).
% Compl_greaterThan
tff(fact_5564_Compl__atMost,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_atMost(A,K)) = set_ord_greaterThan(A,K) ) ).
% Compl_atMost
tff(fact_5565_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
=> ( aa(set(A),A,complete_Sup_Sup(A),set_ord_greaterThan(A,X)) = top_top(A) ) ) ) ).
% Sup_greaterThanAtLeast
tff(fact_5566_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A] : image(A,A,uminus_uminus(A),set_ord_greaterThan(A,X)) = set_ord_lessThan(A,aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_greaterThan
tff(fact_5567_image__uminus__lessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A] : image(A,A,uminus_uminus(A),set_ord_lessThan(A,X)) = set_ord_greaterThan(A,aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_lessThan
tff(fact_5568_last__drop,axiom,
! [A: $tType,N: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( last(A,drop(A,N,Xs)) = last(A,Xs) ) ) ).
% last_drop
tff(fact_5569_greaterThan__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A] : set_ord_greaterThan(A,L) = collect(A,aa(A,fun(A,bool),ord_less(A),L)) ) ).
% greaterThan_def
tff(fact_5570_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)),set_ord_greaterThan(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).
% ivl_disj_un_one(5)
tff(fact_5571_greaterThan__0,axiom,
set_ord_greaterThan(nat,zero_zero(nat)) = image(nat,nat,suc,top_top(set(nat))) ).
% greaterThan_0
tff(fact_5572_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_5573_interval__cases,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [S3: set(A)] :
( ! [A6: A,B4: A,X2: A] :
( pp(member(A,A6,S3))
=> ( pp(member(A,B4,S3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),X2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B4))
=> pp(member(A,X2,S3)) ) ) ) )
=> ? [A6: A,B4: A] :
( ( S3 = bot_bot(set(A)) )
| ( S3 = top_top(set(A)) )
| ( S3 = set_ord_lessThan(A,B4) )
| ( S3 = set_ord_atMost(A,B4) )
| ( S3 = set_ord_greaterThan(A,A6) )
| ( S3 = set_ord_atLeast(A,A6) )
| ( S3 = set_or5935395276787703475ssThan(A,A6,B4) )
| ( S3 = set_or3652927894154168847AtMost(A,A6,B4) )
| ( S3 = set_or7035219750837199246ssThan(A,A6,B4) )
| ( S3 = set_or1337092689740270186AtMost(A,A6,B4) ) ) ) ) ).
% interval_cases
tff(fact_5574_sum__mult__sum__if__inj,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [F: fun(A,B),G: fun(C,B),A5: set(A),B5: set(C)] :
( inj_on(product_prod(A,C),B,product_case_prod(A,C,B,aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ei(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F),G)),product_Sigma(A,C,A5,aTP_Lamp_oe(set(C),fun(A,set(C)),B5)))
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A5)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B5)) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),id(B)),collect(B,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_of(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F),G),A5),B5))) ) ) ) ).
% sum_mult_sum_if_inj
tff(fact_5575_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [F: fun(A,B),P: fun(A,bool),A3: A] :
( inj_on(A,B,F,collect(A,P))
=> ( pp(aa(A,bool,P,A3))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,A3)),aa(A,B,F,Y4))) )
=> ( lattices_ord_arg_min(A,B,F,P) = A3 ) ) ) ) ) ).
% arg_min_inj_eq
tff(fact_5576_atLeast__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [I: A,K: A] :
( pp(member(A,I,set_ord_atLeast(A,K)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),I)) ) ) ).
% atLeast_iff
tff(fact_5577_atLeast__0,axiom,
set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).
% atLeast_0
tff(fact_5578_atLeast__subset__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A,Y: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_ord_atLeast(A,X)),set_ord_atLeast(A,Y)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).
% atLeast_subset_iff
tff(fact_5579_Compl__atLeast,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_atLeast(A,K)) = set_ord_lessThan(A,K) ) ).
% Compl_atLeast
tff(fact_5580_Compl__lessThan,axiom,
! [A: $tType] :
( linorder(A)
=> ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),set_ord_lessThan(A,K)) = set_ord_atLeast(A,K) ) ).
% Compl_lessThan
tff(fact_5581_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( preorder(A)
=> ! [L: A,H2: A,L3: A] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H2)),set_ord_atLeast(A,L3)))
<=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L3),L)) ) ) ) ).
% Icc_subset_Ici_iff
tff(fact_5582_image__uminus__atMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A] : image(A,A,uminus_uminus(A),set_ord_atMost(A,X)) = set_ord_atLeast(A,aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_atMost
tff(fact_5583_image__uminus__atLeast,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X: A] : image(A,A,uminus_uminus(A),set_ord_atLeast(A,X)) = set_ord_atMost(A,aa(A,A,uminus_uminus(A),X)) ) ).
% image_uminus_atLeast
tff(fact_5584_arg__min__nat__lemma,axiom,
! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
( pp(aa(A,bool,P,K))
=> ( pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P)))
& ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,Y3))) ) ) ) ).
% arg_min_nat_lemma
tff(fact_5585_arg__min__nat__le,axiom,
! [A: $tType,P: fun(A,bool),X: A,M: fun(A,nat)] :
( pp(aa(A,bool,P,X))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,X))) ) ).
% arg_min_nat_le
tff(fact_5586_atLeast__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [L: A] : set_ord_atLeast(A,L) = collect(A,aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).
% atLeast_def
tff(fact_5587_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( order(A)
=> ! [P: fun(C,bool),K: C,F: fun(C,A)] :
( pp(aa(C,bool,P,K))
=> ( ! [X2: C] :
( pp(aa(C,bool,P,X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,K)),aa(C,A,F,X2))) )
=> ( aa(C,A,F,lattices_ord_arg_min(C,A,F,P)) = aa(C,A,F,K) ) ) ) ) ).
% arg_min_equality
tff(fact_5588_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [P: fun(A,bool),X: A,F: fun(A,B),Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Y4)),aa(A,B,F,X))) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Y3)),aa(A,B,F,X2))) )
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(A,bool,Q,lattices_ord_arg_min(A,B,F,P))) ) ) ) ) ).
% arg_minI
tff(fact_5589_atLeast__Suc__greaterThan,axiom,
! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = set_ord_greaterThan(nat,K) ).
% atLeast_Suc_greaterThan
tff(fact_5590_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A5: fun(B,set(A)),I6: set(B)] :
? [F2: fun(A,product_prod(B,A))] :
( inj_on(A,product_prod(B,A),F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A5,I6)))
& 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))),image(A,product_prod(B,A),F2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(B,set(A),A5,I6)))),product_Sigma(B,A,I6,A5))) ) ).
% Ex_inj_on_UNION_Sigma
tff(fact_5591_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)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).
% ivl_disj_un_one(8)
tff(fact_5592_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)),set_ord_atLeast(A,A3)),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_5593_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A5: set(A),B5: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A5,aTP_Lamp_og(set(B),fun(A,set(B)),B5))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ).
% card_cartesian_product
tff(fact_5594_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)),set_ord_greaterThan(A,U)) = set_ord_atLeast(A,L) ) ) ) ).
% ivl_disj_un_one(7)
tff(fact_5595_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)),set_ord_atLeast(A,U)) = set_ord_greaterThan(A,L) ) ) ) ).
% ivl_disj_un_one(6)
tff(fact_5596_lists__length__Suc__eq,axiom,
! [A: $tType,A5: set(A),N: nat] : collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_oh(set(A),fun(nat,fun(list(A),bool)),A5),N)) = image(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_oi(list(A),fun(A,list(A)))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_oj(set(A),fun(nat,fun(list(A),bool)),A5),N)),aTP_Lamp_ok(set(A),fun(list(A),set(A)),A5))) ).
% lists_length_Suc_eq
tff(fact_5597_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ord(A)
=> ! [F: fun(B,A),S3: set(B)] : lattic7623131987881927897min_on(B,A,F,S3) = lattices_ord_arg_min(B,A,F,aTP_Lamp_ol(set(B),fun(B,bool),S3)) ) ).
% arg_min_on_def
tff(fact_5598_atLeast__Suc,axiom,
! [K: 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)),set_ord_atLeast(nat,K)),insert(nat,K,bot_bot(set(nat)))) ).
% atLeast_Suc
tff(fact_5599_Compl__Times__UNIV1,axiom,
! [A: $tType,B: $tType,A5: 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_og(set(B),fun(A,set(B)),A5))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_om(set(B),fun(A,set(B)),A5)) ).
% Compl_Times_UNIV1
tff(fact_5600_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A5: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_on(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A5),aTP_Lamp_on(A,set(B))) ).
% Compl_Times_UNIV2
tff(fact_5601_mult__inj__if__coprime__nat,axiom,
! [A: $tType,B: $tType,F: fun(A,nat),A5: set(A),G: fun(B,nat),B5: set(B)] :
( inj_on(A,nat,F,A5)
=> ( inj_on(B,nat,G,B5)
=> ( ! [A6: A,B4: B] :
( pp(member(A,A6,A5))
=> ( pp(member(B,B4,B5))
=> algebr8660921524188924756oprime(nat,aa(A,nat,F,A6),aa(B,nat,G,B4)) ) )
=> inj_on(product_prod(A,B),nat,product_case_prod(A,B,nat,aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_oo(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F),G)),product_Sigma(A,B,A5,aTP_Lamp_og(set(B),fun(A,set(B)),B5))) ) ) ) ).
% mult_inj_if_coprime_nat
tff(fact_5602_coprime__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),C2)
<=> ( algebr8660921524188924756oprime(A,A3,C2)
& algebr8660921524188924756oprime(A,B2,C2) ) ) ) ).
% coprime_mult_left_iff
tff(fact_5603_coprime__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C2,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2))
<=> ( algebr8660921524188924756oprime(A,C2,A3)
& algebr8660921524188924756oprime(A,C2,B2) ) ) ) ).
% coprime_mult_right_iff
tff(fact_5604_coprime__minus__left__iff,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,uminus_uminus(A),A3),B2)
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprime_minus_left_iff
tff(fact_5605_coprime__minus__right__iff,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,aa(A,A,uminus_uminus(A),B2))
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprime_minus_right_iff
tff(fact_5606_coprime__self,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,A3)
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).
% coprime_self
tff(fact_5607_coprime__mod__right__iff,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A3))
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mod_right_iff
tff(fact_5608_coprime__mod__left__iff,axiom,
! [A: $tType] :
( euclid3725896446679973847miring(A)
=> ! [B2: A,A3: A] :
( ( B2 != zero_zero(A) )
=> ( algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B2),B2)
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mod_left_iff
tff(fact_5609_coprime__power__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,N: nat] :
( algebr8660921524188924756oprime(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
<=> ( algebr8660921524188924756oprime(A,A3,B2)
| ( N = zero_zero(nat) ) ) ) ) ).
% coprime_power_right_iff
tff(fact_5610_coprime__power__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,N: nat,B2: A] :
( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),B2)
<=> ( algebr8660921524188924756oprime(A,A3,B2)
| ( N = zero_zero(nat) ) ) ) ) ).
% coprime_power_left_iff
tff(fact_5611_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_5612_coprime__0__right__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,A3,zero_zero(A))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).
% coprime_0_right_iff
tff(fact_5613_coprime__0__left__iff,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] :
( algebr8660921524188924756oprime(A,zero_zero(A),A3)
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).
% coprime_0_left_iff
tff(fact_5614_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(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_right_iff
tff(fact_5615_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(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
& algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).
% coprime_mult_self_left_iff
tff(fact_5616_is__unit__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_5617_arg__min__natI,axiom,
! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
( pp(aa(A,bool,P,K))
=> pp(aa(A,bool,P,lattices_ord_arg_min(A,nat,M,P))) ) ).
% arg_min_natI
tff(fact_5618_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_5619_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_5620_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_5621_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_5622_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).
% coprime_dvd_mult_right_iff
tff(fact_5623_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).
% coprime_dvd_mult_left_iff
tff(fact_5624_divides__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2))
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),C2)) ) ) ) ) ).
% divides_mult
tff(fact_5625_coprime__Suc__0__left,axiom,
! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).
% coprime_Suc_0_left
tff(fact_5626_coprime__Suc__0__right,axiom,
! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).
% coprime_Suc_0_right
tff(fact_5627_coprime__1__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).
% coprime_1_left
tff(fact_5628_coprime__1__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).
% coprime_1_right
tff(fact_5629_coprime__Suc__left__nat,axiom,
! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,N),N) ).
% coprime_Suc_left_nat
tff(fact_5630_coprime__Suc__right__nat,axiom,
! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,N)) ).
% coprime_Suc_right_nat
tff(fact_5631_coprime__commute,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,A3)
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprime_commute
tff(fact_5632_coprime__divisors,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,C2: A,B2: A,D2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D2))
=> ( algebr8660921524188924756oprime(A,C2,D2)
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% coprime_divisors
tff(fact_5633_coprime__crossproduct__nat,axiom,
! [A3: nat,D2: nat,B2: nat,C2: nat] :
( algebr8660921524188924756oprime(nat,A3,D2)
=> ( algebr8660921524188924756oprime(nat,B2,C2)
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D2) )
<=> ( ( A3 = B2 )
& ( C2 = D2 ) ) ) ) ) ).
% coprime_crossproduct_nat
tff(fact_5634_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_right_imp_coprime
tff(fact_5635_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% is_unit_left_imp_coprime
tff(fact_5636_coprime__common__divisor,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A))) ) ) ) ) ).
% coprime_common_divisor
tff(fact_5637_coprime__absorb__right,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y),X))
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y),one_one(A))) ) ) ) ).
% coprime_absorb_right
tff(fact_5638_coprime__imp__coprime,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,D2: A,A3: A,B2: A] :
( algebr8660921524188924756oprime(A,C2,D2)
=> ( ! [E2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),C2)) ) ) )
=> ( ! [E2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),one_one(A)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E2),D2)) ) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% coprime_imp_coprime
tff(fact_5639_coprime__absorb__left,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y))
=> ( algebr8660921524188924756oprime(A,X,Y)
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A))) ) ) ) ).
% coprime_absorb_left
tff(fact_5640_not__coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [C2: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
=> ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).
% not_coprimeI
tff(fact_5641_not__coprimeE,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ~ algebr8660921524188924756oprime(A,A3,B2)
=> ~ ! [C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) ) ) ) ).
% not_coprimeE
tff(fact_5642_coprime__def,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
<=> ! [C4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),one_one(A))) ) ) ) ) ).
% coprime_def
tff(fact_5643_coprimeI,axiom,
! [A: $tType] :
( algebraic_semidom(A)
=> ! [A3: A,B2: A] :
( ! [C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) )
=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprimeI
tff(fact_5644_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X: nat] :
( algebr8660921524188924756oprime(nat,A3,B2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),A3))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),B2))
=> ( X = one_one(nat) ) ) ) ) ).
% coprime_common_divisor_nat
tff(fact_5645_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_5646_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_5647_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_right_right_cancel
tff(fact_5648_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_right_left_cancel
tff(fact_5649_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B2: A,C2: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_left_right_cancel
tff(fact_5650_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [B2: A,C2: A,A3: A] :
( algebr8660921524188924756oprime(A,B2,C2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).
% gcd_mult_left_left_cancel
tff(fact_5651_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel(A)
& semiring_gcd(A) )
=> ! [A3: A,N: A,M: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),N)),M) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),N)),M) )
=> ( algebr8660921524188924756oprime(A,M,N)
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),M) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),M) ) ) ) ) ).
% mult_mod_cancel_right
tff(fact_5652_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel(A)
& semiring_gcd(A) )
=> ! [N: A,A3: A,M: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),N),A3)),M) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),N),B2)),M) )
=> ( algebr8660921524188924756oprime(A,M,N)
=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),M) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),M) ) ) ) ) ).
% mult_mod_cancel_left
tff(fact_5653_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_5654_gcd__coprime__exists,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
=> ? [A11: A,B6: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A11),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
& ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
& algebr8660921524188924756oprime(A,A11,B6) ) ) ) ).
% gcd_coprime_exists
tff(fact_5655_gcd__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,A4: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A4),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
=> ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
=> algebr8660921524188924756oprime(A,A4,B3) ) ) ) ) ).
% gcd_coprime
tff(fact_5656_div__gcd__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( A3 != zero_zero(A) )
| ( B2 != zero_zero(A) ) )
=> algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ) ).
% div_gcd_coprime
tff(fact_5657_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_5658_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_5659_set__encode__vimage__Suc,axiom,
! [A5: set(nat)] : aa(set(nat),nat,nat_set_encode,vimage(nat,nat,suc,A5)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(set(nat),nat,nat_set_encode,A5)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% set_encode_vimage_Suc
tff(fact_5660_floor__rat__def,axiom,
! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_op(rat,fun(int,bool),X)) ).
% floor_rat_def
tff(fact_5661_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B),X: B] :
( sorted_wrt(A,ord_less_eq(A),map(B,A,F,Xs))
=> sorted_wrt(A,ord_less_eq(A),map(B,A,F,linord329482645794927042rt_key(B,A,F,X,Xs))) ) ) ).
% sorted_insort_insert_key
tff(fact_5662_coprime__abs__left__iff,axiom,
! [K: int,L: int] :
( algebr8660921524188924756oprime(int,aa(int,int,abs_abs(int),K),L)
<=> algebr8660921524188924756oprime(int,K,L) ) ).
% coprime_abs_left_iff
tff(fact_5663_coprime__abs__right__iff,axiom,
! [K: int,L: int] :
( algebr8660921524188924756oprime(int,K,aa(int,int,abs_abs(int),L))
<=> algebr8660921524188924756oprime(int,K,L) ) ).
% coprime_abs_right_iff
tff(fact_5664_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))
<=> algebr8660921524188924756oprime(nat,M,N) ) ).
% coprime_int_iff
tff(fact_5665_normalize__stable,axiom,
! [Q4: int,P3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q4))
=> ( algebr8660921524188924756oprime(int,P3,Q4)
=> ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4)) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q4) ) ) ) ).
% normalize_stable
tff(fact_5666_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( algebr8660921524188924756oprime(nat,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)),N)
<=> algebr8660921524188924756oprime(int,K,aa(nat,int,semiring_1_of_nat(int),N)) ) ).
% coprime_nat_abs_left_iff
tff(fact_5667_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( algebr8660921524188924756oprime(nat,N,aa(int,nat,nat2,aa(int,int,abs_abs(int),K)))
<=> algebr8660921524188924756oprime(int,aa(nat,int,semiring_1_of_nat(int),N),K) ) ).
% coprime_nat_abs_right_iff
tff(fact_5668_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A5: set(B),D2: B,B5: set(A)] :
( ( pp(member(B,C2,A5))
=> ( ( pp(member(B,D2,A5))
=> ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_oq(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A5) = top_top(set(A)) ) )
& ( ~ pp(member(B,D2,A5))
=> ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_oq(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A5) = B5 ) ) ) )
& ( ~ pp(member(B,C2,A5))
=> ( ( pp(member(B,D2,A5))
=> ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_oq(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A5) = aa(set(A),set(A),uminus_uminus(set(A)),B5) ) )
& ( ~ pp(member(B,D2,A5))
=> ( vimage(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_oq(B,fun(B,fun(set(A),fun(A,B))),C2),D2),B5),A5) = bot_bot(set(A)) ) ) ) ) ) ).
% vimage_if
tff(fact_5669_vimage__Compl,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A5: set(B)] : vimage(A,B,F,aa(set(B),set(B),uminus_uminus(set(B)),A5)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F,A5)) ).
% vimage_Compl
tff(fact_5670_coprime__crossproduct__int,axiom,
! [A3: int,D2: int,B2: int,C2: int] :
( algebr8660921524188924756oprime(int,A3,D2)
=> ( algebr8660921524188924756oprime(int,B2,C2)
=> ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A3)),aa(int,int,abs_abs(int),C2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D2)) )
<=> ( ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B2) )
& ( aa(int,int,abs_abs(int),C2) = aa(int,int,abs_abs(int),D2) ) ) ) ) ) ).
% coprime_crossproduct_int
tff(fact_5671_finite__vimage__Suc__iff,axiom,
! [F3: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),vimage(nat,nat,suc,F3)))
<=> pp(aa(set(nat),bool,finite_finite2(nat),F3)) ) ).
% finite_vimage_Suc_iff
tff(fact_5672_Nitpick_OThe__psimp,axiom,
! [A: $tType,P: fun(A,bool),X: A] :
( ( P = aa(A,fun(A,bool),fequal(A),X) )
=> ( the(A,P) = X ) ) ).
% Nitpick.The_psimp
tff(fact_5673_vimage__Suc__insert__Suc,axiom,
! [N: nat,A5: set(nat)] : vimage(nat,nat,suc,insert(nat,aa(nat,nat,suc,N),A5)) = insert(nat,N,vimage(nat,nat,suc,A5)) ).
% vimage_Suc_insert_Suc
tff(fact_5674_vimage__Suc__insert__0,axiom,
! [A5: set(nat)] : vimage(nat,nat,suc,insert(nat,zero_zero(nat),A5)) = vimage(nat,nat,suc,A5) ).
% vimage_Suc_insert_0
tff(fact_5675_coprime__common__divisor__int,axiom,
! [A3: int,B2: int,X: int] :
( algebr8660921524188924756oprime(int,A3,B2)
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),A3))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),B2))
=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ) ) ).
% coprime_common_divisor_int
tff(fact_5676_sorted__insort__insert,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),X: A] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_me(A,A),X,Xs)) ) ) ).
% sorted_insort_insert
tff(fact_5677_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F: fun(A,B),D4: set(A),A5: set(B)] :
( inj_on(A,B,F,D4)
=> ( pp(aa(set(B),bool,finite_finite2(B),A5))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F,A5)),D4))),aa(set(B),nat,finite_card(B),A5))) ) ) ).
% card_vimage_inj_on_le
tff(fact_5678_set__decode__div__2,axiom,
! [X: nat] : nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = vimage(nat,nat,suc,nat_set_decode(X)) ).
% set_decode_div_2
tff(fact_5679_old_Orec__nat__def,axiom,
! [T: $tType,X3: T,Xa: fun(nat,fun(T,T)),Xb: nat] : aa(nat,T,rec_nat(T,X3,Xa),Xb) = the(T,rec_set_nat(T,X3,Xa,Xb)) ).
% old.rec_nat_def
tff(fact_5680_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_5681_sgn__rat,axiom,
! [A3: int,B2: int] : aa(rat,rat,sgn_sgn(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),times_times(int),aa(int,int,sgn_sgn(int),A3)),aa(int,int,sgn_sgn(int),B2))) ).
% sgn_rat
tff(fact_5682_minus__rat__cancel,axiom,
! [A3: int,B2: int] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A3)),aa(int,int,uminus_uminus(int),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ).
% minus_rat_cancel
tff(fact_5683_minus__rat,axiom,
! [A3: int,B2: int] : aa(rat,rat,uminus_uminus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),A3)),B2) ).
% minus_rat
tff(fact_5684_divide__rat,axiom,
! [A3: int,B2: int,C2: int,D2: int] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) ).
% divide_rat
tff(fact_5685_mult__rat,axiom,
! [A3: int,B2: int,C2: int,D2: int] : aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),times_times(int),A3),C2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2)) ).
% mult_rat
tff(fact_5686_less__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2 != zero_zero(int) )
=> ( ( D2 != zero_zero(int) )
=> ( 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,aa(int,fun(int,rat),fract,C2),D2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),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),D2)))) ) ) ) ).
% less_rat
tff(fact_5687_le__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2 != zero_zero(int) )
=> ( ( D2 != zero_zero(int) )
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),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),D2)))) ) ) ) ).
% le_rat
tff(fact_5688_add__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2 != zero_zero(int) )
=> ( ( D2 != 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),D2)) = 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),D2)),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),D2)) ) ) ) ).
% add_rat
tff(fact_5689_diff__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2 != zero_zero(int) )
=> ( ( D2 != zero_zero(int) )
=> ( aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),aa(int,rat,aa(int,fun(int,rat),fract,C2),D2)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2)),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),D2)) ) ) ) ).
% diff_rat
tff(fact_5690_Rat__cases,axiom,
! [Q4: rat] :
~ ! [A6: int,B4: int] :
( ( Q4 = aa(int,rat,aa(int,fun(int,rat),fract,A6),B4) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> ~ algebr8660921524188924756oprime(int,A6,B4) ) ) ).
% Rat_cases
tff(fact_5691_Rat__induct,axiom,
! [P: fun(rat,bool),Q4: rat] :
( ! [A6: int,B4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> ( algebr8660921524188924756oprime(int,A6,B4)
=> pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A6),B4))) ) )
=> pp(aa(rat,bool,P,Q4)) ) ).
% Rat_induct
tff(fact_5692_Fract__of__nat__eq,axiom,
! [K: nat] : aa(int,rat,aa(int,fun(int,rat),fract,aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) = aa(nat,rat,semiring_1_of_nat(rat),K) ).
% Fract_of_nat_eq
tff(fact_5693_eq__rat_I3_J,axiom,
! [A3: int,C2: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),A3) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),C2) ).
% eq_rat(3)
tff(fact_5694_eq__rat_I1_J,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2 != zero_zero(int) )
=> ( ( D2 != zero_zero(int) )
=> ( ( aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) = aa(int,rat,aa(int,fun(int,rat),fract,C2),D2) )
<=> ( aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2) = aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2) ) ) ) ) ).
% eq_rat(1)
tff(fact_5695_mult__rat__cancel,axiom,
! [C2: int,A3: int,B2: int] :
( ( C2 != zero_zero(int) )
=> ( aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,aa(int,fun(int,int),times_times(int),C2),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),C2),B2)) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).
% mult_rat_cancel
tff(fact_5696_eq__rat_I2_J,axiom,
! [A3: int] : aa(int,rat,aa(int,fun(int,rat),fract,A3),zero_zero(int)) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).
% eq_rat(2)
tff(fact_5697_Rat__induct__pos,axiom,
! [P: fun(rat,bool),Q4: rat] :
( ! [A6: int,B4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> pp(aa(rat,bool,P,aa(int,rat,aa(int,fun(int,rat),fract,A6),B4))) )
=> pp(aa(rat,bool,P,Q4)) ) ).
% Rat_induct_pos
tff(fact_5698_rat__number__collapse_I6_J,axiom,
! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),zero_zero(int)) = zero_zero(rat) ).
% rat_number_collapse(6)
tff(fact_5699_rat__number__collapse_I1_J,axiom,
! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),K) = zero_zero(rat) ).
% rat_number_collapse(1)
tff(fact_5700_One__rat__def,axiom,
one_one(rat) = aa(int,rat,aa(int,fun(int,rat),fract,one_one(int)),one_one(int)) ).
% One_rat_def
tff(fact_5701_Fract__of__int__eq,axiom,
! [K: int] : aa(int,rat,aa(int,fun(int,rat),fract,K),one_one(int)) = aa(int,rat,ring_1_of_int(rat),K) ).
% Fract_of_int_eq
tff(fact_5702_quotient__of__def,axiom,
! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_or(rat,fun(product_prod(int,int),bool),X)) ).
% quotient_of_def
tff(fact_5703_Rat__cases__nonzero,axiom,
! [Q4: rat] :
( ! [A6: int,B4: int] :
( ( Q4 = aa(int,rat,aa(int,fun(int,rat),fract,A6),B4) )
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> ( ( A6 != zero_zero(int) )
=> ~ algebr8660921524188924756oprime(int,A6,B4) ) ) )
=> ( Q4 = zero_zero(rat) ) ) ).
% Rat_cases_nonzero
tff(fact_5704_Zero__rat__def,axiom,
zero_zero(rat) = aa(int,rat,aa(int,fun(int,rat),fract,zero_zero(int)),one_one(int)) ).
% Zero_rat_def
tff(fact_5705_rat__number__expand_I3_J,axiom,
! [K: num] : aa(num,rat,numeral_numeral(rat),K) = aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),K)),one_one(int)) ).
% rat_number_expand(3)
tff(fact_5706_rat__number__collapse_I3_J,axiom,
! [W2: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(num,int,numeral_numeral(int),W2)),one_one(int)) = aa(num,rat,numeral_numeral(rat),W2) ).
% rat_number_collapse(3)
tff(fact_5707_quotient__of__unique,axiom,
! [R: rat] :
? [X2: product_prod(int,int)] :
( ( R = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),X2)),aa(product_prod(int,int),int,product_snd(int,int),X2)) )
& 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),X2)))
& algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),X2),aa(product_prod(int,int),int,product_snd(int,int),X2))
& ! [Y3: product_prod(int,int)] :
( ( ( R = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Y3)),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
& 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),Y3)))
& algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y3),aa(product_prod(int,int),int,product_snd(int,int),Y3)) )
=> ( Y3 = X2 ) ) ) ).
% quotient_of_unique
tff(fact_5708_Fract_Oabs__eq,axiom,
! [Xa4: int,X: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa4),X) = aa(product_prod(int,int),rat,abs_Rat,if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),X),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,Xa4),X))) ).
% Fract.abs_eq
tff(fact_5709_positive__rat,axiom,
! [A3: int,B2: int] :
( pp(aa(rat,bool,positive,aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B2))) ) ).
% positive_rat
tff(fact_5710_Fract__less__zero__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),zero_zero(int))) ) ) ).
% Fract_less_zero_iff
tff(fact_5711_zero__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A3)) ) ) ).
% zero_less_Fract_iff
tff(fact_5712_one__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A3)) ) ) ).
% one_less_Fract_iff
tff(fact_5713_Fract__less__one__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B2)) ) ) ).
% Fract_less_one_iff
tff(fact_5714_rat__number__collapse_I5_J,axiom,
aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ).
% rat_number_collapse(5)
tff(fact_5715_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_mt(int,fun(int,product_prod(int,int)))),fract)) ).
% Fract.transfer
tff(fact_5716_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_5717_Fract__le__zero__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),zero_zero(rat)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),zero_zero(int))) ) ) ).
% Fract_le_zero_iff
tff(fact_5718_zero__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A3)) ) ) ).
% zero_le_Fract_iff
tff(fact_5719_one__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A3)) ) ) ).
% one_le_Fract_iff
tff(fact_5720_Fract__le__one__iff,axiom,
! [B2: int,A3: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)),one_one(rat)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B2)) ) ) ).
% Fract_le_one_iff
tff(fact_5721_rat__number__expand_I5_J,axiom,
! [K: num] : aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K)) = aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).
% rat_number_expand(5)
tff(fact_5722_rat__number__collapse_I4_J,axiom,
! [W2: num] : aa(int,rat,aa(int,fun(int,rat),fract,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),one_one(int)) = aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W2)) ).
% rat_number_collapse(4)
tff(fact_5723_Rats__cases_H,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [X: A] :
( pp(member(A,X,field_char_0_Rats(A)))
=> ~ ! [A6: int,B4: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
=> ( algebr8660921524188924756oprime(int,A6,B4)
=> ( X != aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A6)),aa(int,A,ring_1_of_int(A),B4)) ) ) ) ) ) ).
% Rats_cases'
tff(fact_5724_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( comm_semiring_0(B)
& comm_semiring_0(A) )
=> ! [A5: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A5,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),A5,bNF_rel_fun(A,B,A,B,A5,A5)),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),A5,bNF_rel_fun(A,B,A,B,A5,A5)),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,A5),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A5,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B5),A5))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B))) ) ) ) ) ).
% horner_sum_transfer
tff(fact_5725_of__rat__neg__numeral__eq,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [W2: num] : field_char_0_of_rat(A,aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),W2))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) ) ).
% of_rat_neg_numeral_eq
tff(fact_5726_Rats__minus__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A] :
( pp(member(A,aa(A,A,uminus_uminus(A),A3),field_char_0_Rats(A)))
<=> pp(member(A,A3,field_char_0_Rats(A))) ) ) ).
% Rats_minus_iff
tff(fact_5727_of__rat__0,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_of_rat(A,zero_zero(rat)) = zero_zero(A) ) ) ).
% of_rat_0
tff(fact_5728_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] :
( ( field_char_0_of_rat(A,A3) = zero_zero(A) )
<=> ( A3 = zero_zero(rat) ) ) ) ).
% of_rat_eq_0_iff
tff(fact_5729_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] :
( ( zero_zero(A) = field_char_0_of_rat(A,A3) )
<=> ( zero_zero(rat) = A3 ) ) ) ).
% zero_eq_of_rat_iff
tff(fact_5730_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_5731_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_5732_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_5733_of__rat__of__nat__eq,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N: nat] : field_char_0_of_rat(A,aa(nat,rat,semiring_1_of_nat(rat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).
% of_rat_of_nat_eq
tff(fact_5734_of__rat__less__0__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)),zero_zero(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),zero_zero(rat))) ) ) ).
% of_rat_less_0_iff
tff(fact_5735_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),field_char_0_of_rat(A,R)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R)) ) ) ).
% zero_less_of_rat_iff
tff(fact_5736_of__rat__le__0__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)),zero_zero(A)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),zero_zero(rat))) ) ) ).
% of_rat_le_0_iff
tff(fact_5737_zero__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),zero_zero(A)),field_char_0_of_rat(A,R)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R)) ) ) ).
% zero_le_of_rat_iff
tff(fact_5738_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_5739_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_5740_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_5741_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_5742_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_5743_Rats__mult,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: A,B2: A] :
( pp(member(A,A3,field_char_0_Rats(A)))
=> ( pp(member(A,B2,field_char_0_Rats(A)))
=> pp(member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2),field_char_0_Rats(A))) ) ) ) ).
% Rats_mult
tff(fact_5744_Rats__0,axiom,
! [A: $tType] :
( field_char_0(A)
=> pp(member(A,zero_zero(A),field_char_0_Rats(A))) ) ).
% Rats_0
tff(fact_5745_Rats__1,axiom,
! [A: $tType] :
( field_char_0(A)
=> pp(member(A,one_one(A),field_char_0_Rats(A))) ) ).
% Rats_1
tff(fact_5746_Rats__of__nat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [N: nat] : pp(member(A,aa(nat,A,semiring_1_of_nat(A),N),field_char_0_Rats(A))) ) ).
% Rats_of_nat
tff(fact_5747_of__rat__minus,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [A3: rat] : field_char_0_of_rat(A,aa(rat,rat,uminus_uminus(rat),A3)) = aa(A,A,uminus_uminus(A),field_char_0_of_rat(A,A3)) ) ).
% of_rat_minus
tff(fact_5748_of__rat__less,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat,S: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),field_char_0_of_rat(A,R)),field_char_0_of_rat(A,S)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),S)) ) ) ).
% of_rat_less
tff(fact_5749_of__rat__less__eq,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [R: rat,S: rat] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),field_char_0_of_rat(A,R)),field_char_0_of_rat(A,S)))
<=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),S)) ) ) ).
% of_rat_less_eq
tff(fact_5750_of__rat__mult,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),times_times(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),field_char_0_of_rat(A,A3)),field_char_0_of_rat(A,B2)) ) ).
% of_rat_mult
tff(fact_5751_list__all2__conv__all__nth,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
& ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),I5)),aa(nat,B,nth(B,Ys2),I5))) ) ) ) ).
% list_all2_conv_all_nth
tff(fact_5752_list__all2__all__nthI,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) )
=> ( ! [N3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),aa(list(A),nat,size_size(list(A)),A3)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A3),N3)),aa(nat,B,nth(B,B2),N3))) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A3),B2)) ) ) ).
% list_all2_all_nthI
tff(fact_5753_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P3: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P3),aa(list(B),nat,size_size(list(B)),Ys2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3))) ) ) ).
% list_all2_nthD2
tff(fact_5754_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys2: list(B),P3: nat] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P3),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3))) ) ) ).
% list_all2_nthD
tff(fact_5755_of__rat__rat,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [B2: int,A3: int] :
( ( B2 != zero_zero(int) )
=> ( field_char_0_of_rat(A,aa(int,rat,aa(int,fun(int,rat),fract,A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2)) ) ) ) ).
% of_rat_rat
tff(fact_5756_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add(B)
& monoid_add(A) )
=> ! [A5: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A5,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),A5,bNF_rel_fun(A,B,A,B,A5,A5)),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,A5),A5),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).
% sum_list_transfer
tff(fact_5757_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( monoid_mult(B)
& monoid_mult(A) )
=> ! [A5: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A5,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),A5,bNF_rel_fun(A,B,A,B,A5,A5)),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,A5),A5),groups5270119922927024881d_list(A)),groups5270119922927024881d_list(B))) ) ) ) ).
% prod_list_transfer
tff(fact_5758_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_bot(A)
=> ordering_top(A,aTP_Lamp_os(A,fun(A,bool)),aTP_Lamp_ot(A,fun(A,bool)),bot_bot(A)) ) ).
% bot.ordering_top_axioms
tff(fact_5759_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_5760_prod__list_OCons,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [X: A,Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),cons(A,X,Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ).
% prod_list.Cons
tff(fact_5761_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_5762_prod__list_Oappend,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A),Ys2: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),append(A,Xs,Ys2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups5270119922927024881d_list(A),Xs)),aa(list(A),A,groups5270119922927024881d_list(A),Ys2)) ) ).
% prod_list.append
tff(fact_5763_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
tff(fact_5764_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( ( A3 != Top )
<=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),Top)) ) ) ).
% ordering_top.not_eq_extremum
tff(fact_5765_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,Top),A3))
<=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
tff(fact_5766_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,Top),A3)) ) ).
% ordering_top.extremum_strict
tff(fact_5767_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A,A3: A] :
( ordering_top(A,Less_eq,Less,Top)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),Top)) ) ).
% ordering_top.extremum
tff(fact_5768_length__dropWhile__le,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_dropWhile_le
tff(fact_5769_sorted__dropWhile,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),P: fun(A,bool)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).
% sorted_dropWhile
tff(fact_5770_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( semiring_1(A)
& semiri3467727345109120633visors(A) )
=> ! [Xs: list(A)] :
( ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = zero_zero(A) )
<=> pp(member(A,zero_zero(A),set2(A,Xs))) ) ) ).
% prod_list_zero_iff
tff(fact_5771_gcd__nat_Oordering__top__axioms,axiom,
ordering_top(nat,dvd_dvd(nat),aTP_Lamp_lu(nat,fun(nat,bool)),zero_zero(nat)) ).
% gcd_nat.ordering_top_axioms
tff(fact_5772_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A)] : aa(list(A),A,groups5270119922927024881d_list(A),Xs) = foldr(A,A,times_times(A),Xs,one_one(A)) ) ).
% prod_list.eq_foldr
tff(fact_5773_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( order_top(A)
=> ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).
% top.ordering_top_axioms
tff(fact_5774_bot__nat__0_Oordering__top__axioms,axiom,
ordering_top(nat,aTP_Lamp_ae(nat,fun(nat,bool)),aTP_Lamp_ad(nat,fun(nat,bool)),zero_zero(nat)) ).
% bot_nat_0.ordering_top_axioms
tff(fact_5775_mlex__eq,axiom,
! [A: $tType,F: fun(A,nat),R2: set(product_prod(A,A))] : mlex_prod(A,F,R2) = collect(product_prod(A,A),product_case_prod(A,A,bool,aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_ou(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F),R2))) ).
% mlex_eq
tff(fact_5776_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S3: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( lattic7623131987881927897min_on(A,B,F,S3) = fChoice(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_ov(set(A),fun(fun(A,B),fun(A,bool)),S3),F)) ) ) ) ).
% arg_min_SOME_Min
tff(fact_5777_prod__decode__triangle__add,axiom,
! [K: nat,M: nat] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M)) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M) ).
% prod_decode_triangle_add
tff(fact_5778_prod__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( aa(nat,product_prod(nat,nat),nat_prod_decode,X) = aa(nat,product_prod(nat,nat),nat_prod_decode,Y) )
<=> ( X = Y ) ) ).
% prod_decode_eq
tff(fact_5779_some__equality,axiom,
! [A: $tType,P: fun(A,bool),A3: A] :
( pp(aa(A,bool,P,A3))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( X2 = A3 ) )
=> ( fChoice(A,P) = A3 ) ) ) ).
% some_equality
tff(fact_5780_some__eq__trivial,axiom,
! [A: $tType,X: A] : fChoice(A,aTP_Lamp_ow(A,fun(A,bool),X)) = X ).
% some_eq_trivial
tff(fact_5781_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,bool),fequal(A),X)) = X ).
% some_sym_eq_trivial
tff(fact_5782_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : fChoice(product_prod(A,B),product_case_prod(A,B,bool,aa(B,fun(A,fun(B,bool)),aTP_Lamp_ox(A,fun(B,fun(A,fun(B,bool))),X),Y))) = aa(B,product_prod(A,B),product_Pair(A,B,X),Y) ).
% Eps_case_prod_eq
tff(fact_5783_prod__decode__inverse,axiom,
! [N: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode,N)) = N ).
% prod_decode_inverse
tff(fact_5784_prod__encode__inverse,axiom,
! [X: product_prod(nat,nat)] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(product_prod(nat,nat),nat,nat_prod_encode,X)) = X ).
% prod_encode_inverse
tff(fact_5785_inj__prod__decode,axiom,
! [A5: set(nat)] : inj_on(nat,product_prod(nat,nat),nat_prod_decode,A5) ).
% inj_prod_decode
tff(fact_5786_someI2,axiom,
! [A: $tType,P: fun(A,bool),A3: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,A3))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).
% someI2
tff(fact_5787_someI__ex,axiom,
! [A: $tType,P: fun(A,bool)] :
( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> pp(aa(A,bool,P,fChoice(A,P))) ) ).
% someI_ex
tff(fact_5788_someI2__ex,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).
% someI2_ex
tff(fact_5789_someI2__bex,axiom,
! [A: $tType,A5: set(A),P: fun(A,bool),Q: fun(A,bool)] :
( ? [X3: A] :
( pp(member(A,X3,A5))
& pp(aa(A,bool,P,X3)) )
=> ( ! [X2: A] :
( ( pp(member(A,X2,A5))
& pp(aa(A,bool,P,X2)) )
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,fChoice(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_oy(set(A),fun(fun(A,bool),fun(A,bool)),A5),P)))) ) ) ).
% someI2_bex
tff(fact_5790_some__eq__ex,axiom,
! [A: $tType,P: fun(A,bool)] :
( pp(aa(A,bool,P,fChoice(A,P)))
<=> ? [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).
% some_eq_ex
tff(fact_5791_some1__equality,axiom,
! [A: $tType,P: fun(A,bool),A3: A] :
( ? [X3: A] :
( pp(aa(A,bool,P,X3))
& ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> ( Y4 = X3 ) ) )
=> ( pp(aa(A,bool,P,A3))
=> ( fChoice(A,P) = A3 ) ) ) ).
% some1_equality
tff(fact_5792_verit__sko__ex_H,axiom,
! [A: $tType,P: fun(A,bool),A5: bool] :
( ( pp(aa(A,bool,P,fChoice(A,P)))
<=> pp(A5) )
=> ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
<=> pp(A5) ) ) ).
% verit_sko_ex'
tff(fact_5793_verit__sko__forall,axiom,
! [A: $tType,P: fun(A,bool)] :
( ! [X_13: A] : pp(aa(A,bool,P,X_13))
<=> pp(aa(A,bool,P,fChoice(A,aTP_Lamp_bf(fun(A,bool),fun(A,bool),P)))) ) ).
% verit_sko_forall
tff(fact_5794_verit__sko__forall_H,axiom,
! [A: $tType,P: fun(A,bool),A5: bool] :
( ( pp(aa(A,bool,P,fChoice(A,aTP_Lamp_bf(fun(A,bool),fun(A,bool),P))))
<=> pp(A5) )
=> ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
<=> pp(A5) ) ) ).
% verit_sko_forall'
tff(fact_5795_verit__sko__forall_H_H,axiom,
! [A: $tType,B5: A,A5: A,P: fun(A,bool)] :
( ( B5 = A5 )
=> ( ( fChoice(A,P) = A5 )
<=> ( fChoice(A,P) = B5 ) ) ) ).
% verit_sko_forall''
tff(fact_5796_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ( X = fChoice(A,P) )
=> ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
<=> pp(aa(A,bool,P,X)) ) ) ).
% verit_sko_ex_indirect
tff(fact_5797_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: fun(A,bool),P2: fun(A,bool)] :
( ( X = fChoice(A,P) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P2,X2)) )
=> ( ? [X_13: A] : pp(aa(A,bool,P2,X_13))
<=> pp(aa(A,bool,P,X)) ) ) ) ).
% verit_sko_ex_indirect2
tff(fact_5798_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ( X = fChoice(A,aTP_Lamp_bf(fun(A,bool),fun(A,bool),P)) )
=> ( ! [X_13: A] : pp(aa(A,bool,P,X_13))
<=> pp(aa(A,bool,P,X)) ) ) ).
% verit_sko_forall_indirect
tff(fact_5799_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: fun(A,bool),P2: fun(A,bool)] :
( ( X = fChoice(A,aTP_Lamp_bf(fun(A,bool),fun(A,bool),P)) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P2,X2)) )
=> ( ! [X_13: A] : pp(aa(A,bool,P2,X_13))
<=> pp(aa(A,bool,P,X)) ) ) ) ).
% verit_sko_forall_indirect2
tff(fact_5800_Nitpick_OEps__psimp,axiom,
! [A: $tType,P: fun(A,bool),X: A,Y: A] :
( pp(aa(A,bool,P,X))
=> ( ~ pp(aa(A,bool,P,Y))
=> ( ( fChoice(A,P) = Y )
=> ( fChoice(A,P) = X ) ) ) ) ).
% Nitpick.Eps_psimp
tff(fact_5801_someI,axiom,
! [A: $tType,P: fun(A,bool),X: A] :
( pp(aa(A,bool,P,X))
=> pp(aa(A,bool,P,fChoice(A,P))) ) ).
% someI
tff(fact_5802_Eps__cong,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,Q,X2)) )
=> ( fChoice(A,P) = fChoice(A,Q) ) ) ).
% Eps_cong
tff(fact_5803_tfl__some,axiom,
! [A: $tType,P4: fun(A,bool),X3: A] :
( pp(aa(A,bool,P4,X3))
=> pp(aa(A,bool,P4,fChoice(A,P4))) ) ).
% tfl_some
tff(fact_5804_some__eq__imp,axiom,
! [A: $tType,P: fun(A,bool),A3: A,B2: A] :
( ( fChoice(A,P) = A3 )
=> ( pp(aa(A,bool,P,B2))
=> pp(aa(A,bool,P,A3)) ) ) ).
% some_eq_imp
tff(fact_5805_some__in__eq,axiom,
! [A: $tType,A5: set(A)] :
( pp(member(A,fChoice(A,aTP_Lamp_a(set(A),fun(A,bool),A5)),A5))
<=> ( A5 != bot_bot(set(A)) ) ) ).
% some_in_eq
tff(fact_5806_exE__some,axiom,
! [A: $tType,P: fun(A,bool),C2: A] :
( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> ( ( C2 = fChoice(A,P) )
=> pp(aa(A,bool,P,C2)) ) ) ).
% exE_some
tff(fact_5807_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),product_case_prod(A,B,bool,aTP_Lamp_oz(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P))) ).
% split_paired_Eps
tff(fact_5808_prod__decode__def,axiom,
nat_prod_decode = nat_prod_decode_aux(zero_zero(nat)) ).
% prod_decode_def
tff(fact_5809_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : fChoice(product_prod(A,B),product_case_prod(A,B,bool,P)) = fChoice(product_prod(A,B),aTP_Lamp_pa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P)) ).
% Eps_case_prod
tff(fact_5810_surj__prod__decode,axiom,
image(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat))) = top_top(set(product_prod(nat,nat))) ).
% surj_prod_decode
tff(fact_5811_mlex__less,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),mlex_prod(A,F,R2))) ) ).
% mlex_less
tff(fact_5812_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F: fun(A,nat),R2: set(product_prod(A,A))] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),mlex_prod(A,F,R2)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
| ( ( aa(A,nat,F,X) = aa(A,nat,F,Y) )
& pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),R2)) ) ) ) ).
% mlex_iff
tff(fact_5813_mlex__leq,axiom,
! [A: $tType,F: fun(A,nat),X: A,Y: A,R2: set(product_prod(A,A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,Y)))
=> ( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),R2))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y),mlex_prod(A,F,R2))) ) ) ).
% mlex_leq
tff(fact_5814_list__decode_Oelims,axiom,
! [X: nat,Y: list(nat)] :
( ( aa(nat,list(nat),nat_list_decode,X) = Y )
=> ( ( ( X = zero_zero(nat) )
=> ( Y != nil(nat) ) )
=> ~ ! [N3: nat] :
( ( X = aa(nat,nat,suc,N3) )
=> ( Y != aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_pb(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N3)) ) ) ) ) ).
% list_decode.elims
tff(fact_5815_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ord(A)
=> ! [F: fun(B,A),P: fun(B,bool)] : lattices_ord_arg_min(B,A,F,P) = fChoice(B,lattic501386751177426532rg_min(B,A,F,P)) ) ).
% arg_min_def
tff(fact_5816_list__decode_Osimps_I2_J,axiom,
! [N: nat] : aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_pb(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).
% list_decode.simps(2)
tff(fact_5817_list__encode__inverse,axiom,
! [X: list(nat)] : aa(nat,list(nat),nat_list_decode,aa(list(nat),nat,nat_list_encode,X)) = X ).
% list_encode_inverse
tff(fact_5818_list__decode__inverse,axiom,
! [N: nat] : aa(list(nat),nat,nat_list_encode,aa(nat,list(nat),nat_list_decode,N)) = N ).
% list_decode_inverse
tff(fact_5819_list__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( aa(nat,list(nat),nat_list_decode,X) = aa(nat,list(nat),nat_list_decode,Y) )
<=> ( X = Y ) ) ).
% list_decode_eq
tff(fact_5820_inj__list__decode,axiom,
! [A5: set(nat)] : inj_on(nat,list(nat),nat_list_decode,A5) ).
% inj_list_decode
tff(fact_5821_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [F: fun(A,B),P: fun(A,bool),X: A,Y: A] :
( pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F,P),X))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y)),aa(A,B,F,X)))
=> ( pp(aa(A,bool,P,Y))
=> pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F,P),Y)) ) ) ) ) ).
% is_arg_min_antimono
tff(fact_5822_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),P: fun(A,bool),X: A] :
( pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F,P),X))
<=> ( pp(aa(A,bool,P,X))
& ! [Y5: A] :
( pp(aa(A,bool,P,Y5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,Y5))) ) ) ) ) ).
% is_arg_min_linorder
tff(fact_5823_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ord(A)
=> ! [F: fun(B,A),P: fun(B,bool),X: B] :
( pp(aa(B,bool,lattic501386751177426532rg_min(B,A,F,P),X))
<=> ( pp(aa(B,bool,P,X))
& ~ ? [Y5: B] :
( pp(aa(B,bool,P,Y5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,Y5)),aa(B,A,F,X))) ) ) ) ) ).
% is_arg_min_def
tff(fact_5824_list__decode_Osimps_I1_J,axiom,
aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ).
% list_decode.simps(1)
tff(fact_5825_is__arg__min__arg__min__nat,axiom,
! [A: $tType,P: fun(A,bool),X: A,M: fun(A,nat)] :
( pp(aa(A,bool,P,X))
=> pp(aa(A,bool,lattic501386751177426532rg_min(A,nat,M,P),lattices_ord_arg_min(A,nat,M,P))) ) ).
% is_arg_min_arg_min_nat
tff(fact_5826_surj__list__decode,axiom,
image(nat,list(nat),nat_list_decode,top_top(set(nat))) = top_top(set(list(nat))) ).
% surj_list_decode
tff(fact_5827_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( order(B)
=> ! [S3: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,finite_finite2(A),S3))
=> ( ( S3 != bot_bot(set(A)) )
=> ? [X_12: A] : pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F,aTP_Lamp_a(set(A),fun(A,bool),S3)),X_12)) ) ) ) ).
% ex_is_arg_min_if_finite
tff(fact_5828_list__decode_Opelims,axiom,
! [X: nat,Y: list(nat)] :
( ( aa(nat,list(nat),nat_list_decode,X) = Y )
=> ( accp(nat,nat_list_decode_rel,X)
=> ( ( ( X = zero_zero(nat) )
=> ( ( Y = nil(nat) )
=> ~ accp(nat,nat_list_decode_rel,zero_zero(nat)) ) )
=> ~ ! [N3: nat] :
( ( X = aa(nat,nat,suc,N3) )
=> ( ( Y = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_pb(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N3)) )
=> ~ accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N3)) ) ) ) ) ) ).
% list_decode.pelims
tff(fact_5829_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A5: fun(C,A),I6: set(C)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),image(C,A,A5,I6)))
=> ( ( I6 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pc(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5),I6))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),image(C,A,A5,I6))))) ) ) ) ) ).
% mono_cSUP
tff(fact_5830_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),image(A,B,F,A5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A5)))) ) ) ) ) ).
% mono_cSup
tff(fact_5831_bdd__above_OI,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A),M7: A] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),M7)) )
=> pp(aa(set(A),bool,condit941137186595557371_above(A),A5)) ) ) ).
% bdd_above.I
tff(fact_5832_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [A5: set(B),F: fun(B,A),M7: A] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),M7)) )
=> pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,F,A5))) ) ) ).
% bdd_above.I2
tff(fact_5833_bdd__above_OE,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,condit941137186595557371_above(A),A5))
=> ~ ! [M8: A] :
~ ! [X3: A] :
( pp(member(A,X3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),M8)) ) ) ) ).
% bdd_above.E
tff(fact_5834_bdd__above_Ounfold,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,condit941137186595557371_above(A),A5))
<=> ? [M9: A] :
! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),M9)) ) ) ) ).
% bdd_above.unfold
tff(fact_5835_cSup__upper2,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A,X6: set(A),Y: A] :
( pp(member(A,X,X6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ) ).
% cSup_upper2
tff(fact_5836_cSup__upper,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A,X6: set(A)] :
( pp(member(A,X,X6))
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).
% cSup_upper
tff(fact_5837_list__decode_Opsimps_I1_J,axiom,
( accp(nat,nat_list_decode_rel,zero_zero(nat))
=> ( aa(nat,list(nat),nat_list_decode,zero_zero(nat)) = nil(nat) ) ) ).
% list_decode.psimps(1)
tff(fact_5838_list__decode_Opinduct,axiom,
! [A0: nat,P: fun(nat,bool)] :
( accp(nat,nat_list_decode_rel,A0)
=> ( ( accp(nat,nat_list_decode_rel,zero_zero(nat))
=> pp(aa(nat,bool,P,zero_zero(nat))) )
=> ( ! [N3: nat] :
( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N3))
=> ( ! [X3: nat,Y3: nat] :
( ( aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X3),Y3) = aa(nat,product_prod(nat,nat),nat_prod_decode,N3) )
=> pp(aa(nat,bool,P,Y3)) )
=> pp(aa(nat,bool,P,aa(nat,nat,suc,N3))) ) )
=> pp(aa(nat,bool,P,A0)) ) ) ) ).
% list_decode.pinduct
tff(fact_5839_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: B,A5: set(B),F: fun(B,A)] :
( pp(member(B,X,A5))
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,F,A5)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)))) ) ) ) ).
% cSUP_upper
tff(fact_5840_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A5: set(B),X: B,U: A] :
( pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,F,A5)))
=> ( pp(member(B,X,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5)))) ) ) ) ) ).
% cSUP_upper2
tff(fact_5841_cSup__le__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S3: set(A),A3: A] :
( ( S3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),S3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S3)),A3))
<=> ! [X5: A] :
( pp(member(A,X5,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),A3)) ) ) ) ) ) ).
% cSup_le_iff
tff(fact_5842_cSup__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [B5: set(A),A5: set(A)] :
( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),A5))
=> ( ! [B4: A] :
( pp(member(A,B4,B5))
=> ? [X3: A] :
( pp(member(A,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B4),X3)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B5)),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ) ).
% cSup_mono
tff(fact_5843_less__cSup__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),Y: A] :
( ( X6 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),X6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
<=> ? [X5: A] :
( pp(member(A,X5,X6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X5)) ) ) ) ) ) ).
% less_cSup_iff
tff(fact_5844_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A5: set(B),Y: A,I: B] :
( pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,F,A5)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),Y))
=> ( pp(member(B,I,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I)),Y)) ) ) ) ) ).
% cSUP_lessD
tff(fact_5845_cSUP__le__iff,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),F: fun(B,A),U: A] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,F,A5)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),U))
<=> ! [X5: B] :
( pp(member(B,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X5)),U)) ) ) ) ) ) ).
% cSUP_le_iff
tff(fact_5846_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),G: fun(C,A),B5: set(C),F: fun(B,A)] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),image(C,A,G,B5)))
=> ( ! [N3: B] :
( pp(member(B,N3,A5))
=> ? [X3: C] :
( pp(member(C,X3,B5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,N3)),aa(C,A,G,X3))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(C,A,G,B5)))) ) ) ) ) ).
% cSUP_mono
tff(fact_5847_cSup__subset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(A),B5: set(A)] :
( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ) ) ) ).
% cSup_subset_mono
tff(fact_5848_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( condit6923001295902523014norder(A)
=> ! [A5: set(B),F: fun(B,A),A3: A] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,F,A5)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))))
<=> ? [X5: B] :
( pp(member(B,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X5))) ) ) ) ) ) ).
% less_cSUP_iff
tff(fact_5849_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( accp(nat,nat_list_decode_rel,aa(nat,nat,suc,N))
=> ( aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),product_case_prod(nat,nat,list(nat),aTP_Lamp_pb(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ).
% list_decode.psimps(2)
tff(fact_5850_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),G: fun(B,A),B5: set(B),F: fun(B,A)] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),image(B,A,G,B5)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B5))
=> ( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,G,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(B,A,F,A5))),aa(set(A),A,complete_Sup_Sup(A),image(B,A,G,B5)))) ) ) ) ) ) ).
% cSUP_subset_mono
tff(fact_5851_cSup__inter__less__eq,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,condit941137186595557371_above(A),A5))
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),B5))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)))) ) ) ) ) ).
% cSup_inter_less_eq
tff(fact_5852_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_5853_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A5))),aa(set(B),B,complete_Inf_Inf(B),image(A,B,F,A5)))) ) ) ) ) ).
% mono_cInf
tff(fact_5854_mono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( condit1219197933456340205attice(B)
& condit1219197933456340205attice(A) )
=> ! [F: fun(A,B),A5: fun(C,A),I6: set(C)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(C,A,A5,I6)))
=> ( ( I6 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),image(C,A,A5,I6)))),aa(set(B),B,complete_Inf_Inf(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pc(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5),I6)))) ) ) ) ) ).
% mono_cINF
tff(fact_5855_bdd__belowI,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A),M: A] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X2)) )
=> pp(aa(set(A),bool,condit1013018076250108175_below(A),A5)) ) ) ).
% bdd_belowI
tff(fact_5856_bdd__below_OI,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A),M7: A] :
( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M7),X2)) )
=> pp(aa(set(A),bool,condit1013018076250108175_below(A),A5)) ) ) ).
% bdd_below.I
tff(fact_5857_bdd__below__uminus,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X6: set(A)] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(A,A,uminus_uminus(A),X6)))
<=> pp(aa(set(A),bool,condit941137186595557371_above(A),X6)) ) ) ).
% bdd_below_uminus
tff(fact_5858_bdd__above__uminus,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X6: set(A)] :
( pp(aa(set(A),bool,condit941137186595557371_above(A),image(A,A,uminus_uminus(A),X6)))
<=> pp(aa(set(A),bool,condit1013018076250108175_below(A),X6)) ) ) ).
% bdd_above_uminus
tff(fact_5859_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [A5: set(B),M7: A,F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M7),aa(B,A,F,X2))) )
=> pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5))) ) ) ).
% bdd_below.I2
tff(fact_5860_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [A5: set(B),M: A,F: fun(B,A)] :
( ! [X2: B] :
( pp(member(B,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F,X2))) )
=> pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5))) ) ) ).
% bdd_belowI2
tff(fact_5861_cInf__lower2,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A,X6: set(A),Y: A] :
( pp(member(A,X,X6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y)) ) ) ) ) ).
% cInf_lower2
tff(fact_5862_cInf__lower,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [X: A,X6: set(A)] :
( pp(member(A,X,X6))
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).
% cInf_lower
tff(fact_5863_bdd__below_Ounfold,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
<=> ? [M9: A] :
! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M9),X5)) ) ) ) ).
% bdd_below.unfold
tff(fact_5864_bdd__below_OE,axiom,
! [A: $tType] :
( preorder(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
=> ~ ! [M8: A] :
~ ! [X3: A] :
( pp(member(A,X3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M8),X3)) ) ) ) ).
% bdd_below.E
tff(fact_5865_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A5: set(B),X: B,U: A] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5)))
=> ( pp(member(B,X,A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),U))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),U)) ) ) ) ) ).
% cINF_lower2
tff(fact_5866_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A5: set(B),X: B] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5)))
=> ( pp(member(B,X,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),aa(B,A,F,X))) ) ) ) ).
% cINF_lower
tff(fact_5867_le__cInf__iff,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S3: set(A),A3: A] :
( ( S3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),S3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S3)))
<=> ! [X5: A] :
( pp(member(A,X5,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),X5)) ) ) ) ) ) ).
% le_cInf_iff
tff(fact_5868_cInf__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [B5: set(A),A5: set(A)] :
( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
=> ( ! [B4: A] :
( pp(member(A,B4,B5))
=> ? [X3: A] :
( pp(member(A,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B4)) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))) ) ) ) ) ).
% cInf_mono
tff(fact_5869_cInf__less__iff,axiom,
! [A: $tType] :
( condit6923001295902523014norder(A)
=> ! [X6: set(A),Y: A] :
( ( X6 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),X6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Y))
<=> ? [X5: A] :
( pp(member(A,X5,X6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y)) ) ) ) ) ) ).
% cInf_less_iff
tff(fact_5870_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A5: set(B),Y: A,I: B] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))))
=> ( pp(member(B,I,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F,I))) ) ) ) ) ).
% less_cINF_D
tff(fact_5871_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [B5: set(B),F: fun(C,A),A5: set(C),G: fun(B,A)] :
( ( B5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(C,A,F,A5)))
=> ( ! [M5: B] :
( pp(member(B,M5,B5))
=> ? [X3: C] :
( pp(member(C,X3,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X3)),aa(B,A,G,M5))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(C,A,F,A5))),aa(set(A),A,complete_Inf_Inf(A),image(B,A,G,B5)))) ) ) ) ) ).
% cINF_mono
tff(fact_5872_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),F: fun(B,A),U: A] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))))
<=> ! [X5: B] :
( pp(member(B,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X5))) ) ) ) ) ) ).
% le_cINF_iff
tff(fact_5873_cInf__superset__mono,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(A),B5: set(A)] :
( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),B5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B5)),aa(set(A),A,complete_Inf_Inf(A),A5))) ) ) ) ) ).
% cInf_superset_mono
tff(fact_5874_cINF__less__iff,axiom,
! [B: $tType,A: $tType] :
( condit6923001295902523014norder(A)
=> ! [A5: set(B),F: fun(B,A),A3: A] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,F,A5)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5))),A3))
<=> ? [X5: B] :
( pp(member(B,X5,A5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X5)),A3)) ) ) ) ) ) ).
% cINF_less_iff
tff(fact_5875_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_5876_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_5877_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_5878_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(B),G: fun(B,A),B5: set(B),F: fun(B,A)] :
( ( A5 != bot_bot(set(B)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),image(B,A,G,B5)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B5))
=> ( ! [X2: B] :
( pp(member(B,X2,B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,G,X2)),aa(B,A,F,X2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(B,A,G,B5))),aa(set(A),A,complete_Inf_Inf(A),image(B,A,F,A5)))) ) ) ) ) ) ).
% cINF_superset_mono
tff(fact_5879_less__eq__cInf__inter,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(A),B5: set(A)] :
( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),B5))
=> ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) != bot_bot(set(A)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)))) ) ) ) ) ).
% less_eq_cInf_inter
tff(fact_5880_cInf__le__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A5: set(A)] :
( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),A5))
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ) ).
% cInf_le_cSup
tff(fact_5881_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_5882_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_mr(int,int)) ).
% Code_Numeral.dup_def
tff(fact_5883_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_5884_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_5885_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_5886_dup_Orep__eq,axiom,
! [X: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,code_dup,X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,X)) ).
% dup.rep_eq
tff(fact_5887_dup_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_integer,code_dup,aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),X)) ).
% dup.abs_eq
tff(fact_5888_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_5889_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_5890_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_5891_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_5892_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,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_5893_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)),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_5894_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_5895_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_5896_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,bit0(M)),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_5897_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_5898_sub_Orep__eq,axiom,
! [X: num,Xa4: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,X),Xa4)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),X)),aa(num,int,numeral_numeral(int),Xa4)) ).
% sub.rep_eq
tff(fact_5899_sub_Oabs__eq,axiom,
! [Xa4: num,X: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,Xa4),X) = 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),Xa4)),aa(num,int,numeral_numeral(int),X))) ).
% sub.abs_eq
tff(fact_5900_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_5901_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_5902_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_5903_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_5904_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_5905_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_5906_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_5907_Sup__int__def,axiom,
! [X6: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X6) = the(int,aTP_Lamp_pd(set(int),fun(int,bool),X6)) ).
% Sup_int_def
tff(fact_5908_lexordp__conv__lexord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
<=> pp(member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),lexord(A,collect(product_prod(A,A),product_case_prod(A,A,bool,ord_less(A)))))) ) ) ).
% lexordp_conv_lexord
tff(fact_5909_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
( transp(A,P)
=> ( sorted_wrt(A,P,Xs)
<=> ! [I5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I5)),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I5)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I5)))) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
tff(fact_5910_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Xs: list(A),Y: A,Ys2: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),cons(A,X,Xs)),cons(A,Y,Ys2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2)) ) ) ) ) ).
% lexordp_simps(3)
tff(fact_5911_lexordp__irreflexive,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A)] :
( ! [X2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Xs)) ) ) ).
% lexordp_irreflexive
tff(fact_5912_lexordp_OCons,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),cons(A,X,Xs)),cons(A,Y,Ys2))) ) ) ).
% lexordp.Cons
tff(fact_5913_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),cons(A,X,Xs)),cons(A,Y,Ys2))) ) ) ) ) ).
% lexordp.Cons_eq
tff(fact_5914_lexordp__append__leftD,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xs: list(A),Us: list(A),Vs: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),append(A,Xs,Us)),append(A,Xs,Vs)))
=> ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A6),A6))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Us),Vs)) ) ) ) ).
% lexordp_append_leftD
tff(fact_5915_Sup__eq__Inf,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A5) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_pe(set(A),fun(A,bool),A5))) ) ).
% Sup_eq_Inf
tff(fact_5916_Inf__eq__Sup,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A5) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_pf(set(A),fun(A,bool),A5))) ) ).
% Inf_eq_Sup
tff(fact_5917_lexordp__induct,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A),P: fun(list(A),fun(list(A),bool))] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
=> ( ! [Y4: A,Ys4: list(A)] : pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),cons(A,Y4,Ys4)))
=> ( ! [X2: A,Xs2: list(A),Y4: A,Ys4: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,cons(A,X2,Xs2)),cons(A,Y4,Ys4))) )
=> ( ! [X2: A,Xs2: list(A),Ys4: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys4))
=> ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),Ys4))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,cons(A,X2,Xs2)),cons(A,X2,Ys4))) ) )
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs),Ys2)) ) ) ) ) ) ).
% lexordp_induct
tff(fact_5918_lexordp__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
=> ( ( ( Xs = nil(A) )
=> ! [Y4: A,Ys5: list(A)] : Ys2 != cons(A,Y4,Ys5) )
=> ( ! [X2: A] :
( ? [Xs4: list(A)] : Xs = cons(A,X2,Xs4)
=> ! [Y4: A] :
( ? [Ys5: list(A)] : Ys2 = cons(A,Y4,Ys5)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4)) ) )
=> ~ ! [X2: A,Xs4: list(A)] :
( ( Xs = cons(A,X2,Xs4) )
=> ! [Ys5: list(A)] :
( ( Ys2 = cons(A,X2,Ys5) )
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs4),Ys5)) ) ) ) ) ) ) ).
% lexordp_cases
tff(fact_5919_lexordp_Osimps,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
<=> ( ? [Y5: A,Ys3: list(A)] :
( ( A1 = nil(A) )
& ( A22 = cons(A,Y5,Ys3) ) )
| ? [X5: A,Y5: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = cons(A,X5,Xs3) )
& ( A22 = cons(A,Y5,Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5)) )
| ? [X5: A,Y5: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = cons(A,X5,Xs3) )
& ( A22 = cons(A,Y5,Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X5))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs3),Ys3)) ) ) ) ) ).
% lexordp.simps
tff(fact_5920_lexordp_Ocases,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),A1),A22))
=> ( ( ( A1 = nil(A) )
=> ! [Y4: A,Ys4: list(A)] : A22 != cons(A,Y4,Ys4) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = cons(A,X2,Xs2)
=> ! [Y4: A] :
( ? [Ys4: list(A)] : A22 = cons(A,Y4,Ys4)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4)) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list(A)] :
( ( A1 = cons(A,X2,Xs2) )
=> ! [Ys4: list(A)] :
( ( A22 = cons(A,Y4,Ys4) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X2))
=> ~ pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs2),Ys4)) ) ) ) ) ) ) ) ) ).
% lexordp.cases
tff(fact_5921_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Us: list(A),Xs: list(A),Ys2: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),append(A,Us,cons(A,X,Xs))),append(A,Us,cons(A,Y,Ys2)))) ) ) ).
% lexordp_append_left_rightI
tff(fact_5922_lexordp__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys2: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys2))
<=> ( ? [X5: A,Vs2: list(A)] : Ys2 = append(A,Xs,cons(A,X5,Vs2))
| ? [Us2: list(A),A8: A,B8: A,Vs2: list(A),Ws: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A8),B8))
& ( Xs = append(A,Us2,cons(A,A8,Vs2)) )
& ( Ys2 = append(A,Us2,cons(A,B8,Ws)) ) ) ) ) ) ).
% lexordp_iff
tff(fact_5923_cInf__cSup,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S3: set(A)] :
( ( S3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),S3))
=> ( aa(set(A),A,complete_Inf_Inf(A),S3) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_pg(set(A),fun(A,bool),S3))) ) ) ) ) ).
% cInf_cSup
tff(fact_5924_cSup__cInf,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [S3: set(A)] :
( ( S3 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,condit941137186595557371_above(A),S3))
=> ( aa(set(A),A,complete_Sup_Sup(A),S3) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_ph(set(A),fun(A,bool),S3))) ) ) ) ) ).
% cSup_cInf
tff(fact_5925_transp__gr,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,aTP_Lamp_pi(A,fun(A,bool))) ) ).
% transp_gr
tff(fact_5926_transp__ge,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,aTP_Lamp_pj(A,fun(A,bool))) ) ).
% transp_ge
tff(fact_5927_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ord(A)
=> ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_lj(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).
% ord_class.lexordp_def
tff(fact_5928_Sup__Inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A5: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),A5)) = aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),collect(set(A),aTP_Lamp_pk(set(set(A)),fun(set(A),bool),A5)))) ) ).
% Sup_Inf
tff(fact_5929_Inf__Sup__le,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A5: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A5))),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),collect(set(A),aTP_Lamp_pk(set(set(A)),fun(set(A),bool),A5)))))) ) ).
% Inf_Sup_le
tff(fact_5930_Sup__Inf__le,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),collect(set(A),aTP_Lamp_pl(set(set(A)),fun(set(A),bool),A5))))),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A5)))) ) ).
% Sup_Inf_le
tff(fact_5931_finite__Inf__Sup,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [A5: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),image(set(A),A,complete_Sup_Sup(A),A5))),aa(set(A),A,complete_Sup_Sup(A),image(set(A),A,complete_Inf_Inf(A),collect(set(A),aTP_Lamp_pm(set(set(A)),fun(set(A),bool),A5)))))) ) ).
% finite_Inf_Sup
tff(fact_5932_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [G: fun(B,A),A5: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),image(set(B),A,aTP_Lamp_pn(fun(B,A),fun(set(B),A),G),A5)) = aa(set(A),A,complete_Inf_Inf(A),image(set(B),A,aTP_Lamp_po(fun(B,A),fun(set(B),A),G),collect(set(B),aTP_Lamp_pp(set(set(B)),fun(set(B),bool),A5)))) ) ).
% SUP_INF_set
tff(fact_5933_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [G: fun(B,A),A5: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),image(set(B),A,aTP_Lamp_po(fun(B,A),fun(set(B),A),G),A5)) = aa(set(A),A,complete_Sup_Sup(A),image(set(B),A,aTP_Lamp_pn(fun(B,A),fun(set(B),A),G),collect(set(B),aTP_Lamp_pp(set(set(B)),fun(set(B),bool),A5)))) ) ).
% INF_SUP_set
tff(fact_5934_lfp__funpow,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),N: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F)) = complete_lattice_lfp(A,F) ) ) ) ).
% lfp_funpow
tff(fact_5935_lfp__Kleene__iter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),K: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),bot_bot(A)) )
=> ( complete_lattice_lfp(A,F) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),bot_bot(A)) ) ) ) ) ).
% lfp_Kleene_iter
tff(fact_5936_transp__le,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,ord_less_eq(A)) ) ).
% transp_le
tff(fact_5937_transp__less,axiom,
! [A: $tType] :
( preorder(A)
=> transp(A,ord_less(A)) ) ).
% transp_less
tff(fact_5938_lfp__ordinal__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),P: fun(A,bool)] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( ! [S4: A] :
( pp(aa(A,bool,P,S4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S4),complete_lattice_lfp(A,F)))
=> pp(aa(A,bool,P,aa(A,A,F,S4))) ) )
=> ( ! [M8: set(A)] :
( ! [X3: A] :
( pp(member(A,X3,M8))
=> pp(aa(A,bool,P,X3)) )
=> pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),M8))) )
=> pp(aa(A,bool,P,complete_lattice_lfp(A,F))) ) ) ) ) ).
% lfp_ordinal_induct
tff(fact_5939_def__lfp__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: A,F: fun(A,A),P: A] :
( ( A5 = complete_lattice_lfp(A,F) )
=> ( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),P))),P))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),P)) ) ) ) ) ).
% def_lfp_induct
tff(fact_5940_lfp__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),P: A] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F)),P))),P))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F)),P)) ) ) ) ).
% lfp_induct
tff(fact_5941_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ord(A)
=> ! [X6: fun(bool,A),Y6: fun(bool,A)] :
( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X6),Y6))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fFalse)),aa(bool,A,Y6,fFalse)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).
% le_rel_bool_arg_iff
tff(fact_5942_lfp__mono,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),G: fun(A,A)] :
( ! [Z6: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,Z6)),aa(A,A,G,Z6)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F)),complete_lattice_lfp(A,G))) ) ) ).
% lfp_mono
tff(fact_5943_lfp__lowerbound,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),A5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,A5)),A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F)),A5)) ) ) ).
% lfp_lowerbound
tff(fact_5944_lfp__greatest,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),A5: A] :
( ! [U2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,U2)),U2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),U2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),complete_lattice_lfp(A,F))) ) ) ).
% lfp_greatest
tff(fact_5945_lfp__lfp,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,fun(A,A))] :
( ! [X2: A,Y4: A,W: A,Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),W)),aa(A,A,aa(A,fun(A,A),F,Y4),Z3))) ) )
=> ( complete_lattice_lfp(A,aTP_Lamp_pq(fun(A,fun(A,A)),fun(A,A),F)) = complete_lattice_lfp(A,aTP_Lamp_pr(fun(A,fun(A,A)),fun(A,A),F)) ) ) ) ).
% lfp_lfp
tff(fact_5946_lfp__eqI,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F3: fun(A,A),X: A] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
=> ( ( aa(A,A,F3,X) = X )
=> ( ! [Z3: A] :
( ( aa(A,A,F3,Z3) = Z3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z3)) )
=> ( complete_lattice_lfp(A,F3) = X ) ) ) ) ) ).
% lfp_eqI
tff(fact_5947_lfp__def,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A)] : complete_lattice_lfp(A,F) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_ps(fun(A,A),fun(A,bool),F))) ) ).
% lfp_def
tff(fact_5948_char__of__take__bit__eq,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat,M: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),N))
=> ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),N),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).
% char_of_take_bit_eq
tff(fact_5949_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),comp(nat,num,int,num_of_nat),nat2)),code_num_of_integer)) ).
% num_of_integer.transfer
tff(fact_5950_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_ml(A,fun(A,bool)),aTP_Lamp_pt(A,fun(A,bool))) ) ).
% Max.semilattice_order_set_axioms
tff(fact_5951_char__of__nat,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: nat] : aa(A,char,unique5772411509450598832har_of(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,char,unique5772411509450598832har_of(nat),N) ) ).
% char_of_nat
tff(fact_5952_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_ay(int,int)),code_integer_of_int)) ).
% integer.id_abs_transfer
tff(fact_5953_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_ay(int,int)),code_int_of_integer)) ).
% integer.rep_transfer
tff(fact_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_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_5961_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_5962_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_5963_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_5964_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_5965_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_5966_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_5967_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_5968_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_mr(int,int)),code_dup)) ).
% dup.transfer
tff(fact_5969_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_5970_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_5971_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_5972_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_5973_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_5974_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_5975_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_5976_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_5977_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_5978_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_5979_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_5980_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_5981_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_5982_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_5983_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_ms(num,fun(num,int))),code_sub)) ).
% sub.transfer
tff(fact_5984_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).
% Inf_fin.semilattice_order_set_axioms
tff(fact_5985_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).
% Min.semilattice_order_set_axioms
tff(fact_5986_UNIV__char__of__nat,axiom,
top_top(set(char)) = image(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).
% UNIV_char_of_nat
tff(fact_5987_inj__on__char__of__nat,axiom,
inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).
% inj_on_char_of_nat
tff(fact_5988_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_pu(A,fun(A,bool)),aTP_Lamp_pv(A,fun(A,bool))) ) ).
% Sup_fin.semilattice_order_set_axioms
tff(fact_5989_char__of__def,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = char2(fNot(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),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),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),bit0(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,bit0(one2)))),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),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_5990_range__nat__of__char,axiom,
image(char,nat,comm_s6883823935334413003f_char(nat),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).
% range_nat_of_char
tff(fact_5991_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_5992_char_Osize_I2_J,axiom,
! [X12: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X7: bool,X8: bool] : aa(char,nat,size_size(char),char2(X12,X22,X32,X42,X52,X62,X7,X8)) = zero_zero(nat) ).
% char.size(2)
tff(fact_5993_nat__of__char__less__256,axiom,
! [C2: char] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).
% nat_of_char_less_256
tff(fact_5994_assert__def,axiom,
! [A: $tType,P: fun(A,bool),X: A] :
( ( pp(aa(A,bool,P,X))
=> ( heap_Time_assert(A,P,X) = aa(A,heap_Time_Heap(A),heap_Time_return(A),X) ) )
& ( ~ pp(aa(A,bool,P,X))
=> ( heap_Time_assert(A,P,X) = heap_Time_raise(A,cons(char,char2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fFalse),cons(char,char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse),cons(char,char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse),cons(char,char2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fFalse),cons(char,char2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse),cons(char,char2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,fFalse),nil(char)))))))) ) ) ) ).
% assert_def
tff(fact_5995_char_Osize__gen,axiom,
! [X12: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X7: bool,X8: bool] : size_char(char2(X12,X22,X32,X42,X52,X62,X7,X8)) = zero_zero(nat) ).
% char.size_gen
tff(fact_5996_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),map(list(A),list(list(A)),aTP_Lamp_pw(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,N,Xs))) ).
% n_lists.simps(2)
tff(fact_5997_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list(A),Ys2: list(A)] :
( ( Xs != nil(A) )
=> ( ( Ys2 != nil(A) )
=> ( ( append(A,Xs,Ys2) = append(A,Ys2,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)) = append(A,Xs,Ys2) ) ) ) ) ) ).
% comm_append_is_replicate
tff(fact_5998_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [A5: set(A)] : uniq(list(A),aTP_Lamp_px(set(A),fun(list(A),bool),A5)) ) ).
% strict_sorted_equal_Uniq
tff(fact_5999_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_6000_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,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).
% integer_of_natural_of_integer
tff(fact_6001_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_6002_card__def,axiom,
! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_py(B,fun(nat,nat)),zero_zero(nat)) ).
% card_def
tff(fact_6003_length__code,axiom,
! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).
% length_code
tff(fact_6004_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list(B)] : aa(list(B),nat,gen_length(B,N),cons(B,X,Xs)) = aa(list(B),nat,gen_length(B,aa(nat,nat,suc,N)),Xs) ).
% gen_length_code(2)
tff(fact_6005_less__than__iff,axiom,
! [X: nat,Y: nat] :
( pp(member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Y),less_than))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ).
% less_than_iff
tff(fact_6006_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,one2),bit0(N)) = code_Neg(bitM(N)) ).
% Code_Numeral.sub_code(4)
tff(fact_6007_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] : aa(code_integer,code_integer,code_dup,code_Neg(N)) = code_Neg(bit0(N)) ).
% Code_Numeral.dup_code(3)
tff(fact_6008_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_6009_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_6010_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_6011_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_6012_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_6013_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_6014_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_6015_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_6016_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_6017_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(bit0(N)) ).
% Code_Numeral.sub_code(5)
tff(fact_6018_pair__less__def,axiom,
fun_pair_less = lex_prod(nat,nat,less_than,less_than) ).
% pair_less_def
tff(fact_6019_Gcd__int__set__eq__fold,axiom,
! [Xs: list(int)] : gcd_Gcd(int,set2(int,Xs)) = fold(int,int,gcd_gcd(int),Xs,zero_zero(int)) ).
% Gcd_int_set_eq_fold
tff(fact_6020_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Xs: list(A)] : gcd_Gcd(A,set2(A,Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).
% Gcd_set_eq_fold
tff(fact_6021_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),set2(A,Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).
% Gcd_fin.set_eq_fold
tff(fact_6022_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( order(B)
& order(D)
& order(C)
& order(A) )
=> ! [A5: fun(A,fun(B,bool)),B5: fun(C,fun(D,bool))] :
( bi_total(A,B,A5)
=> ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(B,fun(B,bool)),bool),bNF_rel_fun(A,B,fun(A,bool),fun(B,bool),A5,bNF_rel_fun(A,B,bool,bool,A5,fequal(bool))),ord_less_eq(A)),ord_less_eq(B)))
=> ( pp(aa(fun(D,fun(D,bool)),bool,aa(fun(C,fun(C,bool)),fun(fun(D,fun(D,bool)),bool),bNF_rel_fun(C,D,fun(C,bool),fun(D,bool),B5,bNF_rel_fun(C,D,bool,bool,B5,fequal(bool))),ord_less_eq(C)),ord_less_eq(D)))
=> pp(aa(fun(fun(B,D),bool),bool,aa(fun(fun(A,C),bool),fun(fun(fun(B,D),bool),bool),bNF_rel_fun(fun(A,C),fun(B,D),bool,bool,bNF_rel_fun(A,B,C,D,A5,B5),fequal(bool)),order_mono(A,C)),order_mono(B,D))) ) ) ) ) ).
% mono_transfer
tff(fact_6023_binomial__def,axiom,
! [N: nat,K: nat] : aa(nat,nat,binomial(N),K) = aa(set(set(nat)),nat,finite_card(set(nat)),collect(set(nat),aa(nat,fun(set(nat),bool),aTP_Lamp_pz(nat,fun(nat,fun(set(nat),bool)),N),K))) ).
% binomial_def
tff(fact_6024_Pow__Compl,axiom,
! [A: $tType,A5: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A5)) = collect(set(A),aTP_Lamp_qa(set(A),fun(set(A),bool),A5)) ).
% Pow_Compl
tff(fact_6025_int_Obi__total,axiom,
bi_total(product_prod(nat,nat),int,pcr_int) ).
% int.bi_total
tff(fact_6026_integer_Obi__total,axiom,
bi_total(int,code_integer,code_pcr_integer) ).
% integer.bi_total
tff(fact_6027_Gcd__nat__set__eq__fold,axiom,
! [Xs: list(nat)] : gcd_Gcd(nat,set2(nat,Xs)) = fold(nat,nat,gcd_gcd(nat),Xs,zero_zero(nat)) ).
% Gcd_nat_set_eq_fold
tff(fact_6028_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)))) = heap_Time_Heap2(A,aTP_Lamp_oa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% Heap_lub_empty
tff(fact_6029_gfp__Kleene__iter,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),K: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),top_top(A)) )
=> ( complete_lattice_gfp(A,F) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),top_top(A)) ) ) ) ) ).
% gfp_Kleene_iter
tff(fact_6030_gfp__gfp,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,fun(A,A))] :
( ! [X2: A,Y4: A,W: A,Z3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X2),W)),aa(A,A,aa(A,fun(A,A),F,Y4),Z3))) ) )
=> ( complete_lattice_gfp(A,aTP_Lamp_qb(fun(A,fun(A,A)),fun(A,A),F)) = complete_lattice_gfp(A,aTP_Lamp_pr(fun(A,fun(A,A)),fun(A,A),F)) ) ) ) ).
% gfp_gfp
tff(fact_6031_gfp__mono,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),G: fun(A,A)] :
( ! [Z6: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,Z6)),aa(A,A,G,Z6)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F)),complete_lattice_gfp(A,G))) ) ) ).
% gfp_mono
tff(fact_6032_gfp__upperbound,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X6: A,F: fun(A,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F,X6)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F))) ) ) ).
% gfp_upperbound
tff(fact_6033_gfp__least,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),X6: A] :
( ! [U2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U2),aa(A,A,F,U2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U2),X6)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F)),X6)) ) ) ).
% gfp_least
tff(fact_6034_gfp__eqI,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F3: fun(A,A),X: A] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F3))
=> ( ( aa(A,A,F3,X) = X )
=> ( ! [Z3: A] :
( ( aa(A,A,F3,Z3) = Z3 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),X)) )
=> ( complete_lattice_gfp(A,F3) = X ) ) ) ) ) ).
% gfp_eqI
tff(fact_6035_gfp__def,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A)] : complete_lattice_gfp(A,F) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_qc(fun(A,A),fun(A,bool),F))) ) ).
% gfp_def
tff(fact_6036_coinduct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),X6: A] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F)))))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),complete_lattice_gfp(A,F))) ) ) ) ).
% coinduct
tff(fact_6037_def__coinduct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A5: A,F: fun(A,A),X6: A] :
( ( A5 = complete_lattice_gfp(A,F) )
=> ( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),A5))))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),A5)) ) ) ) ) ).
% def_coinduct
tff(fact_6038_coinduct__lemma,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [X6: A,F: fun(A,A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X6),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F)))))
=> ( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F))),aa(A,A,F,aa(A,A,aa(A,fun(A,A),sup_sup(A),X6),complete_lattice_gfp(A,F))))) ) ) ) ).
% coinduct_lemma
tff(fact_6039_gfp__ordinal__induct,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),P: fun(A,bool)] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( ! [S4: A] :
( pp(aa(A,bool,P,S4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_gfp(A,F)),S4))
=> pp(aa(A,bool,P,aa(A,A,F,S4))) ) )
=> ( ! [M8: set(A)] :
( ! [X3: A] :
( pp(member(A,X3,M8))
=> pp(aa(A,bool,P,X3)) )
=> pp(aa(A,bool,P,aa(set(A),A,complete_Inf_Inf(A),M8))) )
=> pp(aa(A,bool,P,complete_lattice_gfp(A,F))) ) ) ) ) ).
% gfp_ordinal_induct
tff(fact_6040_gfp__funpow,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A),N: nat] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F)) = complete_lattice_gfp(A,F) ) ) ) ).
% gfp_funpow
tff(fact_6041_lfp__le__gfp,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(A,A)] :
( pp(aa(fun(A,A),bool,order_mono(A,A),F))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),complete_lattice_lfp(A,F)),complete_lattice_gfp(A,F))) ) ) ).
% lfp_le_gfp
tff(fact_6042_Code__Numeral_Osub__code_I2_J,axiom,
! [M: num] : aa(num,code_integer,aa(num,fun(num,code_integer),code_sub,bit0(M)),one2) = aa(num,code_integer,code_Pos,bitM(M)) ).
% Code_Numeral.sub_code(2)
tff(fact_6043_is__singleton__altdef,axiom,
! [A: $tType,A5: set(A)] :
( is_singleton(A,A5)
<=> ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) ) ) ).
% is_singleton_altdef
tff(fact_6044_Code__Numeral_OPos__def,axiom,
code_Pos = numeral_numeral(code_integer) ).
% Code_Numeral.Pos_def
tff(fact_6045_Pos__fold_I2_J,axiom,
! [K: num] : aa(num,code_integer,numeral_numeral(code_integer),bit0(K)) = aa(num,code_integer,code_Pos,bit0(K)) ).
% Pos_fold(2)
tff(fact_6046_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_6047_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_6048_one__integer__code,axiom,
one_one(code_integer) = aa(num,code_integer,code_Pos,one2) ).
% one_integer_code
tff(fact_6049_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,bit0(N)) ).
% Code_Numeral.dup_code(2)
tff(fact_6050_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_6051_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_6052_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_6053_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_6054_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_6055_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_6056_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_6057_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_6058_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_6059_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_6060_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_6061_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_6062_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_6063_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_6064_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_6065_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_6066_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_6067_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_6068_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_6069_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_6070_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_6071_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_6072_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_6073_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_6074_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_6075_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_6076_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,bit0(M)) ).
% Code_Numeral.sub_code(3)
tff(fact_6077_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_6078_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),B5: set(A)] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F),B5)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5))) ) ) ) ) ).
% semilattice_order_set.subset_imp
tff(fact_6079_semilattice__set_OF_Ocong,axiom,
! [A: $tType,F: fun(A,fun(A,A))] : lattic1715443433743089157tice_F(A,F) = lattic1715443433743089157tice_F(A,F) ).
% semilattice_set.F.cong
tff(fact_6080_Inf__fin__def,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ( lattic7752659483105999362nf_fin(A) = lattic1715443433743089157tice_F(A,inf_inf(A)) ) ) ).
% Inf_fin_def
tff(fact_6081_Sup__fin__def,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ( lattic5882676163264333800up_fin(A) = lattic1715443433743089157tice_F(A,sup_sup(A)) ) ) ).
% Sup_fin_def
tff(fact_6082_Max__def,axiom,
! [A: $tType] :
( linorder(A)
=> ( lattic643756798349783984er_Max(A) = lattic1715443433743089157tice_F(A,ord_max(A)) ) ) ).
% Max_def
tff(fact_6083_Min__def,axiom,
! [A: $tType] :
( linorder(A)
=> ( lattic643756798350308766er_Min(A) = lattic1715443433743089157tice_F(A,ord_min(A)) ) ) ).
% Min_def
tff(fact_6084_semilattice__order__set_OcoboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),A3: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)),A3)) ) ) ) ).
% semilattice_order_set.coboundedI
tff(fact_6085_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)))
=> ! [A9: A] :
( pp(member(A,A9,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),A9)) ) ) ) ) ) ).
% semilattice_order_set.boundedE
tff(fact_6086_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),A6)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5))) ) ) ) ) ).
% semilattice_order_set.boundedI
tff(fact_6087_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),X: A] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),X5)) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
tff(fact_6088_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_6089_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_qd(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F),none(A),A5)) ) ) ).
% semilattice_set.eq_fold'
tff(fact_6090_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),insert(A,X,A5)) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ).
% semilattice_set.insert_remove
tff(fact_6091_Min_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> lattic149705377957585745ce_set(A,ord_min(A)) ) ).
% Min.semilattice_set_axioms
tff(fact_6092_Max_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> lattic149705377957585745ce_set(A,ord_max(A)) ) ).
% Max.semilattice_set_axioms
tff(fact_6093_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> lattic149705377957585745ce_set(A,inf_inf(A)) ) ).
% Inf_fin.semilattice_set_axioms
tff(fact_6094_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> lattic149705377957585745ce_set(A,sup_sup(A)) ) ).
% Sup_fin.semilattice_set_axioms
tff(fact_6095_semilattice__order__set_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> lattic149705377957585745ce_set(A,F) ) ).
% semilattice_order_set.axioms(2)
tff(fact_6096_semilattice__set_Oin__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5) ) ) ) ) ).
% semilattice_set.in_idem
tff(fact_6097_semilattice__set_Osingleton,axiom,
! [A: $tType,F: fun(A,fun(A,A)),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),insert(A,X,bot_bot(set(A)))) = X ) ) ).
% semilattice_set.singleton
tff(fact_6098_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),H2: fun(A,A),N6: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( ! [X2: A,Y4: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),F,X2),Y4)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,H2,X2)),aa(A,A,H2,Y4))
=> ( pp(aa(set(A),bool,finite_finite2(A),N6))
=> ( ( N6 != bot_bot(set(A)) )
=> ( aa(A,A,H2,aa(set(A),A,lattic1715443433743089157tice_F(A,F),N6)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),image(A,A,H2,N6)) ) ) ) ) ) ).
% semilattice_set.hom_commute
tff(fact_6099_semilattice__set_Osubset,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),B5: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),B5)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5) ) ) ) ) ) ).
% semilattice_set.subset
tff(fact_6100_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ~ pp(member(A,X,A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
tff(fact_6101_semilattice__set_Oinsert,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)) ) ) ) ) ).
% semilattice_set.insert
tff(fact_6102_semilattice__set_Oclosed,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [X2: A,Y4: A] : pp(member(A,aa(A,A,aa(A,fun(A,A),F,X2),Y4),insert(A,X2,insert(A,Y4,bot_bot(set(A))))))
=> pp(member(A,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5),A5)) ) ) ) ) ).
% semilattice_set.closed
tff(fact_6103_semilattice__set_Ounion,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),B5: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( ( B5 != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5)),aa(set(A),A,lattic1715443433743089157tice_F(A,F),B5)) ) ) ) ) ) ) ).
% semilattice_set.union
tff(fact_6104_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),insert(A,X,A5)) = finite_fold(A,A,F,X,A5) ) ) ) ).
% semilattice_set.eq_fold
tff(fact_6105_semilattice__set_Oinfinite,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5) = the2(A,none(A)) ) ) ) ).
% semilattice_set.infinite
tff(fact_6106_semilattice__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A5: set(A),X: A] :
( lattic149705377957585745ce_set(A,F)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) = bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5) = X ) )
& ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))) != bot_bot(set(A)) )
=> ( aa(set(A),A,lattic1715443433743089157tice_F(A,F),A5) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic1715443433743089157tice_F(A,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ) ) ).
% semilattice_set.remove
tff(fact_6107_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),S3: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( ? [X3: A] :
( pp(member(A,X3,S3))
& ! [Xa3: A] :
( pp(member(A,Xa3,S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa3)) ) )
=> ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_qe(fun(A,B),fun(set(A),fun(B,bool)),F),S3)) = aa(A,B,F,ord_Least(A,aTP_Lamp_qf(set(A),fun(A,bool),S3))) ) ) ) ) ).
% Least_mono
tff(fact_6108_cr__int__def,axiom,
! [X3: product_prod(nat,nat)] : aa(product_prod(nat,nat),fun(int,bool),cr_int,X3) = aa(int,fun(int,bool),fequal(int),aa(product_prod(nat,nat),int,abs_Integ,X3)) ).
% cr_int_def
tff(fact_6109_Least__eq__0,axiom,
! [P: fun(nat,bool)] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ord_Least(nat,P) = zero_zero(nat) ) ) ).
% Least_eq_0
tff(fact_6110_Least__le,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),K: A] :
( pp(aa(A,bool,P,K))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),K)) ) ) ).
% Least_le
tff(fact_6111_Least1I,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool)] :
( ? [X3: A] :
( pp(aa(A,bool,P,X3))
& ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4)) )
& ! [Y4: A] :
( ( pp(aa(A,bool,P,Y4))
& ! [Ya2: A] :
( pp(aa(A,bool,P,Ya2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Ya2)) ) )
=> ( Y4 = X3 ) ) )
=> pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).
% Least1I
tff(fact_6112_Least1__le,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),Z2: A] :
( ? [X3: A] :
( pp(aa(A,bool,P,X3))
& ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4)) )
& ! [Y4: A] :
( ( pp(aa(A,bool,P,Y4))
& ! [Ya2: A] :
( pp(aa(A,bool,P,Ya2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),Ya2)) ) )
=> ( Y4 = X3 ) ) )
=> ( pp(aa(A,bool,P,Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ord_Least(A,P)),Z2)) ) ) ) ).
% Least1_le
tff(fact_6113_LeastI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y3)) )
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ) ).
% LeastI2_order
tff(fact_6114_Least__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A] :
( pp(aa(A,bool,P,X))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
=> ( ord_Least(A,P) = X ) ) ) ) ).
% Least_equality
tff(fact_6115_LeastI2__wellorder,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),A3: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,A3))
=> ( ! [A6: A] :
( pp(aa(A,bool,P,A6))
=> ( ! [B11: A] :
( pp(aa(A,bool,P,B11))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B11)) )
=> pp(aa(A,bool,Q,A6)) ) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2_wellorder
tff(fact_6116_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),Q: fun(A,bool)] :
( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> ( ! [A6: A] :
( pp(aa(A,bool,P,A6))
=> ( ! [B11: A] :
( pp(aa(A,bool,P,B11))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A6),B11)) )
=> pp(aa(A,bool,Q,A6)) ) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2_wellorder_ex
tff(fact_6117_not__less__Least,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [K: A,P: fun(A,bool)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),ord_Least(A,P)))
=> ~ pp(aa(A,bool,P,K)) ) ) ).
% not_less_Least
tff(fact_6118_LeastI,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),K: A] :
( pp(aa(A,bool,P,K))
=> pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).
% LeastI
tff(fact_6119_LeastI2,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),A3: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,A3))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2
tff(fact_6120_LeastI__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool)] :
( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> pp(aa(A,bool,P,ord_Least(A,P))) ) ) ).
% LeastI_ex
tff(fact_6121_LeastI2__ex,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [P: fun(A,bool),Q: fun(A,bool)] :
( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(A,bool,Q,ord_Least(A,P))) ) ) ) ).
% LeastI2_ex
tff(fact_6122_Least__Suc2,axiom,
! [P: fun(nat,bool),N: nat,Q: fun(nat,bool),M: nat] :
( pp(aa(nat,bool,P,N))
=> ( pp(aa(nat,bool,Q,M))
=> ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [K2: nat] :
( pp(aa(nat,bool,P,aa(nat,nat,suc,K2)))
<=> pp(aa(nat,bool,Q,K2)) )
=> ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).
% Least_Suc2
tff(fact_6123_Least__Suc,axiom,
! [P: fun(nat,bool),N: nat] :
( pp(aa(nat,bool,P,N))
=> ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_qg(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).
% Least_Suc
tff(fact_6124_Least__Min,axiom,
! [A: $tType] :
( linorder(A)
=> ! [P: fun(A,bool)] :
( pp(aa(set(A),bool,finite_finite2(A),collect(A,P)))
=> ( ? [X_1: A] : pp(aa(A,bool,P,X_1))
=> ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),collect(A,P)) ) ) ) ) ).
% Least_Min
tff(fact_6125_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
tff(fact_6126_Quotient__int,axiom,
quotient(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ,cr_int) ).
% Quotient_int
tff(fact_6127_list__ex__length,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( list_ex(A,P,Xs)
<=> ? [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N5))) ) ) ).
% list_ex_length
tff(fact_6128_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),I: nat] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
=> ( distinct(A,Xs)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2)))
=> ( map_of(A,B,zip(A,B,Xs,Ys2),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),I)) ) ) ) ) ).
% map_of_zip_nth
tff(fact_6129_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_6130_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_order.axioms(1)
tff(fact_6131_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> semilattice_neutr(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.semilattice_neutr_axioms
tff(fact_6132_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> semilattice_neutr(A,inf_inf(A),top_top(A)) ) ).
% inf_top.semilattice_neutr_axioms
tff(fact_6133_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).
% gcd_nat.semilattice_neutr_axioms
tff(fact_6134_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> semilattice_neutr(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).
% or.semilattice_neutr_axioms
tff(fact_6135_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).
% max_nat.semilattice_neutr_axioms
tff(fact_6136_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list(A),I: nat,M: fun(A,option(B)),Ys2: list(B),Y: B] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I))
=> ( map_upds(A,B,M,Xs,list_update(B,Ys2,I,Y)) = map_upds(A,B,M,Xs,Ys2) ) ) ).
% map_upds_list_update2_drop
tff(fact_6137_natLess__def,axiom,
bNF_Ca8459412986667044542atLess = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,ord_less(nat))) ).
% natLess_def
tff(fact_6138_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [R: set(product_prod(A,A)),As: fun(A,B)] :
( bNF_Ca3754400796208372196lChain(A,B,R,As)
<=> ! [I5: A,J3: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I5),J3),R))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As,I5)),aa(A,B,As,J3))) ) ) ) ).
% relChain_def
tff(fact_6139_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),M: fun(A,option(B)),X: 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)),Ys2)))
=> ( map_upds(A,B,M,append(A,Xs,cons(A,X,nil(A))),Ys2) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys2),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).
% map_upds_append1
tff(fact_6140_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),insert(A,X,bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),insert(A,X,bot_bot(set(A))))) ).
% restrict_upd_same
tff(fact_6141_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] : restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),insert(A,X,bot_bot(set(A))))) = fun_upd(A,option(B),F,X,none(B)) ).
% restrict_complement_singleton_eq
tff(fact_6142_Restr__natLeq,axiom,
! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N)),aTP_Lamp_qh(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N))) ).
% Restr_natLeq
tff(fact_6143_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> folding_insort_key(A,A,ord_less_eq(A),ord_less(A),top_top(set(A)),aTP_Lamp_me(A,A)) ) ).
% sorted_list_of_set.folding_insort_key_axioms
tff(fact_6144_natLeq__def,axiom,
bNF_Ca8665028551170535155natLeq = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,ord_less_eq(nat))) ).
% natLeq_def
tff(fact_6145_Restr__natLeq2,axiom,
! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N),aTP_Lamp_qj(nat,fun(nat,set(nat)),N))) = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N))) ).
% Restr_natLeq2
tff(fact_6146_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( zero(B)
=> ! [F: fun(fun(A,B),C),G: C] :
( ! [X2: fun(A,B)] : aa(fun(A,B),C,F,X2) = G
=> ( aa(fun(A,B),C,F,aTP_Lamp_qk(A,B)) = G ) ) ) ).
% fun_cong_unused_0
tff(fact_6147_natLeq__underS__less,axiom,
! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N)) ).
% natLeq_underS_less
tff(fact_6148_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_6149_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),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).
% inv_o_cancel
tff(fact_6150_inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A] :
( inj_on(A,B,F,A5)
=> ( pp(member(A,X,A5))
=> ( aa(B,A,hilbert_inv_into(A,B,A5,F),aa(A,B,F,X)) = X ) ) ) ).
% inv_into_f_f
tff(fact_6151_inv__identity,axiom,
! [A: $tType,X3: A] : aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),aTP_Lamp_ql(A,A)),X3) = X3 ).
% inv_identity
tff(fact_6152_inv__id,axiom,
! [A: $tType] : hilbert_inv_into(A,A,top_top(set(A)),id(A)) = id(A) ).
% inv_id
tff(fact_6153_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),S3: set(A)] :
( inj_on(A,B,F,A5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),A5))
=> ( image(B,A,hilbert_inv_into(A,B,A5,F),image(A,B,F,S3)) = S3 ) ) ) ).
% inv_into_image_cancel
tff(fact_6154_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),comp(B,C,A,aa(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_6155_inv__into__def,axiom,
! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),X3: B] : aa(B,A,hilbert_inv_into(A,B,A5,F),X3) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_fm(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A5),F),X3)) ).
% inv_into_def
tff(fact_6156_inv__into__def2,axiom,
! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),X: B] : aa(B,A,hilbert_inv_into(A,B,A5,F),X) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_fm(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A5),F),X)) ).
% inv_into_def2
tff(fact_6157_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_6158_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_6159_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_6160_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_6161_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_6162_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_6163_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_6164_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_6165_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_6166_inv__into__injective,axiom,
! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),X: B,Y: B] :
( ( aa(B,A,hilbert_inv_into(A,B,A5,F),X) = aa(B,A,hilbert_inv_into(A,B,A5,F),Y) )
=> ( pp(member(B,X,image(A,B,F,A5)))
=> ( pp(member(B,Y,image(A,B,F,A5)))
=> ( X = Y ) ) ) ) ).
% inv_into_injective
tff(fact_6167_inv__into__into,axiom,
! [A: $tType,B: $tType,X: A,F: fun(B,A),A5: set(B)] :
( pp(member(A,X,image(B,A,F,A5)))
=> pp(member(B,aa(A,B,hilbert_inv_into(B,A,A5,F),X),A5)) ) ).
% inv_into_into
tff(fact_6168_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y: A,F: fun(B,A),A5: set(B)] :
( pp(member(A,Y,image(B,A,F,A5)))
=> ( aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,A5,F),Y)) = Y ) ) ).
% f_inv_into_f
tff(fact_6169_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Y: A] :
( ( image(B,A,F,top_top(set(B))) = top_top(set(A)) )
=> ( aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),Y)) = Y ) ) ).
% surj_f_inv_f
tff(fact_6170_surj__iff__all,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( image(B,A,F,top_top(set(B))) = top_top(set(A)) )
<=> ! [X5: A] : aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),X5)) = X5 ) ).
% surj_iff_all
tff(fact_6171_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A5: set(A)] :
( ( image(B,A,F,top_top(set(B))) = top_top(set(A)) )
=> ( image(B,A,F,image(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),A5)) = A5 ) ) ).
% image_f_inv_f
tff(fact_6172_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
( ( image(B,A,F,top_top(set(B))) = top_top(set(A)) )
=> ( ! [X2: B] : aa(A,B,G,aa(B,A,F,X2)) = X2
=> ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).
% surj_imp_inv_eq
tff(fact_6173_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A,Y: B] :
( inj_on(A,B,F,A5)
=> ( pp(member(A,X,A5))
=> ( ( aa(A,B,F,X) = Y )
=> ( aa(B,A,hilbert_inv_into(A,B,A5,F),Y) = X ) ) ) ) ).
% inv_into_f_eq
tff(fact_6174_inv__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),aa(A,B,F,X)) = X ) ) ).
% inv_f_f
tff(fact_6175_inv__f__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X: A,Y: B] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(A,B,F,X) = Y )
=> ( aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),Y) = X ) ) ) ).
% inv_f_eq
tff(fact_6176_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,B),G: fun(B,A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ! [X2: B] : aa(A,B,F,aa(B,A,G,X2)) = X2
=> ( hilbert_inv_into(A,B,top_top(set(A)),F) = G ) ) ) ).
% inj_imp_inv_eq
tff(fact_6177_inv__equality,axiom,
! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
( ! [X2: A] : aa(B,A,G,aa(A,B,F,X2)) = X2
=> ( ! [Y4: B] : aa(A,B,F,aa(B,A,G,Y4)) = Y4
=> ( hilbert_inv_into(A,B,top_top(set(A)),F) = G ) ) ) ).
% inv_equality
tff(fact_6178_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A5: set(B),A10: set(A),B13: set(A)] :
( ( image(B,A,F,A5) = A10 )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B13),A10))
=> ( image(B,A,F,image(A,B,hilbert_inv_into(B,A,A5,F),B13)) = B13 ) ) ) ).
% image_inv_into_cancel
tff(fact_6179_inv__def,axiom,
! [B: $tType,A: $tType,F: fun(B,A),X3: A] : aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),X3) = fChoice(B,aa(A,fun(B,bool),aTP_Lamp_qm(fun(B,A),fun(A,fun(B,bool)),F),X3)) ).
% inv_def
tff(fact_6180_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B5: set(A),F: fun(B,A),A5: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),image(B,A,F,A5)))
=> inj_on(A,B,hilbert_inv_into(B,A,A5,F),B5) ) ).
% inj_on_inv_into
tff(fact_6181_inj__transfer,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P: fun(A,bool),X: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ! [Y4: B] :
( pp(member(B,Y4,image(A,B,F,top_top(set(A)))))
=> pp(aa(A,bool,P,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),Y4))) )
=> pp(aa(A,bool,P,X)) ) ) ).
% inj_transfer
tff(fact_6182_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),image(A,B,F,A5)) = A5 ) ) ).
% image_inv_f_f
tff(fact_6183_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),top_top(set(B))) = top_top(set(A)) ) ) ).
% inj_imp_surj_inv
tff(fact_6184_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( image(B,A,F,top_top(set(B))) = top_top(set(A)) )
=> inj_on(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),top_top(set(A))) ) ).
% surj_imp_inj_inv
tff(fact_6185_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A5: set(C),X: B] :
( inj_on(A,B,F,image(C,A,G,A5))
=> ( inj_on(C,A,G,A5)
=> ( pp(member(B,X,image(A,B,F,image(C,A,G,A5))))
=> ( aa(B,C,hilbert_inv_into(C,B,A5,aa(fun(C,A),fun(C,B),comp(A,B,C,F),G)),X) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,hilbert_inv_into(C,A,A5,G)),hilbert_inv_into(A,B,image(C,A,G,A5),F)),X) ) ) ) ) ).
% inv_into_comp
tff(fact_6186_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
( ( aa(fun(A,B),fun(A,A),comp(B,A,A,F),G) = id(A) )
=> ( ( aa(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_6187_surj__iff,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( image(B,A,F,top_top(set(B))) = top_top(set(A)) )
<=> ( aa(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_6188_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),comp(B,A,A,hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).
% inj_iff
tff(fact_6189_bijection_Oinv__comp__left,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(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_6190_bijection_Oinv__comp__right,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(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_6191_bijection_Oeq__iff,axiom,
! [A: $tType,F: fun(A,A),A3: A,B2: A] :
( hilbert_bijection(A,F)
=> ( ( aa(A,A,F,A3) = aa(A,A,F,B2) )
<=> ( A3 = B2 ) ) ) ).
% bijection.eq_iff
tff(fact_6192_bijection_OeqI,axiom,
! [A: $tType,F: fun(A,A),A3: A,B2: A] :
( hilbert_bijection(A,F)
=> ( ( aa(A,A,F,A3) = aa(A,A,F,B2) )
=> ( A3 = B2 ) ) ) ).
% bijection.eqI
tff(fact_6193_bijection_Osurj,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( image(A,A,F,top_top(set(A))) = top_top(set(A)) ) ) ).
% bijection.surj
tff(fact_6194_bijection_Oinj,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> inj_on(A,A,F,top_top(set(A))) ) ).
% bijection.inj
tff(fact_6195_bijection_Oinv__right__eq__iff,axiom,
! [A: $tType,F: fun(A,A),B2: A,A3: A] :
( hilbert_bijection(A,F)
=> ( ( B2 = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) )
<=> ( aa(A,A,F,B2) = A3 ) ) ) ).
% bijection.inv_right_eq_iff
tff(fact_6196_bijection_Oinv__left__eq__iff,axiom,
! [A: $tType,F: fun(A,A),A3: A,B2: A] :
( hilbert_bijection(A,F)
=> ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) = B2 )
<=> ( aa(A,A,F,B2) = A3 ) ) ) ).
% bijection.inv_left_eq_iff
tff(fact_6197_bijection_Oeq__inv__iff,axiom,
! [A: $tType,F: fun(A,A),A3: A,B2: A] :
( hilbert_bijection(A,F)
=> ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),B2) )
<=> ( A3 = B2 ) ) ) ).
% bijection.eq_inv_iff
tff(fact_6198_bijection_Oinv__right,axiom,
! [A: $tType,F: fun(A,A),A3: A] :
( hilbert_bijection(A,F)
=> ( aa(A,A,F,aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3)) = A3 ) ) ).
% bijection.inv_right
tff(fact_6199_bijection_Oinv__left,axiom,
! [A: $tType,F: fun(A,A),A3: A] :
( hilbert_bijection(A,F)
=> ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),aa(A,A,F,A3)) = A3 ) ) ).
% bijection.inv_left
tff(fact_6200_bijection_Oeq__invI,axiom,
! [A: $tType,F: fun(A,A),A3: A,B2: A] :
( hilbert_bijection(A,F)
=> ( ( aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),A3) = aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),B2) )
=> ( A3 = B2 ) ) ) ).
% bijection.eq_invI
tff(fact_6201_bijection_Osurj__inv,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( image(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),top_top(set(A))) = top_top(set(A)) ) ) ).
% bijection.surj_inv
tff(fact_6202_bijection_Oinj__inv,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> inj_on(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),top_top(set(A))) ) ).
% bijection.inj_inv
tff(fact_6203_iteratesp_Omono,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A)] : pp(aa(fun(fun(A,bool),fun(A,bool)),bool,order_mono(fun(A,bool),fun(A,bool)),aTP_Lamp_qn(fun(A,A),fun(fun(A,bool),fun(A,bool)),F))) ) ).
% iteratesp.mono
tff(fact_6204_finite__subset__wf,axiom,
! [A: $tType,A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> wf(set(A),collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),bool,aTP_Lamp_qo(set(A),fun(set(A),fun(set(A),bool)),A5)))) ) ).
% finite_subset_wf
tff(fact_6205_wf,axiom,
! [A: $tType] :
( wellorder(A)
=> wf(A,collect(product_prod(A,A),product_case_prod(A,A,bool,ord_less(A)))) ) ).
% wf
tff(fact_6206_ccpo__Sup__least,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A5: set(A),Z2: A] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
=> ( ! [X2: A] :
( pp(member(A,X2,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Z2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z2)) ) ) ) ).
% ccpo_Sup_least
tff(fact_6207_ccpo__Sup__upper,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A5: set(A),X: A] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
=> ( pp(member(A,X,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A5))) ) ) ) ).
% ccpo_Sup_upper
tff(fact_6208_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ~ ? [F4: fun(nat,A)] :
! [I5: nat] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F4,aa(nat,nat,suc,I5))),aa(nat,A,F4,I5)),R)) ) ).
% wf_iff_no_infinite_down_chain
tff(fact_6209_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set(product_prod(A,A)),F: fun(nat,A)] :
( wf(A,R)
=> ~ ! [K2: nat] : pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F,aa(nat,nat,suc,K2))),aa(nat,A,F,K2)),R)) ) ).
% wf_no_infinite_down_chainE
tff(fact_6210_chain__singleton,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),insert(A,X,bot_bot(set(A)))) ) ).
% chain_singleton
tff(fact_6211_wf__if__measure,axiom,
! [A: $tType,P: fun(A,bool),F: fun(A,nat),G: fun(A,A)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,aa(A,A,G,X2))),aa(A,nat,F,X2))) )
=> wf(A,collect(product_prod(A,A),product_case_prod(A,A,bool,aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_qp(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P),G)))) ) ).
% wf_if_measure
tff(fact_6212_wf__bounded__measure,axiom,
! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F: fun(A,nat)] :
( ! [A6: A,B4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B4),A6),R))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A6)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,B4)),aa(A,nat,Ub,A6)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,A6)),aa(A,nat,F,B4))) ) )
=> wf(A,R) ) ).
% wf_bounded_measure
tff(fact_6213_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)
=> ( ! [X2: A,Y4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Y4),transitive_trancl(A,R)))
<=> ~ pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),X2),transitive_rtrancl(A,R))) )
=> ( pp(aa(B,bool,P,K))
=> ? [X2: B] :
( pp(aa(B,bool,P,X2))
& ! [Y3: B] :
( pp(aa(B,bool,P,Y3))
=> pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(B,A,M,X2)),aa(B,A,M,Y3)),transitive_rtrancl(A,R))) ) ) ) ) ) ).
% wf_linord_ex_has_least
tff(fact_6214_reduction__pair__lemma,axiom,
! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( fun_reduction_pair(A,P)
=> ( 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))),R2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P)))
=> ( 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))),S3),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P)))
=> ( wf(A,S3)
=> 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))),sup_sup(set(product_prod(A,A))),R2),S3)) ) ) ) ) ).
% reduction_pair_lemma
tff(fact_6215_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),Ub: fun(A,set(B)),F: fun(A,set(B))] :
( ! [A6: A,B4: A] :
( pp(member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B4),A6),R))
=> ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Ub,A6)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A6)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,B4)),aa(A,set(B),Ub,A6)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F,A6)),aa(A,set(B),F,B4))) ) )
=> wf(A,R) ) ).
% wf_bounded_set
tff(fact_6216_in__chain__finite,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A5: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),A5)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> pp(member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5)) ) ) ) ) ).
% in_chain_finite
tff(fact_6217_iteratesp__def,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X3: fun(A,A)] : comple7512665784863727008ratesp(A,X3) = complete_lattice_lfp(fun(A,bool),aTP_Lamp_qn(fun(A,A),fun(fun(A,bool),fun(A,bool)),X3)) ) ).
% iteratesp_def
tff(fact_6218_wf__int__ge__less__than2,axiom,
! [D2: int] : wf(int,int_ge_less_than2(D2)) ).
% wf_int_ge_less_than2
tff(fact_6219_wf__int__ge__less__than,axiom,
! [D2: int] : wf(int,int_ge_less_than(D2)) ).
% wf_int_ge_less_than
tff(fact_6220_wf__pair__less,axiom,
wf(product_prod(nat,nat),fun_pair_less) ).
% wf_pair_less
tff(fact_6221_wf__less,axiom,
wf(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,ord_less(nat)))) ).
% wf_less
tff(fact_6222_iteratesp_Osimps,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A),A3: A] :
( pp(aa(A,bool,comple7512665784863727008ratesp(A,F),A3))
<=> ( ? [X5: A] :
( ( A3 = aa(A,A,F,X5) )
& pp(aa(A,bool,comple7512665784863727008ratesp(A,F),X5)) )
| ? [M9: set(A)] :
( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M9) )
& comple1602240252501008431_chain(A,ord_less_eq(A),M9)
& ! [X5: A] :
( pp(member(A,X5,M9))
=> pp(aa(A,bool,comple7512665784863727008ratesp(A,F),X5)) ) ) ) ) ) ).
% iteratesp.simps
tff(fact_6223_iteratesp_Ocases,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A),A3: A] :
( pp(aa(A,bool,comple7512665784863727008ratesp(A,F),A3))
=> ( ! [X2: A] :
( ( A3 = aa(A,A,F,X2) )
=> ~ pp(aa(A,bool,comple7512665784863727008ratesp(A,F),X2)) )
=> ~ ! [M8: set(A)] :
( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
=> ( comple1602240252501008431_chain(A,ord_less_eq(A),M8)
=> ~ ! [X3: A] :
( pp(member(A,X3,M8))
=> pp(aa(A,bool,comple7512665784863727008ratesp(A,F),X3)) ) ) ) ) ) ) ).
% iteratesp.cases
tff(fact_6224_iteratesp_OSup,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [M7: set(A),F: fun(A,A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),M7)
=> ( ! [X2: A] :
( pp(member(A,X2,M7))
=> pp(aa(A,bool,comple7512665784863727008ratesp(A,F),X2)) )
=> pp(aa(A,bool,comple7512665784863727008ratesp(A,F),aa(set(A),A,complete_Sup_Sup(A),M7))) ) ) ) ).
% iteratesp.Sup
tff(fact_6225_reduction__pair__def,axiom,
! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A)))] :
( fun_reduction_pair(A,P)
<=> ( wf(A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))
& 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,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))) ) ) ).
% reduction_pair_def
tff(fact_6226_admissible__chfin,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [P: fun(A,bool)] :
( ! [S4: set(A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),S4)
=> pp(aa(set(A),bool,finite_finite2(A),S4)) )
=> comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P) ) ) ).
% admissible_chfin
tff(fact_6227_reduction__pairI,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( wf(A,R2)
=> ( 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,R2,S3)),R2))
=> fun_reduction_pair(A,aa(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)),R2),S3)) ) ) ).
% reduction_pairI
tff(fact_6228_wf__no__loop,axiom,
! [B: $tType,R2: set(product_prod(B,B))] :
( ( relcomp(B,B,B,R2,R2) = bot_bot(set(product_prod(B,B))) )
=> wf(B,R2) ) ).
% wf_no_loop
tff(fact_6229_union__comp__emptyL,axiom,
! [A: $tType,A5: set(product_prod(A,A)),C6: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
( ( relcomp(A,A,A,A5,C6) = bot_bot(set(product_prod(A,A))) )
=> ( ( relcomp(A,A,A,B5,C6) = bot_bot(set(product_prod(A,A))) )
=> ( relcomp(A,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))),A5),B5),C6) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyL
tff(fact_6230_union__comp__emptyR,axiom,
! [A: $tType,A5: set(product_prod(A,A)),B5: set(product_prod(A,A)),C6: set(product_prod(A,A))] :
( ( relcomp(A,A,A,A5,B5) = bot_bot(set(product_prod(A,A))) )
=> ( ( relcomp(A,A,A,A5,C6) = bot_bot(set(product_prod(A,A))) )
=> ( relcomp(A,A,A,A5,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))),B5),C6)) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyR
tff(fact_6231_relpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R2) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R2),R2) ).
% relpow.simps(2)
tff(fact_6232_admissible__disj,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [P: fun(A,bool),Q: fun(A,bool)] :
( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
=> ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Q)
=> comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_qq(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) ) ) ) ).
% admissible_disj
tff(fact_6233_max__ext__compat,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S3: 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,R2,S3)),R2))
=> 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,R2),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,S3)),insert(product_prod(set(A),set(A)),aa(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,R2))) ) ).
% max_ext_compat
tff(fact_6234_min__ext__compat,axiom,
! [A: $tType,R2: set(product_prod(A,A)),S3: 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,R2,S3)),R2))
=> 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,R2),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,S3)),insert(product_prod(set(A),set(A)),aa(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,R2))) ) ).
% min_ext_compat
tff(fact_6235_ntrancl__Suc,axiom,
! [A: $tType,N: nat,R2: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N),R2) = relcomp(A,A,A,transitive_ntrancl(A,N,R2),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))),id2(A)),R2)) ).
% ntrancl_Suc
tff(fact_6236_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_6237_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_6238_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_6239_relpow_Osimps_I1_J,axiom,
! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R2) = id2(A) ).
% relpow.simps(1)
tff(fact_6240_add_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_add(A)
=> monoid(A,plus_plus(A),zero_zero(A)) ) ).
% add.monoid_axioms
tff(fact_6241_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( monoid_mult(A)
=> monoid(A,times_times(A),one_one(A)) ) ).
% mult.monoid_axioms
tff(fact_6242_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.monoid_axioms
tff(fact_6243_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> monoid(A,inf_inf(A),top_top(A)) ) ).
% inf_top.monoid_axioms
tff(fact_6244_gcd__nat_Omonoid__axioms,axiom,
monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).
% gcd_nat.monoid_axioms
tff(fact_6245_or_Omonoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).
% or.monoid_axioms
tff(fact_6246_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).
% xor.monoid_axioms
tff(fact_6247_max__nat_Omonoid__axioms,axiom,
monoid(nat,ord_max(nat),zero_zero(nat)) ).
% max_nat.monoid_axioms
tff(fact_6248_pair__leq__def,axiom,
fun_pair_leq = aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),sup_sup(set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),fun_pair_less),id2(product_prod(nat,nat))) ).
% pair_leq_def
tff(fact_6249_cr__integer__def,axiom,
! [X3: int,Xa: code_integer] :
( pp(aa(code_integer,bool,aa(int,fun(code_integer,bool),code_cr_integer,X3),Xa))
<=> ( X3 = aa(code_integer,int,code_int_of_integer,Xa) ) ) ).
% cr_integer_def
tff(fact_6250_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_6251_integer_Opcr__cr__eq,axiom,
code_pcr_integer = code_cr_integer ).
% integer.pcr_cr_eq
tff(fact_6252_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F,cons(A,X,cons(A,Y,Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,arg_min_list(A,B,F,cons(A,Y,Zs)))),X,arg_min_list(A,B,F,cons(A,Y,Zs))) ) ).
% arg_min_list.simps(2)
tff(fact_6253_iterates_OSup,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [M7: set(A),F: fun(A,A)] :
( comple1602240252501008431_chain(A,ord_less_eq(A),M7)
=> ( ! [X2: A] :
( pp(member(A,X2,M7))
=> pp(member(A,X2,comple6359979572994053840erates(A,F))) )
=> pp(member(A,aa(set(A),A,complete_Sup_Sup(A),M7),comple6359979572994053840erates(A,F))) ) ) ) ).
% iterates.Sup
tff(fact_6254_iterates_Osimps,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A3: A,F: fun(A,A)] :
( pp(member(A,A3,comple6359979572994053840erates(A,F)))
<=> ( ? [X5: A] :
( ( A3 = aa(A,A,F,X5) )
& pp(member(A,X5,comple6359979572994053840erates(A,F))) )
| ? [M9: set(A)] :
( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M9) )
& comple1602240252501008431_chain(A,ord_less_eq(A),M9)
& ! [X5: A] :
( pp(member(A,X5,M9))
=> pp(member(A,X5,comple6359979572994053840erates(A,F))) ) ) ) ) ) ).
% iterates.simps
tff(fact_6255_iterates_Ocases,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [A3: A,F: fun(A,A)] :
( pp(member(A,A3,comple6359979572994053840erates(A,F)))
=> ( ! [X2: A] :
( ( A3 = aa(A,A,F,X2) )
=> ~ pp(member(A,X2,comple6359979572994053840erates(A,F))) )
=> ~ ! [M8: set(A)] :
( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
=> ( comple1602240252501008431_chain(A,ord_less_eq(A),M8)
=> ~ ! [X3: A] :
( pp(member(A,X3,M8))
=> pp(member(A,X3,comple6359979572994053840erates(A,F))) ) ) ) ) ) ) ).
% iterates.cases
tff(fact_6256_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: fun(A,B),Xa4: list(A),Y: A] :
( ( arg_min_list(A,B,X,Xa4) = Y )
=> ( ! [X2: A] :
( ( Xa4 = cons(A,X2,nil(A)) )
=> ( Y != X2 ) )
=> ( ! [X2: A,Y4: A,Zs2: list(A)] :
( ( Xa4 = cons(A,X2,cons(A,Y4,Zs2)) )
=> ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X2)),aa(A,B,X,arg_min_list(A,B,X,cons(A,Y4,Zs2)))),X2,arg_min_list(A,B,X,cons(A,Y4,Zs2))) ) )
=> ~ ( ( Xa4 = nil(A) )
=> ( Y != undefined(A) ) ) ) ) ) ) ).
% arg_min_list.elims
tff(fact_6257_chain__iterates,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A)] :
( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> comple1602240252501008431_chain(A,ord_less_eq(A),comple6359979572994053840erates(A,F)) ) ) ).
% chain_iterates
tff(fact_6258_iterates__le__f,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [X: A,F: fun(A,A)] :
( pp(member(A,X,comple6359979572994053840erates(A,F)))
=> ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F,X))) ) ) ) ).
% iterates_le_f
tff(fact_6259_fixp__induct,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [P: fun(A,bool),F: fun(A,A)] :
( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
=> ( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> ( pp(aa(A,bool,P,aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A)))))
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,P,aa(A,A,F,X2))) )
=> pp(aa(A,bool,P,comple115746919287870866o_fixp(A,F))) ) ) ) ) ) ).
% fixp_induct
tff(fact_6260_iterates__fixp,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A)] :
( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> pp(member(A,comple115746919287870866o_fixp(A,F),comple6359979572994053840erates(A,F))) ) ) ).
% iterates_fixp
tff(fact_6261_fixp__unfold,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A)] :
( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> ( comple115746919287870866o_fixp(A,F) = aa(A,A,F,comple115746919287870866o_fixp(A,F)) ) ) ) ).
% fixp_unfold
tff(fact_6262_fixp__lowerbound,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [F: fun(A,A),Z2: A] :
( comple7038119648293358887notone(A,A,ord_less_eq(A),ord_less_eq(A),F)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,Z2)),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),comple115746919287870866o_fixp(A,F)),Z2)) ) ) ) ).
% fixp_lowerbound
tff(fact_6263_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [X: fun(A,B),Xa4: list(A),Y: A] :
( ( arg_min_list(A,B,X,Xa4) = Y )
=> ( accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),Xa4))
=> ( ! [X2: A] :
( ( Xa4 = cons(A,X2,nil(A)) )
=> ( ( Y = X2 )
=> ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),cons(A,X2,nil(A)))) ) )
=> ( ! [X2: A,Y4: A,Zs2: list(A)] :
( ( Xa4 = cons(A,X2,cons(A,Y4,Zs2)) )
=> ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X2)),aa(A,B,X,arg_min_list(A,B,X,cons(A,Y4,Zs2)))),X2,arg_min_list(A,B,X,cons(A,Y4,Zs2))) )
=> ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),cons(A,X2,cons(A,Y4,Zs2)))) ) )
=> ~ ( ( Xa4 = nil(A) )
=> ( ( Y = undefined(A) )
=> ~ accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),nil(A))) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
tff(fact_6264_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F,linorder_sort_key(B,A,F,Xs))) ) ).
% sorted_sort_key
tff(fact_6265_sorted__sort,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_sort_key(A,A,aTP_Lamp_me(A,A),Xs)) ) ).
% sorted_sort
tff(fact_6266_sorted__sort__id,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> ( linorder_sort_key(A,A,aTP_Lamp_me(A,A),Xs) = Xs ) ) ) ).
% sorted_sort_id
tff(fact_6267_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( pp(member(A,X,set2(A,Xs)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,X,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).
% length_removeAll_less
tff(fact_6268_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
( ( ( pp(aa(set(A),bool,finite_finite2(A),A5))
& pp(aa(set(B),bool,finite_finite2(B),B5)) )
=> ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A5,B5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) )
& ( ~ ( pp(aa(set(A),bool,finite_finite2(A),A5))
& pp(aa(set(B),bool,finite_finite2(B),B5)) )
=> ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A5,B5)) = zero_zero(nat) ) ) ) ).
% card_Plus_conv_if
tff(fact_6269_length__removeAll__less__eq,axiom,
! [A: $tType,X: 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)),removeAll(A,X,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_removeAll_less_eq
tff(fact_6270_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_6271_lcm__altdef__int,axiom,
! [A3: int,B2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) = 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,abs_abs(int),A3)),aa(int,int,abs_abs(int),B2))),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)) ).
% lcm_altdef_int
tff(fact_6272_lcm_Obottom__left__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),zero_zero(A)),A3) = zero_zero(A) ) ).
% lcm.bottom_left_bottom
tff(fact_6273_lcm_Obottom__right__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),zero_zero(A)) = zero_zero(A) ) ).
% lcm.bottom_right_bottom
tff(fact_6274_lcm__neg1,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).
% lcm_neg1
tff(fact_6275_lcm__neg2,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).
% lcm_neg2
tff(fact_6276_lcm__0__iff__int,axiom,
! [M: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = zero_zero(int) )
<=> ( ( M = zero_zero(int) )
| ( N = zero_zero(int) ) ) ) ).
% lcm_0_iff_int
tff(fact_6277_lcm__neg__numeral__1,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [N: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).
% lcm_neg_numeral_1
tff(fact_6278_lcm__neg__numeral__2,axiom,
! [A: $tType] :
( ring_gcd(A)
=> ! [A3: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).
% lcm_neg_numeral_2
tff(fact_6279_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(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).
% lcm_eq_1_iff
tff(fact_6280_lcm__1__iff__int,axiom,
! [M: int,N: int] :
( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = one_one(int) )
<=> ( ( ( M = one_one(int) )
| ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) )
& ( ( N = one_one(int) )
| ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).
% lcm_1_iff_int
tff(fact_6281_lcm__ge__0__int,axiom,
! [X: int,Y: 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),gcd_lcm(int),X),Y))) ).
% lcm_ge_0_int
tff(fact_6282_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> comm_monoid(A,F,Z2) ) ).
% semilattice_neutr.axioms(2)
tff(fact_6283_lcm__neg1__int,axiom,
! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y) ).
% lcm_neg1_int
tff(fact_6284_lcm__neg2__int,axiom,
! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y) ).
% lcm_neg2_int
tff(fact_6285_lcm__pos__int,axiom,
! [M: int,N: int] :
( ( M != zero_zero(int) )
=> ( ( N != zero_zero(int) )
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N))) ) ) ).
% lcm_pos_int
tff(fact_6286_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_6287_zero__eq__lcm__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) )
<=> ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
% zero_eq_lcm_iff
tff(fact_6288_lcm__eq__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: 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) ) ) ) ) ).
% lcm_eq_0_iff
tff(fact_6289_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_6290_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_6291_lcm__mult__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_6292_lcm__mult__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_6293_lcm__div__unit1,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_6294_lcm__div__unit2,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_6295_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> comm_monoid(A,sup_sup(A),bot_bot(A)) ) ).
% sup_bot.comm_monoid_axioms
tff(fact_6296_inf__top_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bounde4346867609351753570nf_top(A)
=> comm_monoid(A,inf_inf(A),top_top(A)) ) ).
% inf_top.comm_monoid_axioms
tff(fact_6297_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> comm_monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).
% or.comm_monoid_axioms
tff(fact_6298_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).
% gcd_nat.comm_monoid_axioms
tff(fact_6299_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( bit_se359711467146920520ations(A)
=> comm_monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).
% xor.comm_monoid_axioms
tff(fact_6300_lcm__cases__int,axiom,
! [X: int,Y: int,P: fun(int,bool)] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y))) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y)))) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y))) ) )
=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y)))) ) )
=> pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y))) ) ) ) ) ).
% lcm_cases_int
tff(fact_6301_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid(nat,ord_max(nat),zero_zero(nat)) ).
% max_nat.comm_monoid_axioms
tff(fact_6302_lcm__unique__int,axiom,
! [D2: int,A3: int,B2: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D2))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D2))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),D2))
& ! [E3: int] :
( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),E3))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),E3)) )
=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D2),E3)) ) )
<=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) ) ) ).
% lcm_unique_int
tff(fact_6303_prod__gcd__lcm__int,axiom,
! [M: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),M)),aa(int,int,abs_abs(int),N)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N)) ).
% prod_gcd_lcm_int
tff(fact_6304_Lcm__int__set__eq__fold,axiom,
! [Xs: list(int)] : gcd_Lcm(int,set2(int,Xs)) = fold(int,int,gcd_lcm(int),Xs,one_one(int)) ).
% Lcm_int_set_eq_fold
tff(fact_6305_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = finite_fold(A,A,gcd_lcm(A),one_one(A),A5) ) )
& ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = zero_zero(A) ) ) ) ) ).
% Lcm_fin.eq_fold
tff(fact_6306_lcm__0__iff__nat,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
| ( N = zero_zero(nat) ) ) ) ).
% lcm_0_iff_nat
tff(fact_6307_lcm__1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
& ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
% lcm_1_iff_nat
tff(fact_6308_lcm__int__int__eq,axiom,
! [M: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),M)),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),gcd_lcm(nat),M),N)) ).
% lcm_int_int_eq
tff(fact_6309_Lcm__UNIV,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,top_top(set(A))) = zero_zero(A) ) ) ).
% Lcm_UNIV
tff(fact_6310_Lcm__empty,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).
% Lcm_empty
tff(fact_6311_Lcm__1__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ( gcd_Lcm(A,A5) = one_one(A) )
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X5),one_one(A))) ) ) ) ).
% Lcm_1_iff
tff(fact_6312_Lcm__fin_Oinfinite,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = zero_zero(A) ) ) ) ).
% Lcm_fin.infinite
tff(fact_6313_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_6314_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),one_one(A)))
<=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = one_one(A) ) ) ) ).
% is_unit_Lcm_fin_iff
tff(fact_6315_lcm__nat__abs__right__eq,axiom,
! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).
% lcm_nat_abs_right_eq
tff(fact_6316_lcm__nat__abs__left__eq,axiom,
! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).
% lcm_nat_abs_left_eq
tff(fact_6317_prod__gcd__lcm__nat,axiom,
! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N)) ).
% prod_gcd_lcm_nat
tff(fact_6318_lcm__pos__nat,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))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N))) ) ) ).
% lcm_pos_nat
tff(fact_6319_Lcm__no__multiple,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ! [M5: A] :
( ( M5 != zero_zero(A) )
=> ? [X3: A] :
( pp(member(A,X3,A5))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),M5)) ) )
=> ( gcd_Lcm(A,A5) = zero_zero(A) ) ) ) ).
% Lcm_no_multiple
tff(fact_6320_Lcm__0__iff_H,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ( gcd_Lcm(A,A5) = zero_zero(A) )
<=> ~ ? [L4: A] :
( ( L4 != zero_zero(A) )
& ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X5),L4)) ) ) ) ) ).
% Lcm_0_iff'
tff(fact_6321_Lcm__eq__0__I,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( pp(member(A,zero_zero(A),A5))
=> ( gcd_Lcm(A,A5) = zero_zero(A) ) ) ) ).
% Lcm_eq_0_I
tff(fact_6322_Lcm__0__iff,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( gcd_Lcm(A,A5) = zero_zero(A) )
<=> pp(member(A,zero_zero(A),A5)) ) ) ) ).
% Lcm_0_iff
tff(fact_6323_Lcm__int__greater__eq__0,axiom,
! [K5: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Lcm(int,K5))) ).
% Lcm_int_greater_eq_0
tff(fact_6324_Lcm__fin__0__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = zero_zero(A) )
<=> pp(member(A,zero_zero(A),A5)) ) ) ) ).
% Lcm_fin_0_iff
tff(fact_6325_lcm__nat__def,axiom,
! [X: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y)) ).
% lcm_nat_def
tff(fact_6326_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] :
( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = one_one(A) )
<=> ( ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X5),one_one(A))) )
& pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).
% Lcm_fin_1_iff
tff(fact_6327_Lcm__no__units,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] : gcd_Lcm(A,A5) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),collect(A,aTP_Lamp_qr(A,bool)))) ) ).
% Lcm_no_units
tff(fact_6328_lcm__int__def,axiom,
! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y)))) ).
% lcm_int_def
tff(fact_6329_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Xs: list(A)] : gcd_Lcm(A,set2(A,Xs)) = fold(A,A,gcd_lcm(A),Xs,one_one(A)) ) ).
% Lcm_set_eq_fold
tff(fact_6330_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),set2(A,Xs)) = fold(A,A,gcd_lcm(A),Xs,one_one(A)) ) ).
% Lcm_fin.set_eq_fold
tff(fact_6331_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_6332_Lcm__int__def,axiom,
! [K5: set(int)] : gcd_Lcm(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int)),K5))) ).
% Lcm_int_def
tff(fact_6333_Lcm__int__eq,axiom,
! [N6: set(nat)] : gcd_Lcm(int,image(nat,int,semiring_1_of_nat(int),N6)) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,N6)) ).
% Lcm_int_eq
tff(fact_6334_Lcm__eq__0__I__nat,axiom,
! [A5: set(nat)] :
( pp(member(nat,zero_zero(nat),A5))
=> ( gcd_Lcm(nat,A5) = zero_zero(nat) ) ) ).
% Lcm_eq_0_I_nat
tff(fact_6335_Lcm__0__iff__nat,axiom,
! [A5: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),A5))
=> ( ( gcd_Lcm(nat,A5) = zero_zero(nat) )
<=> pp(member(nat,zero_zero(nat),A5)) ) ) ).
% Lcm_0_iff_nat
tff(fact_6336_Lcm__nat__infinite,axiom,
! [M7: set(nat)] :
( ~ pp(aa(set(nat),bool,finite_finite2(nat),M7))
=> ( gcd_Lcm(nat,M7) = zero_zero(nat) ) ) ).
% Lcm_nat_infinite
tff(fact_6337_Lcm__nat__empty,axiom,
gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).
% Lcm_nat_empty
tff(fact_6338_Lcm__nat__set__eq__fold,axiom,
! [Xs: list(nat)] : gcd_Lcm(nat,set2(nat,Xs)) = fold(nat,nat,gcd_lcm(nat),Xs,one_one(nat)) ).
% Lcm_nat_set_eq_fold
tff(fact_6339_Lcm__eq__Max__nat,axiom,
! [M7: set(nat)] :
( pp(aa(set(nat),bool,finite_finite2(nat),M7))
=> ( ( M7 != bot_bot(set(nat)) )
=> ( ~ pp(member(nat,zero_zero(nat),M7))
=> ( ! [M5: nat,N3: nat] :
( pp(member(nat,M5,M7))
=> ( pp(member(nat,N3,M7))
=> pp(member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M5),N3),M7)) ) )
=> ( gcd_Lcm(nat,M7) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),M7) ) ) ) ) ) ).
% Lcm_eq_Max_nat
tff(fact_6340_Lcm__nat__def,axiom,
! [M7: set(nat)] :
( ( pp(aa(set(nat),bool,finite_finite2(nat),M7))
=> ( gcd_Lcm(nat,M7) = aa(set(nat),nat,lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat)),M7) ) )
& ( ~ pp(aa(set(nat),bool,finite_finite2(nat),M7))
=> ( gcd_Lcm(nat,M7) = zero_zero(nat) ) ) ) ).
% Lcm_nat_def
tff(fact_6341_Lcm__coprime_H,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ( aa(set(A),nat,finite_card(A),A5) != zero_zero(nat) )
=> ( ! [A6: A,B4: A] :
( pp(member(A,A6,A5))
=> ( pp(member(A,B4,A5))
=> ( ( A6 != B4 )
=> algebr8660921524188924756oprime(A,A6,B4) ) ) )
=> ( gcd_Lcm(A,A5) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_qs(A,A)),A5)) ) ) ) ) ).
% Lcm_coprime'
tff(fact_6342_normalize__idem,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,normal6383669964737779283malize(A),A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% normalize_idem
tff(fact_6343_lcm_Onormalize__bottom,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).
% lcm.normalize_bottom
tff(fact_6344_normalize__eq__0__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% normalize_eq_0_iff
tff(fact_6345_normalize__0,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).
% normalize_0
tff(fact_6346_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,normal6383669964737779283malize(A),B2))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% normalize_mult_normalize_right
tff(fact_6347_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% normalize_mult_normalize_left
tff(fact_6348_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_6349_normalize__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).
% normalize_1
tff(fact_6350_dvd__normalize__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% dvd_normalize_iff
tff(fact_6351_normalize__dvd__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% normalize_dvd_iff
tff(fact_6352_coprime__normalize__left__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,aa(A,A,normal6383669964737779283malize(A),A3),B2)
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprime_normalize_left_iff
tff(fact_6353_coprime__normalize__right__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,aa(A,A,normal6383669964737779283malize(A),B2))
<=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).
% coprime_normalize_right_iff
tff(fact_6354_gcd_Otop__left__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),zero_zero(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% gcd.top_left_normalize
tff(fact_6355_gcd_Otop__right__normalize,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),zero_zero(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% gcd.top_right_normalize
tff(fact_6356_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_6357_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_6358_normalize__mult__unit__right,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_6359_normalize__mult__unit__left,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),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_6360_lcm__mult__gcd,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).
% lcm_mult_gcd
tff(fact_6361_gcd__mult__lcm,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).
% gcd_mult_lcm
tff(fact_6362_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(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)) ) ).
% lcm_mult_distrib'
tff(fact_6363_lcm__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(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)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A3)),C2)) ) ).
% lcm_mult_right
tff(fact_6364_lcm__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(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)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).
% lcm_mult_left
tff(fact_6365_normalize__nat__def,axiom,
normal6383669964737779283malize(nat) = id(nat) ).
% normalize_nat_def
tff(fact_6366_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(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)) ) ).
% gcd_mult_distrib'
tff(fact_6367_gcd__mult__right,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(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)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A3)),C2)) ) ).
% gcd_mult_right
tff(fact_6368_gcd__mult__left,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(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)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).
% gcd_mult_left
tff(fact_6369_dvd__normalize__div,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ) ).
% dvd_normalize_div
tff(fact_6370_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
<=> ( A3 = one_one(A) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
tff(fact_6371_is__unit__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).
% is_unit_normalize
tff(fact_6372_normalize__1__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).
% normalize_1_iff
tff(fact_6373_associated__unit,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).
% associated_unit
tff(fact_6374_normalize__int__def,axiom,
normal6383669964737779283malize(int) = abs_abs(int) ).
% normalize_int_def
tff(fact_6375_semilattice__neutr__set_OF_Ocong,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] : lattic5214292709420241887eutr_F(A,F,Z2) = lattic5214292709420241887eutr_F(A,F,Z2) ).
% semilattice_neutr_set.F.cong
tff(fact_6376_associated__iff__dvd,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).
% associated_iff_dvd
tff(fact_6377_associated__eqI,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
=> ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
=> ( A3 = B2 ) ) ) ) ) ) ).
% associated_eqI
tff(fact_6378_associatedD2,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).
% associatedD2
tff(fact_6379_associatedD1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).
% associatedD1
tff(fact_6380_associatedI,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ) ).
% associatedI
tff(fact_6381_normalize__mult,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(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),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).
% normalize_mult
tff(fact_6382_coprime__crossproduct,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,D2: A,B2: A,C2: A] :
( algebr8660921524188924756oprime(A,A3,D2)
=> ( algebr8660921524188924756oprime(A,B2,C2)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),B2)),aa(A,A,normal6383669964737779283malize(A),D2)) )
<=> ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
& ( aa(A,A,normal6383669964737779283malize(A),C2) = aa(A,A,normal6383669964737779283malize(A),D2) ) ) ) ) ) ) ).
% coprime_crossproduct
tff(fact_6383_lcm__gcd__prod,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).
% lcm_gcd_prod
tff(fact_6384_Gcd__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [C2: A,A5: set(A)] : gcd_Gcd(A,image(A,A,aa(A,fun(A,A),times_times(A),C2),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,A5))) ) ).
% Gcd_mult
tff(fact_6385_lcm__coprime,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( algebr8660921524188924756oprime(A,A3,B2)
=> ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).
% lcm_coprime
tff(fact_6386_Lcm__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A),C2: A] :
( ( A5 != bot_bot(set(A)) )
=> ( gcd_Lcm(A,image(A,A,aa(A,fun(A,A),times_times(A),C2),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,A5))) ) ) ) ).
% Lcm_mult
tff(fact_6387_lcm__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(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)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).
% lcm_gcd
tff(fact_6388_Lcm__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A),B2: A] :
( ( A5 != bot_bot(set(A)) )
=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),image(A,A,aa(A,fun(A,A),times_times(A),B2),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5))) ) ) ) ).
% Lcm_fin_mult
tff(fact_6389_Gcd__fin__mult,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A),B2: A] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),image(A,A,aa(A,fun(A,A),times_times(A),B2),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5))) ) ) ) ).
% Gcd_fin_mult
tff(fact_6390_gcd__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> ( ( B2 != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(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)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ) ) ) ).
% gcd_lcm
tff(fact_6391_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ).
% semilattice_neutr_set.insert_remove
tff(fact_6392_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,X,bot_bot(set(A)))))) ) ) ) ) ).
% semilattice_neutr_set.remove
tff(fact_6393_semilattice__neutr__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> lattic5652469242046573047tr_set(A,F,Z2) ) ).
% semilattice_neutr_set.intro
tff(fact_6394_semilattice__neutr__set_Oaxioms,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_set.axioms
tff(fact_6395_semilattice__neutr__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
<=> semilattice_neutr(A,F,Z2) ) ).
% semilattice_neutr_set_def
tff(fact_6396_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),bot_bot(set(A))) = Z2 ) ) ).
% semilattice_neutr_set.empty
tff(fact_6397_semilattice__neutr__set_Oin__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,X,A5))
=> ( aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)) = aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5) ) ) ) ) ).
% semilattice_neutr_set.in_idem
tff(fact_6398_semilattice__neutr__set_Oinfinite,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5) = Z2 ) ) ) ).
% semilattice_neutr_set.infinite
tff(fact_6399_semilattice__neutr__set_Oeq__fold,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5) = finite_fold(A,A,F,Z2,A5) ) ) ).
% semilattice_neutr_set.eq_fold
tff(fact_6400_semilattice__neutr__set_Osubset,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A),B5: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),B5)),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)) = aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5) ) ) ) ) ).
% semilattice_neutr_set.subset
tff(fact_6401_semilattice__neutr__set_Oinsert,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A),X: A] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),insert(A,X,A5)) = aa(A,A,aa(A,fun(A,A),F,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)) ) ) ) ).
% semilattice_neutr_set.insert
tff(fact_6402_semilattice__neutr__set_Ounion,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A),B5: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> ( aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),B5)) ) ) ) ) ).
% semilattice_neutr_set.union
tff(fact_6403_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(A)] :
( lattic5652469242046573047tr_set(A,F,Z2)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ( A5 != bot_bot(set(A)) )
=> ( ! [X2: A,Y4: A] : pp(member(A,aa(A,A,aa(A,fun(A,A),F,X2),Y4),insert(A,X2,insert(A,Y4,bot_bot(set(A))))))
=> pp(member(A,aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5),A5)) ) ) ) ) ).
% semilattice_neutr_set.closed
tff(fact_6404_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_6405_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_6406_semilattice__order__neutr__set_Osubset__imp,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),B5: set(A)] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B5))
=> ( pp(aa(set(A),bool,finite_finite2(A),B5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),B5)),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5))) ) ) ) ).
% semilattice_order_neutr_set.subset_imp
tff(fact_6407_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_6408_semilattice__order__neutr__set_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> lattic5652469242046573047tr_set(A,F,Z2) ) ).
% semilattice_order_neutr_set.axioms(2)
tff(fact_6409_semilattice__order__neutr__set_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> semila1105856199041335345_order(A,F,Z2,Less_eq,Less) ) ).
% semilattice_order_neutr_set.axioms(1)
tff(fact_6410_semilattice__order__neutr__set_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),X: A] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)))
=> ! [A9: A] :
( pp(member(A,A9,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),A9)) ) ) ) ) ).
% semilattice_order_neutr_set.boundedE
tff(fact_6411_semilattice__order__neutr__set_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),X: A] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( ! [A6: A] :
( pp(member(A,A6,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),A6)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5))) ) ) ) ).
% semilattice_order_neutr_set.boundedI
tff(fact_6412_semilattice__order__neutr__set_OcoboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),A3: A] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(member(A,A3,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)),A3)) ) ) ) ).
% semilattice_order_neutr_set.coboundedI
tff(fact_6413_semilattice__order__neutr__set_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A5: set(A),X: A] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
=> ( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),aa(set(A),A,lattic5214292709420241887eutr_F(A,F,Z2),A5)))
<=> ! [X5: A] :
( pp(member(A,X5,A5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,X),X5)) ) ) ) ) ).
% semilattice_order_neutr_set.bounded_iff
tff(fact_6414_semilattice__order__neutr__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less)
<=> ( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
& lattic5652469242046573047tr_set(A,F,Z2) ) ) ).
% semilattice_order_neutr_set_def
tff(fact_6415_semilattice__order__neutr__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> ( lattic5652469242046573047tr_set(A,F,Z2)
=> lattic3600114342068043075tr_set(A,F,Z2,Less_eq,Less) ) ) ).
% semilattice_order_neutr_set.intro
tff(fact_6416_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_6417_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)),aa(A,A,unit_f5069060285200089521factor(A),A3)) ) ).
% normalize_div
tff(fact_6418_normalize__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,unit_f5069060285200089521factor(A),A3)) = one_one(A) ) ) ) ).
% normalize_unit_factor
tff(fact_6419_unit__factor__simps_I1_J,axiom,
aa(nat,nat,unit_f5069060285200089521factor(nat),zero_zero(nat)) = zero_zero(nat) ).
% unit_factor_simps(1)
tff(fact_6420_unit__factor__idem,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,unit_f5069060285200089521factor(A),aa(A,A,unit_f5069060285200089521factor(A),A3)) = aa(A,A,unit_f5069060285200089521factor(A),A3) ) ).
% unit_factor_idem
tff(fact_6421_unit__factor__simps_I2_J,axiom,
! [N: nat] : aa(nat,nat,unit_f5069060285200089521factor(nat),aa(nat,nat,suc,N)) = one_one(nat) ).
% unit_factor_simps(2)
tff(fact_6422_unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,unit_f5069060285200089521factor(A),A3) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% unit_factor_eq_0_iff
tff(fact_6423_unit__factor__0,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ( aa(A,A,unit_f5069060285200089521factor(A),zero_zero(A)) = zero_zero(A) ) ) ).
% unit_factor_0
tff(fact_6424_unit__factor__1,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ( aa(A,A,unit_f5069060285200089521factor(A),one_one(A)) = one_one(A) ) ) ).
% unit_factor_1
tff(fact_6425_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,unit_f5069060285200089521factor(A),A3)) = A3 ) ).
% normalize_mult_unit_factor
tff(fact_6426_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,unit_f5069060285200089521factor(A),A3)),aa(A,A,normal6383669964737779283malize(A),A3)) = A3 ) ).
% unit_factor_mult_normalize
tff(fact_6427_div__unit__factor,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,unit_f5069060285200089521factor(A),A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).
% div_unit_factor
tff(fact_6428_div__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,normal6383669964737779283malize(A),A3)) = aa(A,A,unit_f5069060285200089521factor(A),A3) ) ).
% div_normalize
tff(fact_6429_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)),aa(A,A,unit_f5069060285200089521factor(A),A3)) = zero_zero(A) )
<=> ( A3 = zero_zero(A) ) ) ) ).
% inv_unit_factor_eq_0_iff
tff(fact_6430_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( aa(A,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),aa(A,A,unit_f5069060285200089521factor(A),B2)),A3) ) ) ) ).
% unit_factor_mult_unit_right
tff(fact_6431_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( aa(A,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),aa(A,A,unit_f5069060285200089521factor(A),B2)) ) ) ) ).
% unit_factor_mult_unit_left
tff(fact_6432_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)),aa(A,A,unit_f5069060285200089521factor(A),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,unit_f5069060285200089521factor(A),B2)) ) ).
% mult_one_div_unit_factor
tff(fact_6433_unit__factor__lcm,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( ( A3 = zero_zero(A) )
| ( B2 = zero_zero(A) ) )
=> ( aa(A,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) ) )
=> ( aa(A,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_6434_unit__factor__normalize,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> ( aa(A,A,unit_f5069060285200089521factor(A),aa(A,A,normal6383669964737779283malize(A),A3)) = one_one(A) ) ) ) ).
% unit_factor_normalize
tff(fact_6435_normalize__unit__factor__eqI,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
=> ( ( aa(A,A,unit_f5069060285200089521factor(A),A3) = aa(A,A,unit_f5069060285200089521factor(A),B2) )
=> ( A3 = B2 ) ) ) ) ).
% normalize_unit_factor_eqI
tff(fact_6436_lcm__mult__distrib,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),aa(A,A,unit_f5069060285200089521factor(A),K)) ) ).
% lcm_mult_distrib
tff(fact_6437_mult__lcm__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(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),gcd_lcm(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(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))),aa(A,A,unit_f5069060285200089521factor(A),C2)) ) ).
% mult_lcm_right
tff(fact_6438_mult__lcm__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,unit_f5069060285200089521factor(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(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))) ) ).
% mult_lcm_left
tff(fact_6439_unit__factor__nat__def,axiom,
! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,nat,unit_f5069060285200089521factor(nat),N) = zero_zero(nat) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,nat,unit_f5069060285200089521factor(nat),N) = one_one(nat) ) ) ) ).
% unit_factor_nat_def
tff(fact_6440_dvd__unit__factor__div,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
=> ( aa(A,A,unit_f5069060285200089521factor(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,unit_f5069060285200089521factor(A),A3)),aa(A,A,unit_f5069060285200089521factor(A),B2)) ) ) ) ).
% dvd_unit_factor_div
tff(fact_6441_unit__factor__self,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,unit_f5069060285200089521factor(A),A3)),A3)) ) ).
% unit_factor_self
tff(fact_6442_unit__factor__int__def,axiom,
unit_f5069060285200089521factor(int) = sgn_sgn(int) ).
% unit_factor_int_def
tff(fact_6443_unit__factor__mult,axiom,
! [A: $tType] :
( normal6328177297339901930cative(A)
=> ! [A3: A,B2: A] : aa(A,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),aa(A,A,unit_f5069060285200089521factor(A),A3)),aa(A,A,unit_f5069060285200089521factor(A),B2)) ) ).
% unit_factor_mult
tff(fact_6444_unit__factor__dvd,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A,B2: A] :
( ( A3 != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,unit_f5069060285200089521factor(A),A3)),B2)) ) ) ).
% unit_factor_dvd
tff(fact_6445_is__unit__unit__factor,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
=> ( aa(A,A,unit_f5069060285200089521factor(A),A3) = A3 ) ) ) ).
% is_unit_unit_factor
tff(fact_6446_gcd__mult__distrib,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),aa(A,A,unit_f5069060285200089521factor(A),K)) ) ).
% gcd_mult_distrib
tff(fact_6447_mult__gcd__right,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(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),gcd_gcd(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(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))),aa(A,A,unit_f5069060285200089521factor(A),C2)) ) ).
% mult_gcd_right
tff(fact_6448_mult__gcd__left,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,unit_f5069060285200089521factor(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(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))) ) ).
% mult_gcd_left
tff(fact_6449_unit__factor__is__unit,axiom,
! [A: $tType] :
( semido2269285787275462019factor(A)
=> ! [A3: A] :
( ( A3 != zero_zero(A) )
=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,unit_f5069060285200089521factor(A),A3)),one_one(A))) ) ) ).
% unit_factor_is_unit
tff(fact_6450_unit__factor__gcd,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A3: A,B2: A] :
( ( ( ( A3 = zero_zero(A) )
& ( B2 = zero_zero(A) ) )
=> ( aa(A,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) ) )
=> ( aa(A,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_6451_coprime__crossproduct_H,axiom,
! [A: $tType] :
( semiri6843258321239162965malize(A)
=> ! [B2: A,D2: A,A3: A,C2: A] :
( ( B2 != zero_zero(A) )
=> ( ( aa(A,A,unit_f5069060285200089521factor(A),B2) = aa(A,A,unit_f5069060285200089521factor(A),D2) )
=> ( algebr8660921524188924756oprime(A,A3,B2)
=> ( algebr8660921524188924756oprime(A,C2,D2)
=> ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
<=> ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
tff(fact_6452_unit__factor__Lcm,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ( ( gcd_Lcm(A,A5) = zero_zero(A) )
=> ( aa(A,A,unit_f5069060285200089521factor(A),gcd_Lcm(A,A5)) = zero_zero(A) ) )
& ( ( gcd_Lcm(A,A5) != zero_zero(A) )
=> ( aa(A,A,unit_f5069060285200089521factor(A),gcd_Lcm(A,A5)) = one_one(A) ) ) ) ) ).
% unit_factor_Lcm
tff(fact_6453_unit__factor__Gcd,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A5: set(A)] :
( ( ( gcd_Gcd(A,A5) = zero_zero(A) )
=> ( aa(A,A,unit_f5069060285200089521factor(A),gcd_Gcd(A,A5)) = zero_zero(A) ) )
& ( ( gcd_Gcd(A,A5) != zero_zero(A) )
=> ( aa(A,A,unit_f5069060285200089521factor(A),gcd_Gcd(A,A5)) = one_one(A) ) ) ) ) ).
% unit_factor_Gcd
tff(fact_6454_normalize__idem__imp__unit__factor__eq,axiom,
! [A: $tType] :
( normal8620421768224518004emidom(A)
=> ! [A3: A] :
( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
=> ( aa(A,A,unit_f5069060285200089521factor(A),A3) = aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ) ) ).
% normalize_idem_imp_unit_factor_eq
tff(fact_6455_unit__factor__Lcm__fin,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] : aa(A,A,unit_f5069060285200089521factor(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)) = aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),zero_zero(A)))) ) ).
% unit_factor_Lcm_fin
tff(fact_6456_unit__factor__Gcd__fin,axiom,
! [A: $tType] :
( semiring_gcd(A)
=> ! [A5: set(A)] : aa(A,A,unit_f5069060285200089521factor(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)) = aa(bool,A,zero_neq_one_of_bool(A),fNot(aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)),zero_zero(A)))) ) ).
% unit_factor_Gcd_fin
tff(fact_6457_times__num__def,axiom,
! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),N) = aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,M)),aa(num,nat,nat_of_num,N))) ).
% times_num_def
tff(fact_6458_nat__of__num__pos,axiom,
! [X: num] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,nat_of_num,X))) ).
% nat_of_num_pos
tff(fact_6459_nat__of__num_Osimps_I2_J,axiom,
! [X: num] : aa(num,nat,nat_of_num,bit0(X)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X)) ).
% nat_of_num.simps(2)
tff(fact_6460_is__num__normalize_I5_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A] :
( neg_numeral_is_num(A,X)
=> neg_numeral_is_num(A,aa(A,A,uminus_uminus(A),X)) ) ) ).
% is_num_normalize(5)
tff(fact_6461_is__num__numeral,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [K: num] : neg_numeral_is_num(A,aa(num,A,numeral_numeral(A),K)) ) ).
% is_num_numeral
tff(fact_6462_nat__of__num__neq__0,axiom,
! [X: num] : aa(num,nat,nat_of_num,X) != zero_zero(nat) ).
% nat_of_num_neq_0
tff(fact_6463_num__eq__iff,axiom,
! [X: num,Y: num] :
( ( X = Y )
<=> ( aa(num,nat,nat_of_num,X) = aa(num,nat,nat_of_num,Y) ) ) ).
% num_eq_iff
tff(fact_6464_nat__of__num__numeral,axiom,
nat_of_num = numeral_numeral(nat) ).
% nat_of_num_numeral
tff(fact_6465_nat__of__num__inverse,axiom,
! [X: num] : aa(nat,num,num_of_nat,aa(num,nat,nat_of_num,X)) = X ).
% nat_of_num_inverse
tff(fact_6466_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_6467_is__num__add__left__commute,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A,Y: A,Z2: A] :
( neg_numeral_is_num(A,X)
=> ( neg_numeral_is_num(A,Y)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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),X),Z2)) ) ) ) ) ).
% is_num_add_left_commute
tff(fact_6468_is__num__add__commute,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A,Y: A] :
( neg_numeral_is_num(A,X)
=> ( neg_numeral_is_num(A,Y)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),X) ) ) ) ) ).
% is_num_add_commute
tff(fact_6469_is__num__normalize_I6_J,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [X: A,Y: A] :
( neg_numeral_is_num(A,X)
=> ( neg_numeral_is_num(A,Y)
=> neg_numeral_is_num(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ) ) ).
% is_num_normalize(6)
tff(fact_6470_nat__of__num__code_I1_J,axiom,
aa(num,nat,nat_of_num,one2) = one_one(nat) ).
% nat_of_num_code(1)
tff(fact_6471_nat__of__num__add,axiom,
! [X: num,Y: num] : aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,Y)) ).
% nat_of_num_add
tff(fact_6472_nat__of__num__mult,axiom,
! [X: num,Y: num] : aa(num,nat,nat_of_num,aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,Y)) ).
% nat_of_num_mult
tff(fact_6473_nat__of__num__inc,axiom,
! [X: num] : aa(num,nat,nat_of_num,inc(X)) = aa(nat,nat,suc,aa(num,nat,nat_of_num,X)) ).
% nat_of_num_inc
tff(fact_6474_less__num__def,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,nat_of_num,M)),aa(num,nat,nat_of_num,N))) ) ).
% less_num_def
tff(fact_6475_less__eq__num__def,axiom,
! [M: num,N: num] :
( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,nat_of_num,M)),aa(num,nat,nat_of_num,N))) ) ).
% less_eq_num_def
tff(fact_6476_nat__of__num__code_I2_J,axiom,
! [N: num] : aa(num,nat,nat_of_num,bit0(N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,N)),aa(num,nat,nat_of_num,N)) ).
% nat_of_num_code(2)
tff(fact_6477_nat__of__num__sqr,axiom,
! [X: num] : aa(num,nat,nat_of_num,sqr(X)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X)) ).
% nat_of_num_sqr
tff(fact_6478_nat__of__num_Osimps_I1_J,axiom,
aa(num,nat,nat_of_num,one2) = aa(nat,nat,suc,zero_zero(nat)) ).
% nat_of_num.simps(1)
tff(fact_6479_nat__of__num_Osimps_I3_J,axiom,
! [X: num] : aa(num,nat,nat_of_num,aa(num,num,bit1,X)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,X)),aa(num,nat,nat_of_num,X))) ).
% nat_of_num.simps(3)
tff(fact_6480_num__of__nat__inverse,axiom,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
=> ( aa(num,nat,nat_of_num,aa(nat,num,num_of_nat,N)) = N ) ) ).
% num_of_nat_inverse
tff(fact_6481_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),aa(num,nat,nat_of_num,M)),aa(num,nat,nat_of_num,N))) ).
% plus_num_def
tff(fact_6482_nat__of__num__code_I3_J,axiom,
! [N: num] : aa(num,nat,nat_of_num,aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,nat_of_num,N)),aa(num,nat,nat_of_num,N))) ).
% nat_of_num_code(3)
tff(fact_6483_is__num_Ocases,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
=> ( ( A3 != one_one(A) )
=> ( ! [X2: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X2) )
=> ~ neg_numeral_is_num(A,X2) )
=> ~ ! [X2: A,Y4: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),Y4) )
=> ( neg_numeral_is_num(A,X2)
=> ~ neg_numeral_is_num(A,Y4) ) ) ) ) ) ) ).
% is_num.cases
tff(fact_6484_is__num_Osimps,axiom,
! [A: $tType] :
( neg_numeral(A)
=> ! [A3: A] :
( neg_numeral_is_num(A,A3)
<=> ( ( A3 = one_one(A) )
| ? [X5: A] :
( ( A3 = aa(A,A,uminus_uminus(A),X5) )
& neg_numeral_is_num(A,X5) )
| ? [X5: A,Y5: A] :
( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Y5) )
& neg_numeral_is_num(A,X5)
& neg_numeral_is_num(A,Y5) ) ) ) ) ).
% is_num.simps
tff(fact_6485_arg__max__nat__lemma,axiom,
! [A: $tType,P: fun(A,bool),K: A,F: fun(A,nat),B2: nat] :
( pp(aa(A,bool,P,K))
=> ( ! [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)),B2)) )
=> ( pp(aa(A,bool,P,lattices_ord_arg_max(A,nat,F,P)))
& ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,Y3)),aa(A,nat,F,lattices_ord_arg_max(A,nat,F,P)))) ) ) ) ) ).
% arg_max_nat_lemma
tff(fact_6486_arg__max__nat__le,axiom,
! [A: $tType,P: fun(A,bool),X: A,F: fun(A,nat),B2: nat] :
( pp(aa(A,bool,P,X))
=> ( ! [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)),B2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,lattices_ord_arg_max(A,nat,F,P)))) ) ) ).
% arg_max_nat_le
tff(fact_6487_arg__max__natI,axiom,
! [A: $tType,P: fun(A,bool),K: A,F: fun(A,nat),B2: nat] :
( pp(aa(A,bool,P,K))
=> ( ! [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)),B2)) )
=> pp(aa(A,bool,P,lattices_ord_arg_max(A,nat,F,P))) ) ) ).
% arg_max_natI
tff(fact_6488_arg__max__equality,axiom,
! [A: $tType,C: $tType] :
( order(A)
=> ! [P: fun(C,bool),K: C,F: fun(C,A)] :
( pp(aa(C,bool,P,K))
=> ( ! [X2: C] :
( pp(aa(C,bool,P,X2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X2)),aa(C,A,F,K))) )
=> ( aa(C,A,F,lattices_ord_arg_max(C,A,F,P)) = aa(C,A,F,K) ) ) ) ) ).
% arg_max_equality
tff(fact_6489_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [P: fun(A,bool),X: A,F: fun(A,B),Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X)),aa(A,B,F,Y4))) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y3))) )
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(A,bool,Q,lattices_ord_arg_max(A,B,F,P))) ) ) ) ) ).
% arg_maxI
tff(fact_6490_arg__max__on__def,axiom,
! [A: $tType,B: $tType] :
( ord(A)
=> ! [F: fun(B,A),S3: set(B)] : lattic1883929316492267755max_on(B,A,F,S3) = lattices_ord_arg_max(B,A,F,aTP_Lamp_ol(set(B),fun(B,bool),S3)) ) ).
% arg_max_on_def
tff(fact_6491_arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ord(A)
=> ! [F: fun(B,A),P: fun(B,bool)] : lattices_ord_arg_max(B,A,F,P) = fChoice(B,lattic501386751176901750rg_max(B,A,F,P)) ) ).
% arg_max_def
tff(fact_6492_is__arg__max__linorder,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [F: fun(A,B),P: fun(A,bool),X: A] :
( pp(aa(A,bool,lattic501386751176901750rg_max(A,B,F,P),X))
<=> ( pp(aa(A,bool,P,X))
& ! [Y5: A] :
( pp(aa(A,bool,P,Y5))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,Y5)),aa(A,B,F,X))) ) ) ) ) ).
% is_arg_max_linorder
tff(fact_6493_is__arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ord(A)
=> ! [F: fun(B,A),P: fun(B,bool),X: B] :
( pp(aa(B,bool,lattic501386751176901750rg_max(B,A,F,P),X))
<=> ( pp(aa(B,bool,P,X))
& ~ ? [Y5: B] :
( pp(aa(B,bool,P,Y5))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X)),aa(B,A,F,Y5))) ) ) ) ) ).
% is_arg_max_def
tff(fact_6494_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_6495_integer__of__natural_Orep__eq,axiom,
! [X: code_natural] : aa(code_integer,int,code_int_of_integer,aa(code_natural,code_integer,code_i5400310926305786745atural,X)) = aa(nat,int,semiring_1_of_nat(int),aa(code_natural,nat,code_nat_of_natural,X)) ).
% integer_of_natural.rep_eq
tff(fact_6496_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_6497_plus__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% plus_natural.rep_eq
tff(fact_6498_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_6499_minus__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% minus_natural.rep_eq
tff(fact_6500_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% times_natural.rep_eq
tff(fact_6501_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_6502_modulo__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% modulo_natural.rep_eq
tff(fact_6503_divide__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% divide_natural.rep_eq
tff(fact_6504_nat__of__natural__min,axiom,
! [K: code_natural,L: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),ord_min(code_natural),K),L)) = aa(nat,nat,aa(nat,fun(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_6505_nat__of__natural__max,axiom,
! [K: code_natural,L: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),ord_max(code_natural),K),L)) = aa(nat,nat,aa(nat,fun(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_6506_division__segment__natural_Orep__eq,axiom,
! [X: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,euclid7384307370059645450egment(code_natural),X)) = aa(nat,nat,euclid7384307370059645450egment(nat),aa(code_natural,nat,code_nat_of_natural,X)) ).
% division_segment_natural.rep_eq
tff(fact_6507_euclidean__size__natural_Orep__eq,axiom,
! [X: code_natural] : aa(code_natural,nat,euclid6346220572633701492n_size(code_natural),X) = aa(nat,nat,euclid6346220572633701492n_size(nat),aa(code_natural,nat,code_nat_of_natural,X)) ).
% euclidean_size_natural.rep_eq
tff(fact_6508_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_6509_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_6510_nat__of__natural__induct,axiom,
! [Y: nat,P: fun(nat,bool)] :
( pp(member(nat,Y,top_top(set(nat))))
=> ( ! [X2: code_natural] : pp(aa(nat,bool,P,aa(code_natural,nat,code_nat_of_natural,X2)))
=> pp(aa(nat,bool,P,Y)) ) ) ).
% nat_of_natural_induct
tff(fact_6511_nat__of__natural__cases,axiom,
! [Y: nat] :
( pp(member(nat,Y,top_top(set(nat))))
=> ~ ! [X2: code_natural] : Y != aa(code_natural,nat,code_nat_of_natural,X2) ) ).
% nat_of_natural_cases
tff(fact_6512_nat__of__natural,axiom,
! [X: code_natural] : pp(member(nat,aa(code_natural,nat,code_nat_of_natural,X),top_top(set(nat)))) ).
% nat_of_natural
tff(fact_6513_set__bit__natural_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% set_bit_natural.rep_eq
tff(fact_6514_unset__bit__natural_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% unset_bit_natural.rep_eq
tff(fact_6515_flip__bit__natural_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% flip_bit_natural.rep_eq
tff(fact_6516_drop__bit__natural_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% drop_bit_natural.rep_eq
tff(fact_6517_bit__natural_Orep__eq,axiom,
! [X: code_natural] : aa(code_natural,fun(nat,bool),bit_se5641148757651400278ts_bit(code_natural),X) = aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),aa(code_natural,nat,code_nat_of_natural,X)) ).
% bit_natural.rep_eq
tff(fact_6518_take__bit__natural_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% take_bit_natural.rep_eq
tff(fact_6519_or__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% or_natural.rep_eq
tff(fact_6520_xor__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% xor_natural.rep_eq
tff(fact_6521_and__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% and_natural.rep_eq
tff(fact_6522_mask__natural_Orep__eq,axiom,
! [X: nat] : aa(code_natural,nat,code_nat_of_natural,aa(nat,code_natural,bit_se2239418461657761734s_mask(code_natural),X)) = aa(nat,nat,bit_se2239418461657761734s_mask(nat),X) ).
% mask_natural.rep_eq
tff(fact_6523_push__bit__natural_Orep__eq,axiom,
! [X: nat,Xa4: 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),X),Xa4)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),X),aa(code_natural,nat,code_nat_of_natural,Xa4)) ).
% push_bit_natural.rep_eq
tff(fact_6524_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_6525_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_6526_nat__of__natural__inject,axiom,
! [X: code_natural,Y: code_natural] :
( ( aa(code_natural,nat,code_nat_of_natural,X) = aa(code_natural,nat,code_nat_of_natural,Y) )
<=> ( X = Y ) ) ).
% nat_of_natural_inject
tff(fact_6527_less__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: code_natural] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),X),Xa4))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4))) ) ).
% less_natural.rep_eq
tff(fact_6528_less__eq__natural_Orep__eq,axiom,
! [X: code_natural,Xa4: code_natural] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),Xa4))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa4))) ) ).
% less_eq_natural.rep_eq
tff(fact_6529_natural__of__integer_Orep__eq,axiom,
! [X: code_integer] : aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_n4118661773612635043nteger,X)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).
% natural_of_integer.rep_eq
tff(fact_6530_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_6531_Nat_Orep__eq,axiom,
! [X: code_integer] : aa(code_natural,nat,code_nat_of_natural,aa(code_integer,code_natural,code_Nat,X)) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,X)) ).
% Nat.rep_eq
tff(fact_6532_Code__Numeral_Osize__nat__def,axiom,
code_size_nat = code_nat_of_natural ).
% Code_Numeral.size_nat_def
tff(fact_6533_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_6534_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_6535_log_Oelims,axiom,
! [X: code_natural,Xa4: code_natural,Y: code_natural] :
( ( log(X,Xa4) = Y )
=> ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa4),X)) )
=> ( Y = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa4),X)) )
=> ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa4),X))) ) ) ) ) ).
% log.elims
tff(fact_6536_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_6537_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_6538_range,axiom,
! [K: code_natural,S: product_prod(code_natural,code_natural)] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),K))
=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),aa(product_prod(code_natural,product_prod(code_natural,code_natural)),code_natural,product_fst(code_natural,product_prod(code_natural,code_natural)),range(K,S))),K)) ) ).
% range
tff(fact_6539_select__weight__member,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A)),S: product_prod(code_natural,code_natural)] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),Xs))))
=> pp(member(A,aa(product_prod(A,product_prod(code_natural,code_natural)),A,product_fst(A,product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),select_weight(A,Xs),S)),set2(A,map(product_prod(code_natural,A),A,product_snd(code_natural,A),Xs)))) ) ).
% select_weight_member
tff(fact_6540_select__weight__drop__zero,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,filter(product_prod(code_natural,A),product_case_prod(code_natural,A,bool,aTP_Lamp_qt(code_natural,fun(A,bool))),Xs)) = select_weight(A,Xs) ).
% select_weight_drop_zero
tff(fact_6541_pick__member,axiom,
! [A: $tType,I: code_natural,Xs: list(product_prod(code_natural,A))] :
( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),aa(list(code_natural),code_natural,groups8242544230860333062m_list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),Xs))))
=> pp(member(A,aa(code_natural,A,pick(A,Xs),I),set2(A,map(product_prod(code_natural,A),A,product_snd(code_natural,A),Xs)))) ) ).
% pick_member
tff(fact_6542_pick_Osimps,axiom,
! [A: $tType,I: code_natural,X: product_prod(code_natural,A),Xs: list(product_prod(code_natural,A))] :
( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
=> ( aa(code_natural,A,pick(A,cons(product_prod(code_natural,A),X,Xs)),I) = aa(product_prod(code_natural,A),A,product_snd(code_natural,A),X) ) )
& ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),I),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X)))
=> ( aa(code_natural,A,pick(A,cons(product_prod(code_natural,A),X,Xs)),I) = aa(code_natural,A,pick(A,Xs),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),I),aa(product_prod(code_natural,A),code_natural,product_fst(code_natural,A),X))) ) ) ) ).
% pick.simps
tff(fact_6543_pick__drop__zero,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : pick(A,filter(product_prod(code_natural,A),product_case_prod(code_natural,A,bool,aTP_Lamp_qt(code_natural,fun(A,bool))),Xs)) = pick(A,Xs) ).
% pick_drop_zero
tff(fact_6544_log_Opelims,axiom,
! [X: code_natural,Xa4: code_natural,Y: code_natural] :
( ( log(X,Xa4) = Y )
=> ( accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,X),Xa4))
=> ~ ( ( ( ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa4),X)) )
=> ( Y = one_one(code_natural) ) )
& ( ~ ( pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less_eq(code_natural),X),one_one(code_natural)))
| pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Xa4),X)) )
=> ( Y = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa4),X))) ) ) )
=> ~ accp(product_prod(code_natural,code_natural),log_rel,aa(code_natural,product_prod(code_natural,code_natural),product_Pair(code_natural,code_natural,X),Xa4)) ) ) ) ).
% log.pelims
tff(fact_6545_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_6546_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)))
=> ( aa(code_natural,A,pick(A,map(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_6547_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_6548_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_6549_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_6550_nat__of__natural__inverse,axiom,
! [X: code_natural] : aa(nat,code_natural,code_natural_of_nat,aa(code_natural,nat,code_nat_of_natural,X)) = X ).
% nat_of_natural_inverse
tff(fact_6551_natural__of__nat__of__nat,axiom,
code_natural_of_nat = semiring_1_of_nat(code_natural) ).
% natural_of_nat_of_nat
tff(fact_6552_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_6553_natural__of__nat__cases,axiom,
! [X: code_natural] :
~ ! [Y4: nat] :
( ( X = aa(nat,code_natural,code_natural_of_nat,Y4) )
=> ~ pp(member(nat,Y4,top_top(set(nat)))) ) ).
% natural_of_nat_cases
tff(fact_6554_natural__of__nat__induct,axiom,
! [P: fun(code_natural,bool),X: 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,X)) ) ).
% natural_of_nat_induct
tff(fact_6555_natural__of__nat__inject,axiom,
! [X: nat,Y: nat] :
( pp(member(nat,X,top_top(set(nat))))
=> ( pp(member(nat,Y,top_top(set(nat))))
=> ( ( aa(nat,code_natural,code_natural_of_nat,X) = aa(nat,code_natural,code_natural_of_nat,Y) )
<=> ( X = Y ) ) ) ) ).
% natural_of_nat_inject
tff(fact_6556_minus__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa4),X)) ).
% minus_natural.abs_eq
tff(fact_6557_one__natural__def,axiom,
one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).
% one_natural_def
tff(fact_6558_divide__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa4),X)) ).
% divide_natural.abs_eq
tff(fact_6559_plus__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa4),X)) ).
% plus_natural.abs_eq
tff(fact_6560_zero__natural__def,axiom,
zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).
% zero_natural_def
tff(fact_6561_less__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xa4),X)) ) ).
% less_natural.abs_eq
tff(fact_6562_less__eq__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa4),X)) ) ).
% less_eq_natural.abs_eq
tff(fact_6563_times__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa4),X)) ).
% times_natural.abs_eq
tff(fact_6564_modulo__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa4),X)) ).
% modulo_natural.abs_eq
tff(fact_6565_take__bit__natural_Oabs__eq,axiom,
! [Xa4: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2584673776208193580ke_bit(code_natural),Xa4),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Xa4),X)) ).
% take_bit_natural.abs_eq
tff(fact_6566_xor__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Xa4),X)) ).
% xor_natural.abs_eq
tff(fact_6567_or__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Xa4),X)) ).
% or_natural.abs_eq
tff(fact_6568_and__natural_Oabs__eq,axiom,
! [Xa4: nat,X: 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,Xa4)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Xa4),X)) ).
% and_natural.abs_eq
tff(fact_6569_bit__natural_Oabs__eq,axiom,
! [X: nat] : aa(code_natural,fun(nat,bool),bit_se5641148757651400278ts_bit(code_natural),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,fun(nat,bool),bit_se5641148757651400278ts_bit(nat),X) ).
% bit_natural.abs_eq
tff(fact_6570_push__bit__natural_Oabs__eq,axiom,
! [Xa4: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4730199178511100633sh_bit(code_natural),Xa4),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Xa4),X)) ).
% push_bit_natural.abs_eq
tff(fact_6571_mask__natural_Oabs__eq,axiom,
! [X: nat] : aa(nat,code_natural,bit_se2239418461657761734s_mask(code_natural),X) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,bit_se2239418461657761734s_mask(nat),X)) ).
% mask_natural.abs_eq
tff(fact_6572_drop__bit__natural_Oabs__eq,axiom,
! [Xa4: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se4197421643247451524op_bit(code_natural),Xa4),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Xa4),X)) ).
% drop_bit_natural.abs_eq
tff(fact_6573_euclidean__size__natural_Oabs__eq,axiom,
! [X: nat] : aa(code_natural,nat,euclid6346220572633701492n_size(code_natural),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,nat,euclid6346220572633701492n_size(nat),X) ).
% euclidean_size_natural.abs_eq
tff(fact_6574_flip__bit__natural_Oabs__eq,axiom,
! [Xa4: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se8732182000553998342ip_bit(code_natural),Xa4),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se8732182000553998342ip_bit(nat),Xa4),X)) ).
% flip_bit_natural.abs_eq
tff(fact_6575_unset__bit__natural_Oabs__eq,axiom,
! [Xa4: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se2638667681897837118et_bit(code_natural),Xa4),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),Xa4),X)) ).
% unset_bit_natural.abs_eq
tff(fact_6576_division__segment__natural_Oabs__eq,axiom,
! [X: nat] : aa(code_natural,code_natural,euclid7384307370059645450egment(code_natural),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,euclid7384307370059645450egment(nat),X)) ).
% division_segment_natural.abs_eq
tff(fact_6577_set__bit__natural_Oabs__eq,axiom,
! [Xa4: nat,X: nat] : aa(code_natural,code_natural,aa(nat,fun(code_natural,code_natural),bit_se5668285175392031749et_bit(code_natural),Xa4),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),Xa4),X)) ).
% set_bit_natural.abs_eq
tff(fact_6578_integer__of__natural_Oabs__eq,axiom,
! [X: nat] : aa(code_natural,code_integer,code_i5400310926305786745atural,aa(nat,code_natural,code_natural_of_nat,X)) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),X)) ).
% integer_of_natural.abs_eq
tff(fact_6579_natural__of__integer_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_natural,code_n4118661773612635043nteger,aa(int,code_integer,code_integer_of_int,X)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,X)) ).
% natural_of_integer.abs_eq
tff(fact_6580_Nat_Oabs__eq,axiom,
! [X: int] : aa(code_integer,code_natural,code_Nat,aa(int,code_integer,code_integer_of_int,X)) = aa(nat,code_natural,code_natural_of_nat,aa(int,nat,nat2,X)) ).
% Nat.abs_eq
tff(fact_6581_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_6582_case__natural__def,axiom,
! [T: $tType,X3: T,Xa: fun(code_natural,T)] : code_case_natural(T,X3,Xa) = code_rec_natural(T,X3,aTP_Lamp_qu(fun(code_natural,T),fun(code_natural,fun(T,T)),Xa)) ).
% case_natural_def
tff(fact_6583_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_6584_rec__natural__def,axiom,
! [T: $tType,X3: T,Xa: fun(code_natural,fun(T,T)),Xb: code_natural] : aa(code_natural,T,code_rec_natural(T,X3,Xa),Xb) = the(T,code_rec_set_natural(T,X3,Xa,Xb)) ).
% rec_natural_def
tff(fact_6585_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_6586_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_6587_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_6588_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_6589_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_6590_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_6591_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_6592_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_6593_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_6594_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_6595_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_6596_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_6597_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_6598_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_6599_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_6600_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_6601_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_6602_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_6603_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_6604_cr__natural__def,axiom,
! [X3: nat,Xa: code_natural] :
( pp(aa(code_natural,bool,aa(nat,fun(code_natural,bool),code_cr_natural,X3),Xa))
<=> ( X3 = aa(code_natural,nat,code_nat_of_natural,Xa) ) ) ).
% cr_natural_def
tff(fact_6605_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_6606_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_6607_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_6608_Suc_Orep__eq,axiom,
! [X: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,code_Suc,X)) = aa(nat,nat,suc,aa(code_natural,nat,code_nat_of_natural,X)) ).
% Suc.rep_eq
tff(fact_6609_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_6610_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_6611_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_6612_Suc_Oabs__eq,axiom,
! [X: nat] : aa(code_natural,code_natural,code_Suc,aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,suc,X)) ).
% Suc.abs_eq
tff(fact_6613_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_6614_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_6615_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F: fun(nat,fun(A,A)),A3: nat,B2: nat,Acc: A] :
( 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))),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)),product_Pair(nat,product_prod(nat,A),A3),aa(A,product_prod(nat,A),product_Pair(nat,A,B2),Acc))))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc) = Acc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
=> ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc) = 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),Acc)) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
tff(fact_6616_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_6617_natural_Opcr__cr__eq,axiom,
code_pcr_natural = code_cr_natural ).
% natural.pcr_cr_eq
tff(fact_6618_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_cp(nat,nat)),code_natural_of_nat)) ).
% natural.id_abs_transfer
tff(fact_6619_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_6620_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_6621_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_6622_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_6623_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_6624_natural_Obi__total,axiom,
bi_total(nat,code_natural,code_pcr_natural) ).
% natural.bi_total
tff(fact_6625_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_cp(nat,nat)),code_nat_of_natural)) ).
% natural.rep_transfer
tff(fact_6626_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_6627_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_6628_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_6629_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_6630_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))))] :
( 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))),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)),product_Pair(nat,product_prod(nat,A),A1),aa(A,product_prod(nat,A),product_Pair(nat,A,A22),A32))))
=> ( ! [F2: fun(nat,fun(A,A)),A6: nat,B4: nat,Acc2: A] :
( 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))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A6),aa(A,product_prod(nat,A),product_Pair(nat,A,B4),Acc2))))
=> ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B4),A6))
=> 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,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A6),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F2,A6),Acc2))) )
=> 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,F2),A6),B4),Acc2)) ) )
=> 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_6631_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_6632_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_6633_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_6634_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_6635_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_6636_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_6637_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_6638_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_6639_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_6640_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_6641_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_6642_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_6643_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_6644_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_6645_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_6646_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: fun(nat,fun(A,A)),Xa4: nat,Xb3: nat,Xc: A,Y: A] :
( ( set_fo6178422350223883121st_nat(A,X,Xa4,Xb3,Xc) = Y )
=> ( 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))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xa4),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb3),Xc))))
=> ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa4))
=> ( Y = Xc ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb3),Xa4))
=> ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa4),one_one(nat)),Xb3,aa(A,A,aa(nat,fun(A,A),X,Xa4),Xc)) ) ) )
=> ~ 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))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xa4),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb3),Xc)))) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
tff(fact_6647_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_6648_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [S5: set(B),T4: set(C),H2: fun(B,C),S3: set(B),T5: set(C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S5))
=> ( pp(aa(set(C),bool,finite_finite2(C),T4))
=> ( bij_betw(B,C,H2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))
=> ( ! [A6: B] :
( pp(member(B,A6,S5))
=> ( aa(C,A,G,aa(B,C,H2,A6)) = one_one(A) ) )
=> ( ! [B4: C] :
( pp(member(C,B4,T4))
=> ( aa(C,A,G,B4) = 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_qv(fun(B,C),fun(fun(C,A),fun(B,A)),H2),G)),S3) = 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_6649_bij__betw__of__nat,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [N6: set(nat),A5: set(A)] :
( bij_betw(nat,A,semiring_1_of_nat(A),N6,A5)
<=> ( image(nat,A,semiring_1_of_nat(A),N6) = A5 ) ) ) ).
% bij_betw_of_nat
tff(fact_6650_bij__betw__Suc,axiom,
! [M7: set(nat),N6: set(nat)] :
( bij_betw(nat,nat,suc,M7,N6)
<=> ( image(nat,nat,suc,M7) = N6 ) ) ).
% bij_betw_Suc
tff(fact_6651_bijection__def,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
<=> bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).
% bijection_def
tff(fact_6652_bijection_Ointro,axiom,
! [A: $tType,F: fun(A,A)] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> hilbert_bijection(A,F) ) ).
% bijection.intro
tff(fact_6653_bijection_Obij,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).
% bijection.bij
tff(fact_6654_bij__betw__funpow,axiom,
! [A: $tType,F: fun(A,A),S3: set(A),N: nat] :
( bij_betw(A,A,F,S3,S3)
=> bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),S3,S3) ) ).
% bij_betw_funpow
tff(fact_6655_bij__fn,axiom,
! [A: $tType,F: fun(A,A),N: nat] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),top_top(set(A)),top_top(set(A))) ) ).
% bij_fn
tff(fact_6656_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_6657_bij__imp__bij__inv,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),top_top(set(B)),top_top(set(A))) ) ).
% bij_imp_bij_inv
tff(fact_6658_bij__inv__eq__iff,axiom,
! [A: $tType,B: $tType,P3: fun(A,B),X: A,Y: B] :
( bij_betw(A,B,P3,top_top(set(A)),top_top(set(B)))
=> ( ( X = aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),P3),Y) )
<=> ( aa(A,B,P3,X) = Y ) ) ) ).
% bij_inv_eq_iff
tff(fact_6659_inv__inv__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( hilbert_inv_into(B,A,top_top(set(B)),hilbert_inv_into(A,B,top_top(set(A)),F)) = F ) ) ).
% inv_inv_eq
tff(fact_6660_bij__betw__inv__into__right,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),A10: set(B),A4: B] :
( bij_betw(A,B,F,A5,A10)
=> ( pp(member(B,A4,A10))
=> ( aa(A,B,F,aa(B,A,hilbert_inv_into(A,B,A5,F),A4)) = A4 ) ) ) ).
% bij_betw_inv_into_right
tff(fact_6661_bij__betw__inv__into__left,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),A10: set(B),A3: A] :
( bij_betw(A,B,F,A5,A10)
=> ( pp(member(A,A3,A5))
=> ( aa(B,A,hilbert_inv_into(A,B,A5,F),aa(A,B,F,A3)) = A3 ) ) ) ).
% bij_betw_inv_into_left
tff(fact_6662_inv__into__inv__into__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),A10: set(B),A3: A] :
( bij_betw(A,B,F,A5,A10)
=> ( pp(member(A,A3,A5))
=> ( aa(A,B,hilbert_inv_into(B,A,A10,hilbert_inv_into(A,B,A5,F)),A3) = aa(A,B,F,A3) ) ) ) ).
% inv_into_inv_into_eq
tff(fact_6663_bij__betw__inv__into,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B5: set(B)] :
( bij_betw(A,B,F,A5,B5)
=> bij_betw(B,A,hilbert_inv_into(A,B,A5,F),B5,A5) ) ).
% bij_betw_inv_into
tff(fact_6664_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),A10: set(B),B5: set(A),B13: set(B)] :
( bij_betw(A,B,F,A5,A10)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),A5))
=> ( ( image(A,B,F,B5) = B13 )
=> bij_betw(B,A,hilbert_inv_into(A,B,A5,F),B13,B5) ) ) ) ).
% bij_betw_inv_into_subset
tff(fact_6665_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( image(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),A5)) = aa(set(B),set(B),uminus_uminus(set(B)),image(A,B,F,A5)) ) ) ).
% bij_image_Compl_eq
tff(fact_6666_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),comp(A,B,C,F),G)) = aa(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_6667_inv__fn,axiom,
! [A: $tType,F: fun(A,A),N: nat] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> ( hilbert_inv_into(A,A,top_top(set(A)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F)) ) ) ).
% inv_fn
tff(fact_6668_mono__inv,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> pp(aa(fun(B,A),bool,order_mono(B,A),hilbert_inv_into(A,B,top_top(set(A)),F))) ) ) ) ).
% mono_inv
tff(fact_6669_bijection_Obij__inv,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> bij_betw(A,A,hilbert_inv_into(A,A,top_top(set(A)),F),top_top(set(A)),top_top(set(A))) ) ).
% bijection.bij_inv
tff(fact_6670_bij__image__Collect__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P: fun(A,bool)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( image(A,B,F,collect(A,P)) = collect(B,aa(fun(A,bool),fun(B,bool),aTP_Lamp_qw(fun(A,B),fun(fun(A,bool),fun(B,bool)),F),P)) ) ) ).
% bij_image_Collect_eq
tff(fact_6671_bij__int__decode,axiom,
bij_betw(nat,int,nat_int_decode,top_top(set(nat)),top_top(set(int))) ).
% bij_int_decode
tff(fact_6672_bij__int__encode,axiom,
bij_betw(int,nat,nat_int_encode,top_top(set(int)),top_top(set(nat))) ).
% bij_int_encode
tff(fact_6673_infinite__imp__bij__betw2,axiom,
! [A: $tType,A5: set(A),A3: A] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ? [H4: fun(A,A)] : bij_betw(A,A,H4,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),insert(A,A3,bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw2
tff(fact_6674_infinite__imp__bij__betw,axiom,
! [A: $tType,A5: set(A),A3: A] :
( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
=> ? [H4: fun(A,A)] : bij_betw(A,A,H4,A5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),insert(A,A3,bot_bot(set(A))))) ) ).
% infinite_imp_bij_betw
tff(fact_6675_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),M7))
=> ? [H4: fun(A,nat)] : bij_betw(A,nat,H4,M7,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M7))) ) ).
% ex_bij_betw_finite_nat
tff(fact_6676_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),M7))
=> ? [H4: fun(nat,A)] : bij_betw(nat,A,H4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M7)),M7) ) ).
% ex_bij_betw_nat_finite
tff(fact_6677_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder(A)
& comple5582772986160207858norder(B) )
=> ! [F: fun(A,B),A5: set(A)] :
( pp(aa(fun(A,B),bool,order_mono(A,B),F))
=> ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A5)) = aa(set(B),B,complete_Inf_Inf(B),image(A,B,F,A5)) ) ) ) ) ).
% mono_bij_Inf
tff(fact_6678_bij__prod__decode,axiom,
bij_betw(nat,product_prod(nat,nat),nat_prod_decode,top_top(set(nat)),top_top(set(product_prod(nat,nat)))) ).
% bij_prod_decode
tff(fact_6679_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),M7: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> bij_betw(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),image(A,B,F,M7),M7) ) ).
% inj_imp_bij_betw_inv
tff(fact_6680_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),M7))
=> ? [H4: fun(nat,A)] : bij_betw(nat,A,H4,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M7)),M7) ) ).
% ex_bij_betw_nat_finite_1
tff(fact_6681_bij__list__decode,axiom,
bij_betw(nat,list(nat),nat_list_decode,top_top(set(nat)),top_top(set(list(nat)))) ).
% bij_list_decode
tff(fact_6682_bij__list__encode,axiom,
bij_betw(list(nat),nat,nat_list_encode,top_top(set(list(nat))),top_top(set(nat))) ).
% bij_list_encode
tff(fact_6683_bij__prod__encode,axiom,
bij_betw(product_prod(nat,nat),nat,nat_prod_encode,top_top(set(product_prod(nat,nat))),top_top(set(nat))) ).
% bij_prod_encode
tff(fact_6684_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A5: set(B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( vimage(A,B,F,A5) = image(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),A5) ) ) ).
% bij_vimage_eq_inv_image
tff(fact_6685_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)))
=> ! [X3: A] : aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F))),X3) = X3 ) ).
% fn_o_inv_fn_is_id
tff(fact_6686_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)))
=> ! [X3: A] : aa(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F))),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),X3) = X3 ) ).
% inv_fn_o_fn_is_id
tff(fact_6687_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [S5: set(B),T4: set(C),H2: fun(B,C),S3: set(B),T5: set(C),G: fun(C,A)] :
( pp(aa(set(B),bool,finite_finite2(B),S5))
=> ( pp(aa(set(C),bool,finite_finite2(C),T4))
=> ( bij_betw(B,C,H2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S3),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T5),T4))
=> ( ! [A6: B] :
( pp(member(B,A6,S5))
=> ( aa(C,A,G,aa(B,C,H2,A6)) = zero_zero(A) ) )
=> ( ! [B4: C] :
( pp(member(C,B4,T4))
=> ( aa(C,A,G,B4) = zero_zero(A) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_qx(fun(B,C),fun(fun(C,A),fun(B,A)),H2),G)),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T5) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
tff(fact_6688_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_6689_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_6690_natLeq__on__wo__rel,axiom,
! [N: nat] : bNF_Wellorder_wo_rel(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N)))) ).
% natLeq_on_wo_rel
tff(fact_6691_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_6692_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),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),comp(B,nat,A,F),G)) ).
% Heap.size_gen_o_map
tff(fact_6693_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice_order(A,ord_max(A),aTP_Lamp_ml(A,fun(A,bool)),aTP_Lamp_pt(A,fun(A,bool))) ) ).
% max.semilattice_order_axioms
tff(fact_6694_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_6695_semilattice__order__set_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ).
% semilattice_order_set.axioms(1)
tff(fact_6696_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),comp(B,C,A,G),F)),V) ).
% Heap.map_comp
tff(fact_6697_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ).
% semilattice_neutr_order.axioms(2)
tff(fact_6698_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_ql(A,A)),T2) = T2 ).
% Heap.map_ident
tff(fact_6699_semilattice__order_Omono,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A,D2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),D2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),aa(A,A,aa(A,fun(A,A),F,C2),D2))) ) ) ) ).
% semilattice_order.mono
tff(fact_6700_semilattice__order_OorderE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).
% semilattice_order.orderE
tff(fact_6701_semilattice__order_OorderI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).
% semilattice_order.orderI
tff(fact_6702_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb1
tff(fact_6703_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb2
tff(fact_6704_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb3
tff(fact_6705_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3))
=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb4
tff(fact_6706_semilattice__order_OboundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2)) ) ) ) ).
% semilattice_order.boundedE
tff(fact_6707_semilattice__order_OboundedI,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2))) ) ) ) ).
% semilattice_order.boundedI
tff(fact_6708_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ) ).
% semilattice_order.order_iff
tff(fact_6709_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),A3)) ) ).
% semilattice_order.cobounded1
tff(fact_6710_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2)) ) ).
% semilattice_order.cobounded2
tff(fact_6711_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = A3 ) ) ) ).
% semilattice_order.absorb_iff1
tff(fact_6712_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
<=> ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = B2 ) ) ) ).
% semilattice_order.absorb_iff2
tff(fact_6713_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2)) ) ) ) ).
% semilattice_order.bounded_iff
tff(fact_6714_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2)) ) ) ).
% semilattice_order.coboundedI1
tff(fact_6715_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2)) ) ) ).
% semilattice_order.coboundedI2
tff(fact_6716_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)))
=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2)) ) ) ) ).
% semilattice_order.strict_boundedE
tff(fact_6717_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( ( A3 = aa(A,A,aa(A,fun(A,A),F,A3),B2) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
tff(fact_6718_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,C2: A,B2: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2)) ) ) ).
% semilattice_order.strict_coboundedI1
tff(fact_6719_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),B2: A,C2: A,A3: A] :
( semilattice_order(A,F,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2)) ) ) ).
% semilattice_order.strict_coboundedI2
tff(fact_6720_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_6721_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_6722_semilattice__order__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( lattic4895041142388067077er_set(A,F,Less_eq,Less)
<=> ( semilattice_order(A,F,Less_eq,Less)
& lattic149705377957585745ce_set(A,F) ) ) ).
% semilattice_order_set_def
tff(fact_6723_semilattice__order__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
=> ( lattic149705377957585745ce_set(A,F)
=> lattic4895041142388067077er_set(A,F,Less_eq,Less) ) ) ).
% semilattice_order_set.intro
tff(fact_6724_semilattice__neutr__order__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila1105856199041335345_order(A,F,Z2,Less_eq,Less)
<=> ( semilattice_neutr(A,F,Z2)
& semilattice_order(A,F,Less_eq,Less) ) ) ).
% semilattice_neutr_order_def
tff(fact_6725_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_neutr(A,F,Z2)
=> ( semilattice_order(A,F,Less_eq,Less)
=> semila1105856199041335345_order(A,F,Z2,Less_eq,Less) ) ) ).
% semilattice_neutr_order.intro
tff(fact_6726_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semilattice_order(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).
% inf.semilattice_order_axioms
tff(fact_6727_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice_order(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).
% min.semilattice_order_axioms
tff(fact_6728_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice_order(A,sup_sup(A),aTP_Lamp_pu(A,fun(A,bool)),aTP_Lamp_pv(A,fun(A,bool))) ) ).
% sup.semilattice_order_axioms
tff(fact_6729_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( ord_lexordp_eq(A,A1,A22)
<=> ( ? [Ys3: list(A)] :
( ( A1 = nil(A) )
& ( A22 = Ys3 ) )
| ? [X5: A,Y5: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = cons(A,X5,Xs3) )
& ( A22 = cons(A,Y5,Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5)) )
| ? [X5: A,Y5: A,Xs3: list(A),Ys3: list(A)] :
( ( A1 = cons(A,X5,Xs3) )
& ( A22 = cons(A,Y5,Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X5))
& ord_lexordp_eq(A,Xs3,Ys3) ) ) ) ) ).
% lexordp_eq.simps
tff(fact_6730_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Xs: list(A),Y: A,Ys2: list(A)] :
( ord_lexordp_eq(A,cons(A,X,Xs),cons(A,Y,Ys2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
& ord_lexordp_eq(A,Xs,Ys2) ) ) ) ) ).
% lexordp_eq_simps(4)
tff(fact_6731_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ord_lexordp_eq(A,cons(A,X,Xs),cons(A,Y,Ys2)) ) ) ).
% lexordp_eq.Cons
tff(fact_6732_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ord(A)
=> ! [X: A,Y: A,Xs: list(A),Ys2: list(A)] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( ord_lexordp_eq(A,Xs,Ys2)
=> ord_lexordp_eq(A,cons(A,X,Xs),cons(A,Y,Ys2)) ) ) ) ) ).
% lexordp_eq.Cons_eq
tff(fact_6733_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ord(A)
=> ! [A1: list(A),A22: list(A)] :
( ord_lexordp_eq(A,A1,A22)
=> ( ( A1 != nil(A) )
=> ( ! [X2: A] :
( ? [Xs2: list(A)] : A1 = cons(A,X2,Xs2)
=> ! [Y4: A] :
( ? [Ys4: list(A)] : A22 = cons(A,Y4,Ys4)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4)) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list(A)] :
( ( A1 = cons(A,X2,Xs2) )
=> ! [Ys4: list(A)] :
( ( A22 = cons(A,Y4,Ys4) )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X2))
=> ~ ord_lexordp_eq(A,Xs2,Ys4) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
tff(fact_6734_bdd__below__primitive__def,axiom,
! [A: $tType] :
( preorder(A)
=> ( condit1013018076250108175_below(A) = condit16957441358409770ng_bdd(A,aTP_Lamp_pj(A,fun(A,bool))) ) ) ).
% bdd_below_primitive_def
tff(fact_6735_Quotient__natural,axiom,
quotient(nat,code_natural,fequal(nat),code_natural_of_nat,code_nat_of_natural,code_cr_natural) ).
% Quotient_natural
tff(fact_6736_bdd__above__primitive__def,axiom,
! [A: $tType] :
( preorder(A)
=> ( condit941137186595557371_above(A) = condit16957441358409770ng_bdd(A,ord_less_eq(A)) ) ) ).
% bdd_above_primitive_def
tff(fact_6737_card_Ofolding__on__axioms,axiom,
! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_nv(A,fun(nat,nat))) ).
% card.folding_on_axioms
tff(fact_6738_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> condit622319405099724424ng_bdd(A,aTP_Lamp_pj(A,fun(A,bool)),aTP_Lamp_pi(A,fun(A,bool))) ) ).
% bdd_below.preordering_bdd_axioms
tff(fact_6739_bdd__above_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> condit622319405099724424ng_bdd(A,ord_less_eq(A),ord_less(A)) ) ).
% bdd_above.preordering_bdd_axioms
tff(fact_6740_Quotient__integer,axiom,
quotient(int,code_integer,fequal(int),code_integer_of_int,code_int_of_integer,code_cr_integer) ).
% Quotient_integer
tff(fact_6741_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_6742_finite__subset__Union__chain,axiom,
! [A: $tType,A5: set(A),B12: set(set(A)),A14: set(set(A))] :
( pp(aa(set(A),bool,finite_finite2(A),A5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12)))
=> ( ( B12 != bot_bot(set(set(A))) )
=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),B12))
=> ~ ! [B7: set(A)] :
( pp(member(set(A),B7,B12))
=> ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B7)) ) ) ) ) ) ).
% finite_subset_Union_chain
tff(fact_6743_wait__def,axiom,
! [N: nat] : heap_Time_wait(N) = heap_Time_Heap2(product_unit,aTP_Lamp_qy(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ).
% wait_def
tff(fact_6744_subset_Ochain__empty,axiom,
! [A: $tType,A5: set(set(A))] : pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),bot_bot(set(set(A))))) ).
% subset.chain_empty
tff(fact_6745_subset__chain__def,axiom,
! [A: $tType,A14: set(set(A)),C7: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),C7))
<=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C7),A14))
& ! [X5: set(A)] :
( pp(member(set(A),X5,C7))
=> ! [Xa2: set(A)] :
( pp(member(set(A),Xa2,C7))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X5),Xa2))
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa2),X5)) ) ) ) ) ) ).
% subset_chain_def
tff(fact_6746_subset__Zorn,axiom,
! [A: $tType,A5: set(set(A))] :
( ! [C8: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C8))
=> ? [X3: set(A)] :
( pp(member(set(A),X3,A5))
& ! [Xa3: set(A)] :
( pp(member(set(A),Xa3,C8))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Xa3),X3)) ) ) )
=> ? [X2: set(A)] :
( pp(member(set(A),X2,A5))
& ! [Xa: set(A)] :
( pp(member(set(A),Xa,A5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X2),Xa))
=> ( Xa = X2 ) ) ) ) ) ).
% subset_Zorn
tff(fact_6747_subset__chain__insert,axiom,
! [A: $tType,A14: set(set(A)),B5: set(A),B12: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),insert(set(A),B5,B12)))
<=> ( pp(member(set(A),B5,A14))
& ! [X5: set(A)] :
( pp(member(set(A),X5,B12))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X5),B5))
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B5),X5)) ) )
& pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),B12)) ) ) ).
% subset_chain_insert
tff(fact_6748_subset__Zorn_H,axiom,
! [A: $tType,A5: set(set(A))] :
( ! [C8: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C8))
=> pp(member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8),A5)) )
=> ? [X2: set(A)] :
( pp(member(set(A),X2,A5))
& ! [Xa: set(A)] :
( pp(member(set(A),Xa,A5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X2),Xa))
=> ( Xa = X2 ) ) ) ) ) ).
% subset_Zorn'
tff(fact_6749_subset_Ochain__extend,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A)),Z2: set(A)] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
=> ( pp(member(set(A),Z2,A5))
=> ( ! [X2: set(A)] :
( pp(member(set(A),X2,C6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X2),Z2)) )
=> pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),insert(set(A),Z2,bot_bot(set(set(A))))),C6))) ) ) ) ).
% subset.chain_extend
tff(fact_6750_subset_Ochain__total,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A)),X: set(A),Y: set(A)] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
=> ( pp(member(set(A),X,C6))
=> ( pp(member(set(A),Y,C6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X),Y))
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y),X)) ) ) ) ) ).
% subset.chain_total
tff(fact_6751_subset_OchainI,axiom,
! [A: $tType,C6: set(set(A)),A5: set(set(A))] :
( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C6),A5))
=> ( ! [X2: set(A),Y4: set(A)] :
( pp(member(set(A),X2,C6))
=> ( pp(member(set(A),Y4,C6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X2),Y4))
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Y4),X2)) ) ) )
=> pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6)) ) ) ).
% subset.chainI
tff(fact_6752_subset_Ochain__def,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
<=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C6),A5))
& ! [X5: set(A)] :
( pp(member(set(A),X5,C6))
=> ! [Xa2: set(A)] :
( pp(member(set(A),Xa2,C6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),X5),Xa2))
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool)),aa(fun(set(A),fun(set(A),bool)),fun(fun(set(A),fun(set(A),bool)),fun(set(A),fun(set(A),bool))),sup_sup(fun(set(A),fun(set(A),bool))),ord_less(set(A))),fequal(set(A))),Xa2),X5)) ) ) ) ) ) ).
% subset.chain_def
tff(fact_6753_uminus__unit__def,axiom,
! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).
% uminus_unit_def
tff(fact_6754_subset__Zorn__nonempty,axiom,
! [A: $tType,A14: set(set(A))] :
( ( A14 != bot_bot(set(set(A))) )
=> ( ! [C9: set(set(A))] :
( ( C9 != bot_bot(set(set(A))) )
=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),C9))
=> pp(member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9),A14)) ) )
=> ? [X2: set(A)] :
( pp(member(set(A),X2,A14))
& ! [Xa: set(A)] :
( pp(member(set(A),Xa,A14))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X2),Xa))
=> ( Xa = X2 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
tff(fact_6755_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( wellorder(A)
=> ! [P: fun(fun(A,B),fun(A,fun(B,bool)))] :
( ! [R3: B,F2: fun(A,B),G5: fun(A,B),X2: A] :
( ! [Y3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X2))
=> ( aa(A,B,F2,Y3) = aa(A,B,G5,Y3) ) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X2),R3))
<=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G5),X2),R3)) ) )
=> ( ! [X2: A,F2: fun(A,B)] :
( ! [Y3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X2))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),Y3),aa(A,B,F2,Y3))) )
=> ? [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X2),X_1)) )
=> ? [F2: fun(A,B)] :
! [X3: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F2),X3),aa(A,B,F2,X3))) ) ) ) ).
% dependent_wellorder_choice
tff(fact_6756_Union__in__chain,axiom,
! [A: $tType,B12: set(set(A)),A14: set(set(A))] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B12))
=> ( ( B12 != bot_bot(set(set(A))) )
=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),B12))
=> pp(member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12),B12)) ) ) ) ).
% Union_in_chain
tff(fact_6757_Inter__in__chain,axiom,
! [A: $tType,B12: set(set(A)),A14: set(set(A))] :
( pp(aa(set(set(A)),bool,finite_finite2(set(A)),B12))
=> ( ( B12 != bot_bot(set(set(A))) )
=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A14,ord_less(set(A))),B12))
=> pp(member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12),B12)) ) ) ) ).
% Inter_in_chain
tff(fact_6758_chain__subset__alt__def,axiom,
! [A: $tType,C6: set(set(A))] :
( chain_subset(A,C6)
<=> pp(aa(set(set(A)),bool,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C6)) ) ).
% chain_subset_alt_def
tff(fact_6759_chains__alt__def,axiom,
! [A: $tType,A5: set(set(A))] : chains(A,A5) = collect(set(set(A)),pred_chain(set(A),A5,ord_less(set(A)))) ).
% chains_alt_def
tff(fact_6760_sup2CI,axiom,
! [A: $tType,B: $tType,B5: fun(A,fun(B,bool)),X: A,Y: B,A5: fun(A,fun(B,bool))] :
( ( ~ pp(aa(B,bool,aa(A,fun(B,bool),B5,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y)) )
=> 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))),A5),B5),X),Y)) ) ).
% sup2CI
tff(fact_6761_sup2I2,axiom,
! [A: $tType,B: $tType,B5: fun(A,fun(B,bool)),X: A,Y: B,A5: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),B5,X),Y))
=> 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))),A5),B5),X),Y)) ) ).
% sup2I2
tff(fact_6762_sup2I1,axiom,
! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),X: A,Y: B,B5: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y))
=> 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))),A5),B5),X),Y)) ) ).
% sup2I1
tff(fact_6763_sup2E,axiom,
! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),B5: fun(A,fun(B,bool)),X: A,Y: 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))),A5),B5),X),Y))
=> ( ~ pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),B5,X),Y)) ) ) ).
% sup2E
tff(fact_6764_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C6: set(A)] :
( pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
=> ( ~ pred_maxchain(A,A5,P,C6)
=> ( pp(aa(set(A),bool,pred_chain(A,A5,P),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_qz(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A5),P),C6))))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),C6),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_qz(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A5),P),C6)))) ) ) ) ).
% pred_on.not_maxchain_Some
tff(fact_6765_subset__maxchain__max,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A)),X6: set(A)] :
( pred_maxchain(set(A),A5,ord_less(set(A)),C6)
=> ( pp(member(set(A),X6,A5))
=> ( 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)),C6)),X6))
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6) = X6 ) ) ) ) ).
% subset_maxchain_max
tff(fact_6766_subset_OHausdorff,axiom,
! [A: $tType,A5: set(set(A))] :
? [X_12: set(set(A))] : pred_maxchain(set(A),A5,ord_less(set(A)),X_12) ).
% subset.Hausdorff
tff(fact_6767_subset_Omaxchain__def,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A))] :
( pred_maxchain(set(A),A5,ord_less(set(A)),C6)
<=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
& ~ ? [S6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),S6))
& pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),C6),S6)) ) ) ) ).
% subset.maxchain_def
tff(fact_6768_subset_Omaxchain__imp__chain,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A))] :
( pred_maxchain(set(A),A5,ord_less(set(A)),C6)
=> pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6)) ) ).
% subset.maxchain_imp_chain
tff(fact_6769_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
=> ( ~ pred_maxchain(set(A),A5,ord_less(set(A)),C6)
=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_ra(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A5),C6))))
& pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),C6),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_ra(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A5),C6)))) ) ) ) ).
% subset.not_maxchain_Some
tff(fact_6770_pred__on_Omaxchain__def,axiom,
! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C6: set(A)] :
( pred_maxchain(A,A5,P,C6)
<=> ( pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
& ~ ? [S6: set(A)] :
( pp(aa(set(A),bool,pred_chain(A,A5,P),S6))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),C6),S6)) ) ) ) ).
% pred_on.maxchain_def
tff(fact_6771_pred__on_Osuc__def,axiom,
! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C6: set(A)] :
( ( ( ~ pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
| pred_maxchain(A,A5,P,C6) )
=> ( pred_suc(A,A5,P,C6) = C6 ) )
& ( ~ ( ~ pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
| pred_maxchain(A,A5,P,C6) )
=> ( pred_suc(A,A5,P,C6) = fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_qz(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A5),P),C6)) ) ) ) ).
% pred_on.suc_def
tff(fact_6772_subset_Osuc__def,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A))] :
( ( ( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
| pred_maxchain(set(A),A5,ord_less(set(A)),C6) )
=> ( pred_suc(set(A),A5,ord_less(set(A)),C6) = C6 ) )
& ( ~ ( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
| pred_maxchain(set(A),A5,ord_less(set(A)),C6) )
=> ( pred_suc(set(A),A5,ord_less(set(A)),C6) = fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_ra(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A5),C6)) ) ) ) ).
% subset.suc_def
tff(fact_6773_subset_Onot__chain__suc,axiom,
! [A: $tType,A5: set(set(A)),X6: set(set(A))] :
( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),X6))
=> ( pred_suc(set(A),A5,ord_less(set(A)),X6) = X6 ) ) ).
% subset.not_chain_suc
tff(fact_6774_subset_Omaxchain__suc,axiom,
! [A: $tType,A5: set(set(A)),X6: set(set(A))] :
( pred_maxchain(set(A),A5,ord_less(set(A)),X6)
=> ( pred_suc(set(A),A5,ord_less(set(A)),X6) = X6 ) ) ).
% subset.maxchain_suc
tff(fact_6775_subset_Ochain__suc,axiom,
! [A: $tType,A5: set(set(A)),X6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),X6))
=> pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),pred_suc(set(A),A5,ord_less(set(A)),X6))) ) ).
% subset.chain_suc
tff(fact_6776_subset_Osubset__suc,axiom,
! [A: $tType,X6: set(set(A)),Y6: set(set(A)),A5: set(set(A))] :
( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),Y6))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),pred_suc(set(A),A5,ord_less(set(A)),Y6))) ) ).
% subset.subset_suc
tff(fact_6777_subset_Osuc__subset,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),pred_suc(set(A),A5,ord_less(set(A)),X6))) ).
% subset.suc_subset
tff(fact_6778_subset_Osuc__in__carrier,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] :
( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),A5))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A5,ord_less(set(A)),X6)),A5)) ) ).
% subset.suc_in_carrier
tff(fact_6779_subset_Ochain__sucD,axiom,
! [A: $tType,A5: set(set(A)),X6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),X6))
=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A5,ord_less(set(A)),X6)),A5))
& pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),pred_suc(set(A),A5,ord_less(set(A)),X6))) ) ) ).
% subset.chain_sucD
tff(fact_6780_subset_Osuc__not__equals,axiom,
! [A: $tType,A5: set(set(A)),C6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
=> ( ~ pred_maxchain(set(A),A5,ord_less(set(A)),C6)
=> ( pred_suc(set(A),A5,ord_less(set(A)),C6) != C6 ) ) ) ).
% subset.suc_not_equals
tff(fact_6781_alloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H2: heap_ext(product_unit)] : array_alloc(A,Xs,H2) = aa(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,H2))),array_set(A,array2(A,lim(product_unit,H2)),Xs,lim_update(product_unit,aTP_Lamp_rb(heap_ext(product_unit),fun(nat,nat),H2),H2))) ) ).
% alloc_def
tff(fact_6782_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_ms(num,fun(num,int))) ).
% Code_Numeral.sub_def
tff(fact_6783_present__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H2: heap_ext(product_unit),A3: array(A)] :
( pp(aa(array(A),bool,array_present(A,H2),A3))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,A3)),lim(product_unit,H2))) ) ) ).
% present_def
tff(fact_6784_array_Osize_I2_J,axiom,
! [A: $tType,X: nat] : aa(array(A),nat,size_size(array(A)),array2(A,X)) = zero_zero(nat) ).
% array.size(2)
tff(fact_6785_array_Osize__gen,axiom,
! [A: $tType,Xa4: fun(A,nat),X: nat] : size_array(A,Xa4,array2(A,X)) = zero_zero(nat) ).
% array.size_gen
tff(fact_6786_noteq__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) ) ) ) ) ).
% noteq_def
tff(fact_6787_get__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H2: heap_ext(product_unit),A3: array(A)] : array_get(A,H2,A3) = map(nat,A,from_nat(A),arrays(product_unit,H2,typerep_typerep(A,type2(A)),addr_of_array(A,A3))) ) ).
% get_def
tff(fact_6788_Heap_Oempty__def,axiom,
empty = heap_ext2(product_unit,aTP_Lamp_rc(typerep,fun(nat,list(nat))),aTP_Lamp_rd(typerep,fun(nat,nat)),zero_zero(nat),product_Unity) ).
% Heap.empty_def
tff(fact_6789_coset__def,axiom,
! [A: $tType,Xs: list(A)] : coset(A,Xs) = aa(set(A),set(A),uminus_uminus(set(A)),set2(A,Xs)) ).
% coset_def
tff(fact_6790_compl__coset,axiom,
! [A: $tType,Xs: list(A)] : aa(set(A),set(A),uminus_uminus(set(A)),coset(A,Xs)) = set2(A,Xs) ).
% compl_coset
tff(fact_6791_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),product_Pair(A,A,A3),B2),transitive_trancl(A,collect(product_prod(A,A),product_case_prod(A,A,bool,R))))) ) ).
% Nitpick.tranclp_unfold
tff(fact_6792_Field__natLeq__on,axiom,
! [N: nat] : field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N)))) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N)) ).
% Field_natLeq_on
tff(fact_6793_less__nat__rel,axiom,
ord_less(nat) = transitive_tranclp(nat,aTP_Lamp_nl(nat,fun(nat,bool))) ).
% less_nat_rel
tff(fact_6794_tranclp__power,axiom,
! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,P),X),Y))
<=> ? [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
& pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N5),P),X),Y)) ) ) ).
% tranclp_power
tff(fact_6795_underS__Field3,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( ( field2(A,R) != bot_bot(set(A)) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R,A3)),field2(A,R))) ) ).
% underS_Field3
tff(fact_6796_underS__Field2,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] :
( pp(member(A,A3,field2(A,R)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),order_underS(A,R,A3)),field2(A,R))) ) ).
% underS_Field2
tff(fact_6797_natLeq__on__Well__order,axiom,
! [N: nat] : order_well_order_on(nat,field2(nat,collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N)))),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N)))) ).
% natLeq_on_Well_order
tff(fact_6798_Quotient3__int,axiom,
quotient3(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ) ).
% Quotient3_int
tff(fact_6799_natLeq__on__well__order__on,axiom,
! [N: nat] : order_well_order_on(nat,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),N)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),N)))) ).
% natLeq_on_well_order_on
tff(fact_6800_embedS__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,field2(A,R),R)
=> ( bNF_Wellorder_embedS(A,B,R,R4,F)
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),image(A,B,F,field2(A,R))),field2(B,R4))) ) ) ).
% embedS_Field
tff(fact_6801_list__all__length,axiom,
! [A: $tType,P: fun(A,bool),Xs: list(A)] :
( list_all(A,P,Xs)
<=> ! [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N5))) ) ) ).
% list_all_length
tff(fact_6802_ofilter__subset__embedS,axiom,
! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B5: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A5)
=> ( order_ofilter(A,R,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
<=> bNF_Wellorder_embedS(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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_re(set(A),fun(A,set(A)),A5))),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B5,aTP_Lamp_re(set(A),fun(A,set(A)),B5))),id(A)) ) ) ) ) ).
% ofilter_subset_embedS
tff(fact_6803_embedS__iff,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R4: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,field2(A,R),R)
=> ( bNF_Wellorder_embed(A,B,R,R4,F)
=> ( bNF_Wellorder_embedS(A,B,R,R4,F)
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),image(A,B,F,field2(A,R))),field2(B,R4))) ) ) ) ).
% embedS_iff
tff(fact_6804_ofilterIncl__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),bool,aTP_Lamp_rf(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R))) ).
% ofilterIncl_def
tff(fact_6805_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B5: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A5)
=> ( order_ofilter(A,R,B5)
=> ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),B5))
<=> bNF_Wellorder_embedS(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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_re(set(A),fun(A,set(A)),A5))),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B5,aTP_Lamp_re(set(A),fun(A,set(A)),B5))),id(A)) )
& ( ( A5 = B5 )
<=> bNF_Wellorder_iso(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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_re(set(A),fun(A,set(A)),A5))),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B5,aTP_Lamp_re(set(A),fun(A,set(A)),B5))),id(A)) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
tff(fact_6806_ofilter__ordLess,axiom,
! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),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))),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_re(set(A),fun(A,set(A)),A5)))),R),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).
% ofilter_ordLess
tff(fact_6807_ofilter__subset__ordLess,axiom,
! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B5: set(A)] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(A,R,A5)
=> ( order_ofilter(A,R,B5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),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))),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_re(set(A),fun(A,set(A)),A5)))),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))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,B5,aTP_Lamp_re(set(A),fun(A,set(A)),B5)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).
% ofilter_subset_ordLess
tff(fact_6808_bsqr__ofilter,axiom,
! [A: $tType,R: set(product_prod(A,A)),D4: set(product_prod(A,A))] :
( order_well_order_on(A,field2(A,R),R)
=> ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R),D4)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less(set(product_prod(A,A))),D4),product_Sigma(A,A,field2(A,R),aTP_Lamp_rg(set(product_prod(A,A)),fun(A,set(A)),R))))
=> ( ~ ? [A6: A] : field2(A,R) = order_under(A,R,A6)
=> ? [A7: set(A)] :
( order_ofilter(A,R,A7)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A7),field2(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))),D4),product_Sigma(A,A,A7,aTP_Lamp_re(set(A),fun(A,set(A)),A7)))) ) ) ) ) ) ).
% bsqr_ofilter
tff(fact_6809_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R4: 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))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R4),R),bNF_We4044943003108391690rdLess(A,B)))
<=> ? [P5: set(product_prod(B,B))] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),field2(B,P5)),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))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R4),P5),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))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B)),P5),R),bNF_We4044943003108391690rdLess(B,B))) ) ) ).
% internalize_ordLess
tff(fact_6810_of__int__code_I1_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: num] : aa(int,A,ring_1_of_int(A),neg(K)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).
% of_int_code(1)
tff(fact_6811_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_6812_less__int__code_I3_J,axiom,
! [L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),neg(L))) ).
% less_int_code(3)
tff(fact_6813_less__int__code_I7_J,axiom,
! [K: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),zero_zero(int))) ).
% less_int_code(7)
tff(fact_6814_less__eq__int__code_I7_J,axiom,
! [K: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),zero_zero(int))) ).
% less_eq_int_code(7)
tff(fact_6815_less__eq__int__code_I3_J,axiom,
! [L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),neg(L))) ).
% less_eq_int_code(3)
tff(fact_6816_nat__code_I1_J,axiom,
! [K: num] : aa(int,nat,nat2,neg(K)) = zero_zero(nat) ).
% nat_code(1)
tff(fact_6817_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_6818_less__eq__int__code_I9_J,axiom,
! [K: num,L: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),neg(L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),L),K)) ) ).
% less_eq_int_code(9)
tff(fact_6819_less__int__code_I9_J,axiom,
! [K: num,L: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),neg(L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),L),K)) ) ).
% less_int_code(9)
tff(fact_6820_Int_Osub__code_I4_J,axiom,
! [N: num] : sub(one2,bit0(N)) = neg(bitM(N)) ).
% Int.sub_code(4)
tff(fact_6821_Int_Osub__code_I5_J,axiom,
! [N: num] : sub(one2,aa(num,num,bit1,N)) = neg(bit0(N)) ).
% Int.sub_code(5)
tff(fact_6822_Int_Osub__code_I1_J,axiom,
sub(one2,one2) = zero_zero(int) ).
% Int.sub_code(1)
tff(fact_6823_Int_Osub__def,axiom,
! [M: num,N: num] : sub(M,N) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) ).
% Int.sub_def
tff(fact_6824_minus__int__code_I6_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M)),neg(N)) = sub(N,M) ).
% minus_int_code(6)
tff(fact_6825_Int_Osub__code_I9_J,axiom,
! [M: num,N: num] : sub(bit0(M),aa(num,num,bit1,N)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),dup(sub(M,N))),one_one(int)) ).
% Int.sub_code(9)
tff(fact_6826_Int_Osub__code_I8_J,axiom,
! [M: num,N: num] : sub(aa(num,num,bit1,M),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_6827_Int_Odup__code_I1_J,axiom,
dup(zero_zero(int)) = zero_zero(int) ).
% Int.dup_code(1)
tff(fact_6828_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_6829_Int_Odup__code_I3_J,axiom,
! [N: num] : dup(neg(N)) = neg(bit0(N)) ).
% Int.dup_code(3)
tff(fact_6830_Int_Osub__code_I6_J,axiom,
! [M: num,N: num] : sub(bit0(M),bit0(N)) = dup(sub(M,N)) ).
% Int.sub_code(6)
tff(fact_6831_Int_Osub__code_I7_J,axiom,
! [M: num,N: num] : sub(aa(num,num,bit1,M),aa(num,num,bit1,N)) = dup(sub(M,N)) ).
% Int.sub_code(7)
tff(fact_6832_Int_Osub__code_I2_J,axiom,
! [M: num] : sub(bit0(M),one2) = aa(num,int,pos,bitM(M)) ).
% Int.sub_code(2)
tff(fact_6833_Int_Osub__code_I3_J,axiom,
! [M: num] : sub(aa(num,num,bit1,M),one2) = aa(num,int,pos,bit0(M)) ).
% Int.sub_code(3)
tff(fact_6834_less__int__code_I8_J,axiom,
! [K: num,L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),aa(num,int,pos,L))) ).
% less_int_code(8)
tff(fact_6835_less__int__code_I6_J,axiom,
! [K: num,L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),neg(L))) ).
% less_int_code(6)
tff(fact_6836_less__eq__int__code_I6_J,axiom,
! [K: num,L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),neg(L))) ).
% less_eq_int_code(6)
tff(fact_6837_less__eq__int__code_I8_J,axiom,
! [K: num,L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),aa(num,int,pos,L))) ).
% less_eq_int_code(8)
tff(fact_6838_Int_ONeg__def,axiom,
! [N: num] : neg(N) = aa(int,int,uminus_uminus(int),aa(num,int,pos,N)) ).
% Int.Neg_def
tff(fact_6839_uminus__int__code_I2_J,axiom,
! [M: num] : aa(int,int,uminus_uminus(int),aa(num,int,pos,M)) = neg(M) ).
% uminus_int_code(2)
tff(fact_6840_uminus__int__code_I3_J,axiom,
! [M: num] : aa(int,int,uminus_uminus(int),neg(M)) = aa(num,int,pos,M) ).
% uminus_int_code(3)
tff(fact_6841_nat__code_I3_J,axiom,
! [K: num] : aa(int,nat,nat2,aa(num,int,pos,K)) = aa(num,nat,nat_of_num,K) ).
% nat_code(3)
tff(fact_6842_times__int__code_I3_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,M)),aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_int_code(3)
tff(fact_6843_Int_OPos__def,axiom,
pos = numeral_numeral(int) ).
% Int.Pos_def
tff(fact_6844_less__eq__int__code_I2_J,axiom,
! [L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(num,int,pos,L))) ).
% less_eq_int_code(2)
tff(fact_6845_less__eq__int__code_I4_J,axiom,
! [K: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),zero_zero(int))) ).
% less_eq_int_code(4)
tff(fact_6846_of__int__code_I3_J,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,pos,K)) = aa(num,A,numeral_numeral(A),K) ) ).
% of_int_code(3)
tff(fact_6847_one__int__code,axiom,
one_one(int) = aa(num,int,pos,one2) ).
% one_int_code
tff(fact_6848_plus__int__code_I3_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,M)),aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% plus_int_code(3)
tff(fact_6849_less__int__code_I4_J,axiom,
! [K: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),zero_zero(int))) ).
% less_int_code(4)
tff(fact_6850_less__int__code_I2_J,axiom,
! [L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(num,int,pos,L))) ).
% less_int_code(2)
tff(fact_6851_less__eq__int__code_I5_J,axiom,
! [K: num,L: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),aa(num,int,pos,L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),K),L)) ) ).
% less_eq_int_code(5)
tff(fact_6852_less__int__code_I5_J,axiom,
! [K: num,L: num] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),aa(num,int,pos,L)))
<=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),K),L)) ) ).
% less_int_code(5)
tff(fact_6853_minus__int__code_I3_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,M)),aa(num,int,pos,N)) = sub(M,N) ).
% minus_int_code(3)
tff(fact_6854_Int_Odup__code_I2_J,axiom,
! [N: num] : dup(aa(num,int,pos,N)) = aa(num,int,pos,bit0(N)) ).
% Int.dup_code(2)
tff(fact_6855_minus__int__code_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,M)),neg(N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% minus_int_code(4)
tff(fact_6856_minus__int__code_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M)),aa(num,int,pos,N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).
% minus_int_code(5)
tff(fact_6857_times__int__code_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,M)),neg(N)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_int_code(4)
tff(fact_6858_times__int__code_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M)),aa(num,int,pos,N)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_int_code(5)
tff(fact_6859_times__int__code_I6_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M)),neg(N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).
% times_int_code(6)
tff(fact_6860_plus__int__code_I5_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M)),aa(num,int,pos,N)) = sub(N,M) ).
% plus_int_code(5)
tff(fact_6861_plus__int__code_I4_J,axiom,
! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,M)),neg(N)) = sub(M,N) ).
% plus_int_code(4)
tff(fact_6862_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_6863_Least__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_rh(fun(A,bool),fun(A,bool),P)) ) ).
% Least_def
tff(fact_6864_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_6865_Greatest__def,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_ri(fun(A,bool),fun(A,bool),P)) ) ).
% Greatest_def
tff(fact_6866_ord_OLeast__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),P: fun(A,bool)] : aa(fun(A,bool),A,least(A,Less_eq),P) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_rj(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P)) ).
% ord.Least_def
tff(fact_6867_ord_OLeast_Ocong,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] : least(A,Less_eq) = least(A,Less_eq) ).
% ord.Least.cong
tff(fact_6868_GreatestI__nat,axiom,
! [P: fun(nat,bool),K: nat,B2: nat] :
( pp(aa(nat,bool,P,K))
=> ( ! [Y4: nat] :
( pp(aa(nat,bool,P,Y4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
=> pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).
% GreatestI_nat
tff(fact_6869_Greatest__le__nat,axiom,
! [P: fun(nat,bool),K: nat,B2: nat] :
( pp(aa(nat,bool,P,K))
=> ( ! [Y4: nat] :
( pp(aa(nat,bool,P,Y4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),order_Greatest(nat,P))) ) ) ).
% Greatest_le_nat
tff(fact_6870_GreatestI__ex__nat,axiom,
! [P: fun(nat,bool),B2: nat] :
( ? [X_1: nat] : pp(aa(nat,bool,P,X_1))
=> ( ! [Y4: nat] :
( pp(aa(nat,bool,P,Y4))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
=> pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).
% GreatestI_ex_nat
tff(fact_6871_GreatestI2__order,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
=> ( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> ( ! [Y3: A] :
( pp(aa(A,bool,P,Y3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X2)) )
=> pp(aa(A,bool,Q,X2)) ) )
=> pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).
% GreatestI2_order
tff(fact_6872_Greatest__equality,axiom,
! [A: $tType] :
( order(A)
=> ! [P: fun(A,bool),X: A] :
( pp(aa(A,bool,P,X))
=> ( ! [Y4: A] :
( pp(aa(A,bool,P,Y4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
=> ( order_Greatest(A,P) = X ) ) ) ) ).
% Greatest_equality
tff(fact_6873_sum__encode__def,axiom,
! [X: sum_sum(nat,nat)] : aa(sum_sum(nat,nat),nat,nat_sum_encode,X) = sum_case_sum(nat,nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aTP_Lamp_rk(nat,nat),X) ).
% sum_encode_def
tff(fact_6874_disjnt__ge__max,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y6: set(A),X6: set(A)] :
( pp(aa(set(A),bool,finite_finite2(A),Y6))
=> ( ! [X2: A] :
( pp(member(A,X2,X6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y6)),X2)) )
=> disjnt(A,X6,Y6) ) ) ) ).
% disjnt_ge_max
tff(fact_6875_inj__sum__encode,axiom,
! [A5: set(sum_sum(nat,nat))] : inj_on(sum_sum(nat,nat),nat,nat_sum_encode,A5) ).
% inj_sum_encode
tff(fact_6876_sum__encode__eq,axiom,
! [X: sum_sum(nat,nat),Y: sum_sum(nat,nat)] :
( ( aa(sum_sum(nat,nat),nat,nat_sum_encode,X) = aa(sum_sum(nat,nat),nat,nat_sum_encode,Y) )
<=> ( X = Y ) ) ).
% sum_encode_eq
tff(fact_6877_bij__sum__encode,axiom,
bij_betw(sum_sum(nat,nat),nat,nat_sum_encode,top_top(set(sum_sum(nat,nat))),top_top(set(nat))) ).
% bij_sum_encode
tff(fact_6878_surj__sum__encode,axiom,
image(sum_sum(nat,nat),nat,nat_sum_encode,top_top(set(sum_sum(nat,nat)))) = top_top(set(nat)) ).
% surj_sum_encode
tff(fact_6879_int__encode__def,axiom,
! [I: int] : aa(int,nat,nat_int_encode,I) = aa(sum_sum(nat,nat),nat,nat_sum_encode,if(sum_sum(nat,nat),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I),aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),aa(int,nat,nat2,I)),aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),I)),one_one(int)))))) ).
% int_encode_def
tff(fact_6880_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top.axioms(2)
tff(fact_6881_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [A5: set(B),B5: set(C),G: fun(sum_sum(B,C),A)] :
( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( pp(aa(set(C),bool,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),groups7121269368397514597t_prod(sum_sum(B,C),A),G),sum_Plus(B,C,A5,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),aa(fun(B,sum_sum(B,C)),fun(B,A),comp(sum_sum(B,C),A,B,G),sum_Inl(B,C))),A5)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),comp(sum_sum(B,C),A,C,G),sum_Inr(C,B))),B5)) ) ) ) ) ).
% prod.Plus
tff(fact_6882_ordering__top__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Top: A] :
( ordering_top_axioms(A,Less_eq,Top)
<=> ! [A8: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),Top)) ) ).
% ordering_top_axioms_def
tff(fact_6883_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Top: A] :
( ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),Top))
=> ordering_top_axioms(A,Less_eq,Top) ) ).
% ordering_top_axioms.intro
tff(fact_6884_ndepth__Push__Node__aux,axiom,
! [A: $tType,I: nat,F: fun(nat,sum_sum(A,nat)),K: nat] :
( ( case_nat(sum_sum(A,nat),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,I)),F,K) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_rl(fun(nat,sum_sum(A,nat)),fun(nat,bool),F)))),K)) ) ).
% ndepth_Push_Node_aux
tff(fact_6885_le__sum__encode__Inl,axiom,
! [X: nat,Y: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),Y)))) ) ).
% le_sum_encode_Inl
tff(fact_6886_le__sum__encode__Inr,axiom,
! [X: nat,Y: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),Y)))) ) ).
% le_sum_encode_Inr
tff(fact_6887_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X12: A] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X12)) = aa(nat,nat,suc,zero_zero(nat)) ).
% sum.size(3)
tff(fact_6888_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X22: B] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).
% sum.size(4)
tff(fact_6889_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),X) != zero_zero(nat) ).
% sum.size_neq
tff(fact_6890_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),X) != zero_zero(nat) ).
% prod.size_neq
tff(fact_6891_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa4: fun(A,nat),X: fun(B,nat),X12: A] : basic_BNF_size_sum(A,B,Xa4,X,aa(A,sum_sum(A,B),sum_Inl(A,B),X12)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa4,X12)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(1)
tff(fact_6892_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa4: fun(A,nat),X: fun(B,nat),X22: B] : basic_BNF_size_sum(A,B,Xa4,X,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,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(2)
tff(fact_6893_sum__decode__def,axiom,
! [N: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inl(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
=> ( aa(nat,sum_sum(nat,nat),nat_sum_decode,N) = aa(nat,sum_sum(nat,nat),sum_Inr(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).
% sum_decode_def
tff(fact_6894_sum__decode__inverse,axiom,
! [N: nat] : aa(sum_sum(nat,nat),nat,nat_sum_encode,aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) = N ).
% sum_decode_inverse
tff(fact_6895_sum__encode__inverse,axiom,
! [X: sum_sum(nat,nat)] : aa(nat,sum_sum(nat,nat),nat_sum_decode,aa(sum_sum(nat,nat),nat,nat_sum_encode,X)) = X ).
% sum_encode_inverse
tff(fact_6896_sum__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( aa(nat,sum_sum(nat,nat),nat_sum_decode,X) = aa(nat,sum_sum(nat,nat),nat_sum_decode,Y) )
<=> ( X = Y ) ) ).
% sum_decode_eq
tff(fact_6897_inj__sum__decode,axiom,
! [A5: set(nat)] : inj_on(nat,sum_sum(nat,nat),nat_sum_decode,A5) ).
% inj_sum_decode
tff(fact_6898_surj__sum__decode,axiom,
image(nat,sum_sum(nat,nat),nat_sum_decode,top_top(set(nat))) = top_top(set(sum_sum(nat,nat))) ).
% surj_sum_decode
tff(fact_6899_bij__sum__decode,axiom,
bij_betw(nat,sum_sum(nat,nat),nat_sum_decode,top_top(set(nat)),top_top(set(sum_sum(nat,nat)))) ).
% bij_sum_decode
tff(fact_6900_int__decode__def,axiom,
! [N: nat] : aa(nat,int,nat_int_decode,N) = sum_case_sum(nat,int,nat,semiring_1_of_nat(int),aTP_Lamp_rm(nat,int),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ).
% int_decode_def
tff(fact_6901_Node__def,axiom,
! [A: $tType,B: $tType] : old_Node(B,A) = collect(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_rn(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool)) ).
% Node_def
tff(fact_6902_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)),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat),aTP_Lamp_ro(nat,sum_sum(A,nat))),A3),old_Node(A,B))) ).
% Node_K0_I
tff(fact_6903_Push__neq__K0,axiom,
! [A: $tType,K: nat,F: fun(nat,sum_sum(A,nat))] :
~ ! [X2: nat] : old_Push(A,aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,K)),F,X2) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).
% Push_neq_K0
tff(fact_6904_ndepth__def,axiom,
! [B: $tType,A: $tType,N: old_node(A,B)] : old_ndepth(A,B,N) = aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat,product_case_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),nat,aTP_Lamp_rq(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat))),old_Rep_Node(A,B,N)) ).
% ndepth_def
tff(fact_6905_ndepth__K0,axiom,
! [A: $tType,B: $tType,X: 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)),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),aTP_Lamp_rr(nat,sum_sum(B,nat))),X))) = zero_zero(nat) ).
% ndepth_K0
tff(fact_6906_ndepth__Push__Node,axiom,
! [B: $tType,A: $tType,I: nat,N: old_node(A,B)] : old_ndepth(A,B,aa(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,I))),N)) = aa(nat,nat,suc,old_ndepth(A,B,N)) ).
% ndepth_Push_Node
tff(fact_6907_Atom__def,axiom,
! [B: $tType,A: $tType,X3: sum_sum(A,nat)] : old_Atom(A,B,X3) = insert(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)),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),aTP_Lamp_rr(nat,sum_sum(B,nat))),X3)),bot_bot(set(old_node(A,B)))) ).
% Atom_def
tff(fact_6908_Scons__def,axiom,
! [B: $tType,A: $tType,M7: set(old_node(A,B)),N6: set(old_node(A,B))] : old_Scons(A,B,M7,N6) = 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))),image(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))),M7)),image(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)))),N6)) ).
% Scons_def
tff(fact_6909_ntrunc__0,axiom,
! [B: $tType,A: $tType,M7: set(old_node(A,B))] : old_ntrunc(A,B,zero_zero(nat),M7) = bot_bot(set(old_node(A,B))) ).
% ntrunc_0
tff(fact_6910_ntrunc__Atom,axiom,
! [B: $tType,A: $tType,K: nat,A3: sum_sum(A,nat)] : old_ntrunc(A,B,aa(nat,nat,suc,K),old_Atom(A,B,A3)) = old_Atom(A,B,A3) ).
% ntrunc_Atom
tff(fact_6911_ntrunc__Scons,axiom,
! [B: $tType,A: $tType,K: nat,M7: set(old_node(A,B)),N6: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,K),old_Scons(A,B,M7,N6)) = old_Scons(A,B,old_ntrunc(A,B,K,M7),old_ntrunc(A,B,K,N6)) ).
% ntrunc_Scons
tff(fact_6912_ntrunc__def,axiom,
! [B: $tType,A: $tType,K: nat,N6: set(old_node(A,B))] : old_ntrunc(A,B,K,N6) = collect(old_node(A,B),aa(set(old_node(A,B)),fun(old_node(A,B),bool),aTP_Lamp_rs(nat,fun(set(old_node(A,B)),fun(old_node(A,B),bool)),K),N6)) ).
% ntrunc_def
tff(fact_6913_ntrunc__one__In0,axiom,
! [B: $tType,A: $tType,M7: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In0(A,B,M7)) = bot_bot(set(old_node(A,B))) ).
% ntrunc_one_In0
tff(fact_6914_ntrunc__one__In1,axiom,
! [B: $tType,A: $tType,M7: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat)),old_In1(A,B,M7)) = bot_bot(set(old_node(A,B))) ).
% ntrunc_one_In1
tff(fact_6915_ntrunc__In0,axiom,
! [B: $tType,A: $tType,K: nat,M7: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,aa(nat,nat,suc,K)),old_In0(A,B,M7)) = old_In0(A,B,old_ntrunc(A,B,aa(nat,nat,suc,K),M7)) ).
% ntrunc_In0
tff(fact_6916_ntrunc__In1,axiom,
! [B: $tType,A: $tType,K: nat,M7: set(old_node(A,B))] : old_ntrunc(A,B,aa(nat,nat,suc,aa(nat,nat,suc,K)),old_In1(A,B,M7)) = old_In1(A,B,old_ntrunc(A,B,aa(nat,nat,suc,K),M7)) ).
% ntrunc_In1
tff(fact_6917_ntrunc__Leaf,axiom,
! [B: $tType,A: $tType,K: nat,A3: A] : old_ntrunc(A,B,aa(nat,nat,suc,K),old_Leaf(A,B,A3)) = old_Leaf(A,B,A3) ).
% ntrunc_Leaf
tff(fact_6918_In1__def,axiom,
! [B: $tType,A: $tType,M7: set(old_node(A,B))] : old_In1(A,B,M7) = old_Scons(A,B,old_Numb(A,B,one_one(nat)),M7) ).
% In1_def
tff(fact_6919_ntrunc__Numb,axiom,
! [A: $tType,B: $tType,K: nat,I: nat] : old_ntrunc(A,B,aa(nat,nat,suc,K),old_Numb(A,B,I)) = old_Numb(A,B,I) ).
% ntrunc_Numb
tff(fact_6920_In0__def,axiom,
! [B: $tType,A: $tType,M7: set(old_node(A,B))] : old_In0(A,B,M7) = old_Scons(A,B,old_Numb(A,B,zero_zero(nat)),M7) ).
% In0_def
tff(fact_6921_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
( ordering(A,Less_eq,Less)
=> ( ordering_top_axioms(A,Less_eq,Top)
=> ordering_top(A,Less_eq,Less,Top) ) ) ).
% ordering_top.intro
tff(fact_6922_ordering__top__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
<=> ( ordering(A,Less_eq,Less)
& ordering_top_axioms(A,Less_eq,Top) ) ) ).
% ordering_top_def
tff(fact_6923_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),Top: A] :
( ordering_top(A,Less_eq,Less,Top)
=> ordering(A,Less_eq,Less) ) ).
% ordering_top.axioms(1)
tff(fact_6924_ordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
| ( A6 = B4 ) ) )
=> ( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),A6)) )
=> ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),A6))
=> ( ! [A6: A,B4: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),C3)) ) )
=> ordering(A,Less_eq,Less) ) ) ) ) ).
% ordering_strictI
tff(fact_6925_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( ( A3 != B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2)) ) ) ) ).
% ordering.not_eq_order_implies_strict
tff(fact_6926_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
tff(fact_6927_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
tff(fact_6928_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
tff(fact_6929_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3))
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
tff(fact_6930_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( ordering(A,Less_eq,Less)
=> ( ( A3 = B2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3)) ) ) ) ).
% ordering.eq_iff
tff(fact_6931_ordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,aTP_Lamp_rt(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_rt(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
=> ordering(A,Less_eq,Less) ) ).
% ordering_dualI
tff(fact_6932_order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.ordering_axioms
tff(fact_6933_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( order(A)
=> ordering(A,aTP_Lamp_ru(A,fun(A,bool)),aTP_Lamp_rv(A,fun(A,bool))) ) ).
% dual_order.ordering_axioms
tff(fact_6934_nth__item_Oelims,axiom,
! [A: $tType] :
( countable(A)
=> ! [X: nat,Y: set(old_node(A,product_unit))] :
( ( nth_item(A,X) = Y )
=> ( ( ( X = zero_zero(nat) )
=> ( Y != undefined(set(old_node(A,product_unit))) ) )
=> ~ ! [N3: nat] :
( ( X = aa(nat,nat,suc,N3) )
=> ( Y != sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_ry(nat,set(old_node(A,product_unit))),aTP_Lamp_sc(nat,set(old_node(A,product_unit))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N3)) ) ) ) ) ) ).
% nth_item.elims
tff(fact_6935_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_6936_group__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A6) = A6
=> ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A6)),A6) = Z2
=> group_axioms(A,F,Z2,Inverse) ) ) ).
% group_axioms.intro
tff(fact_6937_group__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,Inverse: fun(A,A)] :
( group_axioms(A,F,Z2,Inverse)
<=> ( ! [A8: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A8) = A8
& ! [A8: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A8)),A8) = Z2 ) ) ).
% group_axioms_def
tff(fact_6938_nth__item_Osimps_I1_J,axiom,
! [A: $tType] :
( countable(A)
=> ( nth_item(A,zero_zero(nat)) = undefined(set(old_node(A,product_unit))) ) ) ).
% nth_item.simps(1)
tff(fact_6939_nth__item_Osimps_I2_J,axiom,
! [A: $tType] :
( countable(A)
=> ! [N: nat] : nth_item(A,aa(nat,nat,suc,N)) = sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_ry(nat,set(old_node(A,product_unit))),aTP_Lamp_sc(nat,set(old_node(A,product_unit))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ) ).
% nth_item.simps(2)
tff(fact_6940_nth__item_Opelims,axiom,
! [A: $tType] :
( countable(A)
=> ! [X: nat,Y: set(old_node(A,product_unit))] :
( ( nth_item(A,X) = Y )
=> ( accp(nat,nth_item_rel,X)
=> ( ( ( X = zero_zero(nat) )
=> ( ( Y = undefined(set(old_node(A,product_unit))) )
=> ~ accp(nat,nth_item_rel,zero_zero(nat)) ) )
=> ~ ! [N3: nat] :
( ( X = aa(nat,nat,suc,N3) )
=> ( ( Y = sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_ry(nat,set(old_node(A,product_unit))),aTP_Lamp_sc(nat,set(old_node(A,product_unit))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N3)) )
=> ~ accp(nat,nth_item_rel,aa(nat,nat,suc,N3)) ) ) ) ) ) ) ).
% nth_item.pelims
tff(fact_6941_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( countable(A)
=> ! [N: nat] :
( accp(nat,nth_item_rel,aa(nat,nat,suc,N))
=> ( nth_item(A,aa(nat,nat,suc,N)) = sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_ry(nat,set(old_node(A,product_unit))),aTP_Lamp_sc(nat,set(old_node(A,product_unit))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N)) ) ) ) ).
% nth_item.psimps(2)
tff(fact_6942_nth__item_Opsimps_I1_J,axiom,
! [A: $tType] :
( countable(A)
=> ( 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_6943_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_6944_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_6945_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semigroup(A,sup_sup(A)) ) ).
% sup.semigroup_axioms
tff(fact_6946_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_6947_semigroup_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A6: A,B4: A,C3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A6),B4)),C3) = aa(A,A,aa(A,fun(A,A),F,A6),aa(A,A,aa(A,fun(A,A),F,B4),C3))
=> semigroup(A,F) ) ).
% semigroup.intro
tff(fact_6948_semigroup__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semigroup(A,F)
<=> ! [A8: A,B8: A,C4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A8),B8)),C4) = aa(A,A,aa(A,fun(A,A),F,A8),aa(A,A,aa(A,fun(A,A),F,B8),C4)) ) ).
% semigroup_def
tff(fact_6949_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_6950_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_6951_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_min(A)) ) ).
% min.semigroup_axioms
tff(fact_6952_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semigroup(A,ord_max(A)) ) ).
% max.semigroup_axioms
tff(fact_6953_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_add(A)
=> semigroup(A,plus_plus(A)) ) ).
% add.semigroup_axioms
tff(fact_6954_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( semigroup_mult(A)
=> semigroup(A,times_times(A)) ) ).
% mult.semigroup_axioms
tff(fact_6955_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semigroup(A,inf_inf(A)) ) ).
% inf.semigroup_axioms
tff(fact_6956_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_6957_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_6958_monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( monoid_axioms(A,F,Z2)
<=> ( ! [A8: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A8) = A8
& ! [A8: A] : aa(A,A,aa(A,fun(A,A),F,A8),Z2) = A8 ) ) ).
% monoid_axioms_def
tff(fact_6959_monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,Z2),A6) = A6
=> ( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),Z2) = A6
=> monoid_axioms(A,F,Z2) ) ) ).
% monoid_axioms.intro
tff(fact_6960_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_6961_nths__nths,axiom,
! [A: $tType,Xs: list(A),A5: set(nat),B5: set(nat)] : nths(A,nths(A,Xs,A5),B5) = nths(A,Xs,collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_se(set(nat),fun(set(nat),fun(nat,bool)),A5),B5))) ).
% nths_nths
tff(fact_6962_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,Less_eq,Less)
=> ordering_axioms(A,Less_eq,Less) ) ).
% ordering.axioms(2)
tff(fact_6963_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Less_eq: fun(A,fun(A,bool))] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
& ( A6 != B4 ) ) )
=> ( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),A6))
=> ( A6 = B4 ) ) )
=> ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_axioms.intro
tff(fact_6964_ordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering_axioms(A,Less_eq,Less)
<=> ( ! [A8: A,B8: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A8),B8))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),B8))
& ( A8 != B8 ) ) )
& ! [A8: A,B8: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),B8))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B8),A8))
=> ( A8 = B8 ) ) ) ) ) ).
% ordering_axioms_def
tff(fact_6965_ordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& ordering_axioms(A,Less_eq,Less) ) ) ).
% ordering_def
tff(fact_6966_ordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( partial_preordering(A,Less_eq)
=> ( ordering_axioms(A,Less_eq,Less)
=> ordering(A,Less_eq,Less) ) ) ).
% ordering.intro
tff(fact_6967_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,aTP_Lamp_pj(A,fun(A,bool))) ) ).
% dual_order.partial_preordering_axioms
tff(fact_6968_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> partial_preordering(A,ord_less_eq(A)) ) ).
% order.partial_preordering_axioms
tff(fact_6969_partial__preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] :
( partial_preordering(A,Less_eq)
<=> ( ! [A8: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),A8))
& ! [A8: A,B8: A,C4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),B8))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B8),C4))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),C4)) ) ) ) ) ).
% partial_preordering_def
tff(fact_6970_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( partial_preordering(A,Less_eq)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),C2)) ) ) ) ).
% partial_preordering.trans
tff(fact_6971_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool))] :
( ! [A6: A] : pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),A6))
=> ( ! [A6: A,B4: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),C3)) ) )
=> partial_preordering(A,Less_eq) ) ) ).
% partial_preordering.intro
tff(fact_6972_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),A3: A] :
( partial_preordering(A,Less_eq)
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),A3)) ) ).
% partial_preordering.refl
tff(fact_6973_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% ordering.axioms(1)
tff(fact_6974_Abs__int__cases,axiom,
! [X: int] :
~ ! [Y4: set(product_prod(nat,nat))] :
( ( X = aa(set(product_prod(nat,nat)),int,abs_int,Y4) )
=> ~ pp(member(set(product_prod(nat,nat)),Y4,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool)))) ) ).
% Abs_int_cases
tff(fact_6975_Abs__int__induct,axiom,
! [P: fun(int,bool),X: int] :
( ! [Y4: set(product_prod(nat,nat))] :
( pp(member(set(product_prod(nat,nat)),Y4,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))))
=> pp(aa(int,bool,P,aa(set(product_prod(nat,nat)),int,abs_int,Y4))) )
=> pp(aa(int,bool,P,X)) ) ).
% Abs_int_induct
tff(fact_6976_Abs__int__inject,axiom,
! [X: set(product_prod(nat,nat)),Y: set(product_prod(nat,nat))] :
( pp(member(set(product_prod(nat,nat)),X,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))))
=> ( pp(member(set(product_prod(nat,nat)),Y,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))))
=> ( ( aa(set(product_prod(nat,nat)),int,abs_int,X) = aa(set(product_prod(nat,nat)),int,abs_int,Y) )
<=> ( X = Y ) ) ) ) ).
% Abs_int_inject
tff(fact_6977_Abs__Integ__def,axiom,
abs_Integ = quot_abs(product_prod(nat,nat),int,intrel,abs_int) ).
% Abs_Integ_def
tff(fact_6978_Abs__int__inverse,axiom,
! [Y: set(product_prod(nat,nat))] :
( pp(member(set(product_prod(nat,nat)),Y,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))))
=> ( aa(int,set(product_prod(nat,nat)),rep_int,aa(set(product_prod(nat,nat)),int,abs_int,Y)) = Y ) ) ).
% Abs_int_inverse
tff(fact_6979_Rep__int__inject,axiom,
! [X: int,Y: int] :
( ( aa(int,set(product_prod(nat,nat)),rep_int,X) = aa(int,set(product_prod(nat,nat)),rep_int,Y) )
<=> ( X = Y ) ) ).
% Rep_int_inject
tff(fact_6980_Rep__int__inverse,axiom,
! [X: int] : aa(set(product_prod(nat,nat)),int,abs_int,aa(int,set(product_prod(nat,nat)),rep_int,X)) = X ).
% Rep_int_inverse
tff(fact_6981_Rep__int,axiom,
! [X: int] : pp(member(set(product_prod(nat,nat)),aa(int,set(product_prod(nat,nat)),rep_int,X),collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool)))) ).
% Rep_int
tff(fact_6982_Rep__int__cases,axiom,
! [Y: set(product_prod(nat,nat))] :
( pp(member(set(product_prod(nat,nat)),Y,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))))
=> ~ ! [X2: int] : Y != aa(int,set(product_prod(nat,nat)),rep_int,X2) ) ).
% Rep_int_cases
tff(fact_6983_Rep__int__induct,axiom,
! [Y: set(product_prod(nat,nat)),P: fun(set(product_prod(nat,nat)),bool)] :
( pp(member(set(product_prod(nat,nat)),Y,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))))
=> ( ! [X2: int] : pp(aa(set(product_prod(nat,nat)),bool,P,aa(int,set(product_prod(nat,nat)),rep_int,X2)))
=> pp(aa(set(product_prod(nat,nat)),bool,P,Y)) ) ) ).
% Rep_int_induct
tff(fact_6984_Rep__Integ__def,axiom,
rep_Integ = quot_rep(int,product_prod(nat,nat),rep_int) ).
% Rep_Integ_def
tff(fact_6985_type__definition__int,axiom,
type_definition(int,set(product_prod(nat,nat)),rep_int,abs_int,collect(set(product_prod(nat,nat)),aTP_Lamp_sf(set(product_prod(nat,nat)),bool))) ).
% type_definition_int
tff(fact_6986_Domainp__cr__rat,axiom,
! [X3: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,domainp(product_prod(int,int),rat,pcr_rat),X3))
<=> ( aa(product_prod(int,int),int,product_snd(int,int),X3) != zero_zero(int) ) ) ).
% Domainp_cr_rat
tff(fact_6987_semilattice__neutr_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice(A,F)
=> ( comm_monoid(A,F,Z2)
=> semilattice_neutr(A,F,Z2) ) ) ).
% semilattice_neutr.intro
tff(fact_6988_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> semilattice(A,inf_inf(A)) ) ).
% inf.semilattice_axioms
tff(fact_6989_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
=> semilattice(A,F) ) ).
% semilattice_order.axioms(1)
tff(fact_6990_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_max(A)) ) ).
% max.semilattice_axioms
tff(fact_6991_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> semilattice(A,ord_min(A)) ) ).
% min.semilattice_axioms
tff(fact_6992_semilattice_Oright__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).
% semilattice.right_idem
tff(fact_6993_semilattice_Oleft__idem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).
% semilattice.left_idem
tff(fact_6994_semilattice_Oidem,axiom,
! [A: $tType,F: fun(A,fun(A,A)),A3: A] :
( semilattice(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,A3),A3) = A3 ) ) ).
% semilattice.idem
tff(fact_6995_semilattice__set_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> lattic149705377957585745ce_set(A,F) ) ).
% semilattice_set.intro
tff(fact_6996_semilattice__set_Oaxioms,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( lattic149705377957585745ce_set(A,F)
=> semilattice(A,F) ) ).
% semilattice_set.axioms
tff(fact_6997_semilattice__set__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( lattic149705377957585745ce_set(A,F)
<=> semilattice(A,F) ) ).
% semilattice_set_def
tff(fact_6998_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
=> semilattice(A,F) ) ).
% semilattice_neutr.axioms(1)
tff(fact_6999_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> semilattice(A,sup_sup(A)) ) ).
% sup.semilattice_axioms
tff(fact_7000_semilattice__neutr__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( semilattice_neutr(A,F,Z2)
<=> ( semilattice(A,F)
& comm_monoid(A,F,Z2) ) ) ).
% semilattice_neutr_def
tff(fact_7001_semilattice__order_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice(A,F)
=> ( semila6385135966242565138axioms(A,F,Less_eq,Less)
=> semilattice_order(A,F,Less_eq,Less) ) ) ).
% semilattice_order.intro
tff(fact_7002_semilattice__order__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
<=> ( semilattice(A,F)
& semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).
% semilattice_order_def
tff(fact_7003_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),F: fun(A,fun(A,A)),Less: fun(A,fun(A,bool))] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
<=> ( A6 = aa(A,A,aa(A,fun(A,A),F,A6),B4) ) )
=> ( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
<=> ( ( A6 = aa(A,A,aa(A,fun(A,A),F,A6),B4) )
& ( A6 != B4 ) ) )
=> semila6385135966242565138axioms(A,F,Less_eq,Less) ) ) ).
% semilattice_order_axioms.intro
tff(fact_7004_semilattice__order__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semila6385135966242565138axioms(A,F,Less_eq,Less)
<=> ( ! [A8: A,B8: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),B8))
<=> ( A8 = aa(A,A,aa(A,fun(A,A),F,A8),B8) ) )
& ! [A8: A,B8: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A8),B8))
<=> ( ( A8 = aa(A,A,aa(A,fun(A,A),F,A8),B8) )
& ( A8 != B8 ) ) ) ) ) ).
% semilattice_order_axioms_def
tff(fact_7005_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( semilattice_order(A,F,Less_eq,Less)
=> semila6385135966242565138axioms(A,F,Less_eq,Less) ) ).
% semilattice_order.axioms(2)
tff(fact_7006_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_7007_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_7008_comm__monoid__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( comm_monoid_axioms(A,F,Z2)
<=> ! [A8: A] : aa(A,A,aa(A,fun(A,A),F,A8),Z2) = A8 ) ).
% comm_monoid_axioms_def
tff(fact_7009_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),Z2) = A6
=> comm_monoid_axioms(A,F,Z2) ) ).
% comm_monoid_axioms.intro
tff(fact_7010_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_7011_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_7012_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_7013_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> abel_semigroup(A,F) ) ).
% semilattice.axioms(1)
tff(fact_7014_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> abel_semigroup(A,sup_sup(A)) ) ).
% sup.abel_semigroup_axioms
tff(fact_7015_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_7016_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_7017_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_7018_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> abel_semigroup(A,Conj) ) ).
% abstract_boolean_algebra.axioms(1)
tff(fact_7019_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> abel_semigroup(A,Disj) ) ).
% abstract_boolean_algebra.axioms(2)
tff(fact_7020_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_min(A)) ) ).
% min.abel_semigroup_axioms
tff(fact_7021_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> abel_semigroup(A,plus_plus(A)) ) ).
% add.abel_semigroup_axioms
tff(fact_7022_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( linorder(A)
=> abel_semigroup(A,ord_max(A)) ) ).
% max.abel_semigroup_axioms
tff(fact_7023_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> abel_semigroup(A,times_times(A)) ) ).
% mult.abel_semigroup_axioms
tff(fact_7024_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> abel_semigroup(A,inf_inf(A)) ) ).
% inf.abel_semigroup_axioms
tff(fact_7025_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_7026_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_7027_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_7028_abel__semigroup__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_s757365448890700780axioms(A,F)
<=> ! [A8: A,B8: A] : aa(A,A,aa(A,fun(A,A),F,A8),B8) = aa(A,A,aa(A,fun(A,A),F,B8),A8) ) ).
% abel_semigroup_axioms_def
tff(fact_7029_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A6: A,B4: A] : aa(A,A,aa(A,fun(A,A),F,A6),B4) = aa(A,A,aa(A,fun(A,A),F,B4),A6)
=> abel_s757365448890700780axioms(A,F) ) ).
% abel_semigroup_axioms.intro
tff(fact_7030_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_7031_semilattice_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( abel_semigroup(A,F)
=> ( semilattice_axioms(A,F)
=> semilattice(A,F) ) ) ).
% semilattice.intro
tff(fact_7032_semilattice__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
<=> ( abel_semigroup(A,F)
& semilattice_axioms(A,F) ) ) ).
% semilattice_def
tff(fact_7033_semilattice__axioms_Ointro,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( ! [A6: A] : aa(A,A,aa(A,fun(A,A),F,A6),A6) = A6
=> semilattice_axioms(A,F) ) ).
% semilattice_axioms.intro
tff(fact_7034_semilattice__axioms__def,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice_axioms(A,F)
<=> ! [A8: A] : aa(A,A,aa(A,fun(A,A),F,A8),A8) = A8 ) ).
% semilattice_axioms_def
tff(fact_7035_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> semilattice_axioms(A,F) ) ).
% semilattice.axioms(2)
tff(fact_7036_abstract__boolean__algebra__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
<=> ( abel_semigroup(A,Conj)
& abel_semigroup(A,Disj)
& boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ).
% abstract_boolean_algebra_def
tff(fact_7037_abstract__boolean__algebra_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( abel_semigroup(A,Conj)
=> ( abel_semigroup(A,Disj)
=> ( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
=> boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ) ) ).
% abstract_boolean_algebra.intro
tff(fact_7038_abstract__boolean__algebra__axioms__def,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One)
<=> ( ! [X5: A,Y5: A,Z7: A] : aa(A,A,aa(A,fun(A,A),Conj,X5),aa(A,A,aa(A,fun(A,A),Disj,Y5),Z7)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X5),Y5)),aa(A,A,aa(A,fun(A,A),Conj,X5),Z7))
& ! [X5: A,Y5: A,Z7: A] : aa(A,A,aa(A,fun(A,A),Disj,X5),aa(A,A,aa(A,fun(A,A),Conj,Y5),Z7)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X5),Y5)),aa(A,A,aa(A,fun(A,A),Disj,X5),Z7))
& ! [X5: A] : aa(A,A,aa(A,fun(A,A),Conj,X5),One) = X5
& ! [X5: A] : aa(A,A,aa(A,fun(A,A),Disj,X5),Zero) = X5
& ! [X5: A] : aa(A,A,aa(A,fun(A,A),Conj,X5),aa(A,A,Compl,X5)) = Zero
& ! [X5: A] : aa(A,A,aa(A,fun(A,A),Disj,X5),aa(A,A,Compl,X5)) = One ) ) ).
% abstract_boolean_algebra_axioms_def
tff(fact_7039_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),One: A,Zero: A,Compl: fun(A,A)] :
( ! [X2: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,aa(A,fun(A,A),Disj,Y4),Z3)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X2),Y4)),aa(A,A,aa(A,fun(A,A),Conj,X2),Z3))
=> ( ! [X2: A,Y4: A,Z3: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),aa(A,A,aa(A,fun(A,A),Conj,Y4),Z3)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,X2),Y4)),aa(A,A,aa(A,fun(A,A),Disj,X2),Z3))
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),One) = X2
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),Zero) = X2
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Conj,X2),aa(A,A,Compl,X2)) = Zero
=> ( ! [X2: A] : aa(A,A,aa(A,fun(A,A),Disj,X2),aa(A,A,Compl,X2)) = One
=> boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
tff(fact_7040_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
=> boolea6902313364301356556axioms(A,Conj,Disj,Compl,Zero,One) ) ).
% abstract_boolean_algebra.axioms(3)
tff(fact_7041_typerep_Osize__neq,axiom,
! [X: typerep] : aa(typerep,nat,size_size(typerep),X) != zero_zero(nat) ).
% typerep.size_neq
tff(fact_7042_tuple__isomorphism_Osize__neq,axiom,
! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism(A,B,C)] : aa(tuple_isomorphism(A,B,C),nat,size_size(tuple_isomorphism(A,B,C)),X) != zero_zero(nat) ).
% tuple_isomorphism.size_neq
tff(fact_7043_tuple__isomorphism_Osize_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,X12: fun(A,product_prod(B,C)),X22: fun(product_prod(B,C),A)] : aa(tuple_isomorphism(A,B,C),nat,size_size(tuple_isomorphism(A,B,C)),tuple_1188178415141063261rphism(A,B,C,X12,X22)) = aa(nat,nat,suc,zero_zero(nat)) ).
% tuple_isomorphism.size(2)
tff(fact_7044_typerep_Osize_I2_J,axiom,
! [X12: literal,X22: list(typerep)] : aa(typerep,nat,size_size(typerep),typerep3(X12,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(typerep,size_size(typerep),X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% typerep.size(2)
tff(fact_7045_typerep_Osize__gen,axiom,
! [X12: literal,X22: list(typerep)] : aa(typerep,nat,size_typerep,typerep3(X12,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(typerep,size_typerep,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% typerep.size_gen
tff(fact_7046_tuple__isomorphism_Osize__gen,axiom,
! [B: $tType,C: $tType,A: $tType,Xb3: fun(A,nat),Xa4: fun(B,nat),X: fun(C,nat),X12: fun(A,product_prod(B,C)),X22: fun(product_prod(B,C),A)] : tuple_9181185373184732606rphism(A,B,C,Xb3,Xa4,X,tuple_1188178415141063261rphism(A,B,C,X12,X22)) = aa(nat,nat,suc,zero_zero(nat)) ).
% tuple_isomorphism.size_gen
tff(fact_7047_typerep__fun__box__def,axiom,
! [A: $tType,B: $tType] :
( ( typerep2(B)
& typerep2(A) )
=> ! [T5: itself(fun_box(A,B))] : typerep_typerep(fun_box(A,B),T5) = typerep3(literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fFalse,fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,fTrue,zero_zero(literal)))))))))))))))),cons(typerep,typerep_typerep(A,type2(A)),cons(typerep,typerep_typerep(B,type2(B)),nil(typerep)))) ) ).
% typerep_fun_box_def
tff(fact_7048_typerep__word__def,axiom,
! [A: $tType] :
( typerep2(A)
=> ! [T5: itself(word(A))] : typerep_typerep(word(A),T5) = typerep3(literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fFalse,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,zero_zero(literal))))))))))))),cons(typerep,typerep_typerep(A,type2(A)),nil(typerep))) ) ).
% typerep_word_def
tff(fact_7049_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,aTP_Lamp_pj(A,fun(A,bool)),aTP_Lamp_pi(A,fun(A,bool))) ) ).
% dual_order.preordering_axioms
tff(fact_7050_irreflp__greater,axiom,
! [A: $tType] :
( preorder(A)
=> irreflp(A,aTP_Lamp_pi(A,fun(A,bool))) ) ).
% irreflp_greater
tff(fact_7051_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,Less_eq,Less)
=> partial_preordering(A,Less_eq) ) ).
% preordering.axioms(1)
tff(fact_7052_irreflp__less,axiom,
! [A: $tType] :
( preorder(A)
=> irreflp(A,ord_less(A)) ) ).
% irreflp_less
tff(fact_7053_preordering__dualI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,aTP_Lamp_rt(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_rt(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
=> preordering(A,Less_eq,Less) ) ).
% preordering_dualI
tff(fact_7054_preordering_Oasym,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),A3)) ) ) ).
% preordering.asym
tff(fact_7055_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A] :
( preordering(A,Less_eq,Less)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),A3)) ) ).
% preordering.irrefl
tff(fact_7056_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2)) ) ) ) ).
% preordering.strict_trans
tff(fact_7057_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2)) ) ) ) ).
% preordering.strict_trans1
tff(fact_7058_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A,C2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),C2)) ) ) ) ).
% preordering.strict_trans2
tff(fact_7059_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B2),A3)) ) ) ) ).
% preordering.strict_iff_not
tff(fact_7060_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),A3: A,B2: A] :
( preordering(A,Less_eq,Less)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A3),B2)) ) ) ).
% preordering.strict_implies_order
tff(fact_7061_preordering__strictI,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
| ( A6 = B4 ) ) )
=> ( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),A6)) )
=> ( ! [A6: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),A6))
=> ( ! [A6: A,B4: A,C3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less,B4),C3))
=> pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),C3)) ) )
=> preordering(A,Less_eq,Less) ) ) ) ) ).
% preordering_strictI
tff(fact_7062_order_Opreordering__axioms,axiom,
! [A: $tType] :
( preorder(A)
=> preordering(A,ord_less_eq(A),ord_less(A)) ) ).
% order.preordering_axioms
tff(fact_7063_preordering__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,Less_eq,Less)
<=> ( partial_preordering(A,Less_eq)
& preordering_axioms(A,Less_eq,Less) ) ) ).
% preordering_def
tff(fact_7064_preordering_Ointro,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( partial_preordering(A,Less_eq)
=> ( preordering_axioms(A,Less_eq,Less)
=> preordering(A,Less_eq,Less) ) ) ).
% preordering.intro
tff(fact_7065_preordering__axioms__def,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering_axioms(A,Less_eq,Less)
<=> ! [A8: A,B8: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A8),B8))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A8),B8))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B8),A8)) ) ) ) ).
% preordering_axioms_def
tff(fact_7066_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: fun(A,fun(A,bool)),Less_eq: fun(A,fun(A,bool))] :
( ! [A6: A,B4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),Less,A6),B4))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,A6),B4))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),Less_eq,B4),A6)) ) )
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering_axioms.intro
tff(fact_7067_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
( preordering(A,Less_eq,Less)
=> preordering_axioms(A,Less_eq,Less) ) ).
% preordering.axioms(2)
tff(fact_7068_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,P3: nat,K: nat,G: fun(nat,A),H2: 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_sg(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z2),K),G),H2)),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_sh(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H2)),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_7069_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)) = image(A,B,F,heap_Time_set_Heap(A,V)) ).
% Heap.set_map
tff(fact_7070_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),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_7071_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),comp(nat,A,nat,G),aTP_Lamp_cq(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_7072_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),comp(nat,A,nat,G),aTP_Lamp_cq(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_7073_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),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
tff(fact_7074_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),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
tff(fact_7075_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),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_7076_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_7077_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),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_cm(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ) ).
% comm_monoid_set.atMost_shift
tff(fact_7078_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),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_cm(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ) ).
% comm_monoid_set.lessThan_Suc_shift
tff(fact_7079_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),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_cm(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ) ).
% comm_monoid_set.atMost_Suc_shift
tff(fact_7080_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_7081_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_7082_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_7083_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_si(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_sk(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_7084_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_cm(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ) ).
% comm_monoid_set.atLeast1_atMost_eq
tff(fact_7085_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),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),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_sl(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F),G)),set_ord_atMost(nat,N)) ) ) ).
% comm_monoid_set.in_pairs_0
tff(fact_7086_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_7087_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_7088_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_7089_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_7090_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,D2: B,G: fun(B,A),H2: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X2))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),D2))
=> ( aa(B,A,G,X2) = aa(B,A,H2,X2) ) ) )
=> ( 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),H2),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ) ).
% comm_monoid_set.ivl_cong
tff(fact_7091_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_sm(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),A3)),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_sk(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F),Z2),A3),N)),set_ord_lessThan(nat,N)) ) ) ).
% comm_monoid_set.nested_swap'
tff(fact_7092_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_cm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.shift_bounds_cl_Suc_ivl
tff(fact_7093_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_cm(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.shift_bounds_Suc_ivl
tff(fact_7094_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),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),set_ord_lessThan(nat,N))),aa(nat,A,G,N)) ) ) ).
% comm_monoid_set.lessThan_Suc
tff(fact_7095_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),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),set_ord_atMost(nat,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ).
% comm_monoid_set.atMost_Suc
tff(fact_7096_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_sn(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_7097_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_so(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_7098_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_7099_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_sp(fun(nat,A),fun(nat,fun(nat,A)),G),N)),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),set_ord_lessThan(nat,N)) ) ) ).
% comm_monoid_set.nat_diff_reindex
tff(fact_7100_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_sq(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),F),Z2),G),K)),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),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ) ).
% comm_monoid_set.nat_group
tff(fact_7101_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_7102_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_7103_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_7104_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_7105_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_7106_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_7107_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_7108_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_cm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).
% comm_monoid_set.Suc_reindex_ivl
tff(fact_7109_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap(A),Xa4: 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,X)))
=> ( pp(member(A,Za,heap_Time_set_Heap(A,Xa4)))
=> ( ( 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),X) = 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),Xa4) )
=> ( X = Xa4 ) ) ) ).
% Heap.inj_map_strong
tff(fact_7110_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X)))
=> ( 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),X) = 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),X) ) ) ).
% Heap.map_cong0
tff(fact_7111_Heap_Omap__cong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ( X = 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),X) = 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_7112_comm__monoid__set_OIf__cases,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A5: set(B),P: fun(B,bool),H2: fun(B,A),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(set(B),bool,finite_finite2(B),A5))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_sr(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H2),G)),A5) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),collect(B,P)))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),aa(set(B),set(B),uminus_uminus(set(B)),collect(B,P))))) ) ) ) ).
% comm_monoid_set.If_cases
tff(fact_7113_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),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
tff(fact_7114_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),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
tff(fact_7115_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_7116_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),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),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_sl(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_7117_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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_gz(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_st(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),G)),set_ord_lessThan(nat,N)) ) ) ).
% comm_monoid_set.triangle_reindex
tff(fact_7118_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),product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_gw(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_st(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),G)),set_ord_atMost(nat,N)) ) ) ).
% comm_monoid_set.triangle_reindex_eq
tff(fact_7119_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),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_7120_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),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_7121_Heap_Oin__rel,axiom,
! [A: $tType,B: $tType,R2: 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),R2),A3),B2))
<=> ? [Z7: heap_Time_Heap(product_prod(A,B))] :
( pp(member(heap_Time_Heap(product_prod(A,B)),Z7,collect(heap_Time_Heap(product_prod(A,B)),aTP_Lamp_su(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R2))))
& ( 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)),Z7) = 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)),Z7) = B2 ) ) ) ).
% Heap.in_rel
tff(fact_7122_Heap_Opred__set,axiom,
! [A: $tType,P: fun(A,bool),X3: 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),X3))
<=> ! [Xa2: A] :
( pp(member(A,Xa2,heap_Time_set_Heap(A,X3)))
=> pp(aa(A,bool,P,Xa2)) ) ) ).
% Heap.pred_set
tff(fact_7123_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R2: 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),R2)) = aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),domainp(A,B,R2)) ).
% Heap.Domainp_rel
tff(fact_7124_Heap_Opred__True,axiom,
! [A: $tType,X3: 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_sv(A,bool)),X3)) ).
% Heap.pred_True
tff(fact_7125_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_7126_Heap_Orel__refl,axiom,
! [B: $tType,Ra: fun(B,fun(B,bool)),X: heap_Time_Heap(B)] :
( ! [X2: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra,X2),X2))
=> 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),Ra),X),X)) ) ).
% Heap.rel_refl
tff(fact_7127_Heap_Orel__transp,axiom,
! [A: $tType,R2: fun(A,fun(A,bool))] :
( transp(A,R2)
=> 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),R2)) ) ).
% Heap.rel_transp
tff(fact_7128_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_7129_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,bool)),Ra: 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))),R2),Ra))
=> 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),R2)),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra))) ) ).
% Heap.rel_mono
tff(fact_7130_Heap_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(B,bool))] :
( bi_total(A,B,R2)
=> bi_total(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),R2)) ) ).
% Heap.bi_total_rel
tff(fact_7131_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_7132_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R2: 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,R2,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),R2),fequal(bool))),heap_Time_pred_Heap(A)),heap_Time_pred_Heap(B))) ).
% Heap.pred_transfer
tff(fact_7133_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F7: $tType,E4: $tType,Rb: fun(A,fun(E4,bool)),Sd: fun(B,fun(F7,bool))] : pp(aa(fun(fun(E4,F7),fun(heap_Time_Heap(E4),heap_Time_Heap(F7))),bool,aa(fun(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B))),fun(fun(fun(E4,F7),fun(heap_Time_Heap(E4),heap_Time_Heap(F7))),bool),bNF_rel_fun(fun(A,B),fun(E4,F7),fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(E4),heap_Time_Heap(F7)),bNF_rel_fun(A,E4,B,F7,Rb,Sd),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(E4),heap_Time_Heap(B),heap_Time_Heap(F7),aa(fun(A,fun(E4,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(E4),bool)),heap_Time_rel_Heap(A,E4),Rb),aa(fun(B,fun(F7,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(F7),bool)),heap_Time_rel_Heap(B,F7),Sd))),heap_Time_map_Heap(A,B)),heap_Time_map_Heap(E4,F7))) ).
% Heap.map_transfer
tff(fact_7134_Heap_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ( X = 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_sw(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),X) = 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_7135_Heap_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: 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),X)),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_sx(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y)) ) ).
% Heap.rel_map(1)
tff(fact_7136_Heap_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: 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),X),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_sy(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y)) ) ).
% Heap.rel_map(2)
tff(fact_7137_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R2: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T5: fun(A,fun(B,bool))] :
( quotient(A,B,R2,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),R2),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_7138_Heap_Opred__map,axiom,
! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X: 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),X)))
<=> 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),comp(B,bool,A,Q),F)),X)) ) ).
% Heap.pred_map
tff(fact_7139_Heap_Opred__mono__strong,axiom,
! [A: $tType,P: fun(A,bool),X: 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),X))
=> ( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X)))
=> ( 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),X)) ) ) ).
% Heap.pred_mono_strong
tff(fact_7140_Heap_Orel__refl__strong,axiom,
! [A: $tType,X: heap_Time_Heap(A),Ra: fun(A,fun(A,bool))] :
( ! [Z3: A] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra,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),Ra),X),X)) ) ).
% Heap.rel_refl_strong
tff(fact_7141_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R2: fun(A,fun(B,bool)),X: heap_Time_Heap(A),Y: heap_Time_Heap(B),Ra: 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),R2),X),Y))
=> ( ! [Z3: A,Yb: B] :
( pp(member(A,Z3,heap_Time_set_Heap(A,X)))
=> ( pp(member(B,Yb,heap_Time_set_Heap(B,Y)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,Z3),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,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),Ra),X),Y)) ) ) ).
% Heap.rel_mono_strong
tff(fact_7142_Heap_Opred__cong,axiom,
! [A: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),P: fun(A,bool),Pa: fun(A,bool)] :
( ( X = 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),X))
<=> 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_7143_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),Y: heap_Time_Heap(B),Xa4: heap_Time_Heap(B),R2: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( ( X = Ya )
=> ( ( Y = Xa4 )
=> ( ! [Z3: A,Yb: B] :
( pp(member(A,Z3,heap_Time_set_Heap(A,Ya)))
=> ( pp(member(B,Yb,heap_Time_set_Heap(B,Xa4)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R2,Z3),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,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),R2),X),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),Ra),Ya),Xa4)) ) ) ) ) ).
% Heap.rel_cong
tff(fact_7144_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_mt(int,fun(int,product_prod(int,int)))) ).
% Fract_def
tff(fact_7145_subset_Osuc__Union__closed__suc,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A)),Y6: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(member(set(set(A)),Y6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ( pred_suc(set(A),A5,ord_less(set(A)),Y6) = Y6 )
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),Y6)) ) ) ) ).
% subset.suc_Union_closed_suc
tff(fact_7146_predicate2I,axiom,
! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
( ! [X2: A,Y4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X2),Y4))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X2),Y4)) )
=> 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))),P),Q)) ) ).
% predicate2I
tff(fact_7147_predicate1I,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
( ! [X2: A] :
( pp(aa(A,bool,P,X2))
=> pp(aa(A,bool,Q,X2)) )
=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q)) ) ).
% predicate1I
tff(fact_7148_of__int__eq__id,axiom,
ring_1_of_int(int) = id(int) ).
% of_int_eq_id
tff(fact_7149_predicate1D,axiom,
! [A: $tType,P: fun(A,bool),Q: fun(A,bool),X: A] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
=> ( pp(aa(A,bool,P,X))
=> pp(aa(A,bool,Q,X)) ) ) ).
% predicate1D
tff(fact_7150_predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),X: A,Y: 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))),P),Q))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).
% predicate2D
tff(fact_7151_rev__predicate1D,axiom,
! [A: $tType,P: fun(A,bool),X: A,Q: fun(A,bool)] :
( pp(aa(A,bool,P,X))
=> ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
=> pp(aa(A,bool,Q,X)) ) ) ).
% rev_predicate1D
tff(fact_7152_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Y: B,Q: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
=> ( 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))),P),Q))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).
% rev_predicate2D
tff(fact_7153_subset_Osuc__Union__closed_Ocases,axiom,
! [A: $tType,A3: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),A3,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ! [X9: set(set(A))] :
( ( A3 = pred_suc(set(A),A5,ord_less(set(A)),X9) )
=> ~ pp(member(set(set(A)),X9,pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) )
=> ~ ! [X9: set(set(set(A)))] :
( ( A3 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X9) )
=> ~ pp(member(set(set(set(A))),X9,pow2(set(set(A)),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))) ) ) ) ).
% subset.suc_Union_closed.cases
tff(fact_7154_subset_Osuc__Union__closed_Osimps,axiom,
! [A: $tType,A3: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),A3,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
<=> ( ? [X10: set(set(A))] :
( ( A3 = pred_suc(set(A),A5,ord_less(set(A)),X10) )
& pp(member(set(set(A)),X10,pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) )
| ? [X10: set(set(set(A)))] :
( ( A3 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X10) )
& pp(member(set(set(set(A))),X10,pow2(set(set(A)),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))) ) ) ) ).
% subset.suc_Union_closed.simps
tff(fact_7155_subset_Osuc__Union__closed_Osuc,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> pp(member(set(set(A)),pred_suc(set(A),A5,ord_less(set(A)),X6),pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ) ).
% subset.suc_Union_closed.suc
tff(fact_7156_subset_Oeq__suc__Union,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ( pred_suc(set(A),A5,ord_less(set(A)),X6) = X6 )
<=> ( X6 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))) ) ) ) ).
% subset.eq_suc_Union
tff(fact_7157_subset_Osuc__Union__closed__Union,axiom,
! [A: $tType,A5: set(set(A))] : pp(member(set(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))),pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ).
% subset.suc_Union_closed_Union
tff(fact_7158_subset_Osuc__Union__closed__in__carrier,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),A5)) ) ).
% subset.suc_Union_closed_in_carrier
tff(fact_7159_subset_Osuc__Union__closed__total,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A)),Y6: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(member(set(set(A)),Y6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),Y6))
| pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Y6),X6)) ) ) ) ).
% subset.suc_Union_closed_total
tff(fact_7160_subset_Osuc__Union__closed__induct,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A)),Q: fun(set(set(A)),bool)] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ! [X9: set(set(A))] :
( pp(member(set(set(A)),X9,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(aa(set(set(A)),bool,Q,X9))
=> pp(aa(set(set(A)),bool,Q,pred_suc(set(A),A5,ord_less(set(A)),X9))) ) )
=> ( ! [X9: set(set(set(A)))] :
( pp(aa(set(set(set(A))),bool,aa(set(set(set(A))),fun(set(set(set(A))),bool),ord_less_eq(set(set(set(A)))),X9),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ! [X3: set(set(A))] :
( pp(member(set(set(A)),X3,X9))
=> pp(aa(set(set(A)),bool,Q,X3)) )
=> pp(aa(set(set(A)),bool,Q,aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X9))) ) )
=> pp(aa(set(set(A)),bool,Q,X6)) ) ) ) ).
% subset.suc_Union_closed_induct
tff(fact_7161_subset_Osuc__Union__closed__cases,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ! [Y7: set(set(A))] :
( ( X6 = pred_suc(set(A),A5,ord_less(set(A)),Y7) )
=> ~ pp(member(set(set(A)),Y7,pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) )
=> ~ ! [Y7: set(set(set(A)))] :
( ( X6 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),Y7) )
=> ~ pp(aa(set(set(set(A))),bool,aa(set(set(set(A))),fun(set(set(set(A))),bool),ord_less_eq(set(set(set(A)))),Y7),pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ) ) ) ).
% subset.suc_Union_closed_cases
tff(fact_7162_subset_Osuc__Union__closed_OUnion,axiom,
! [A: $tType,X6: set(set(set(A))),A5: set(set(A))] :
( pp(aa(set(set(set(A))),bool,aa(set(set(set(A))),fun(set(set(set(A))),bool),ord_less_eq(set(set(set(A)))),X6),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> pp(member(set(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X6),pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ) ).
% subset.suc_Union_closed.Union
tff(fact_7163_subset_Osuc__Union__closed_Ointros_I2_J,axiom,
! [A: $tType,Xa5: set(set(set(A))),A5: set(set(A))] :
( pp(aa(set(set(set(A))),bool,aa(set(set(set(A))),fun(set(set(set(A))),bool),ord_less_eq(set(set(set(A)))),Xa5),pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> pp(member(set(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),Xa5),pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ) ).
% subset.suc_Union_closed.intros(2)
tff(fact_7164_subset_Osuc__Union__closed__chain,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),X6)) ) ).
% subset.suc_Union_closed_chain
tff(fact_7165_subset_Osuc__Union__closed__empty,axiom,
! [A: $tType,A5: set(set(A))] : pp(member(set(set(A)),bot_bot(set(set(A))),pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ).
% subset.suc_Union_closed_empty
tff(fact_7166_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_7167_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_7168_subset_Osuc__Union__closed__subsetD,axiom,
! [A: $tType,Y6: set(set(A)),X6: set(set(A)),A5: set(set(A))] :
( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Y6),X6))
=> ( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(member(set(set(A)),Y6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ( X6 = Y6 )
| pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A5,ord_less(set(A)),Y6)),X6)) ) ) ) ) ).
% subset.suc_Union_closed_subsetD
tff(fact_7169_subset_Osuc__Union__closed__total_H,axiom,
! [A: $tType,X6: set(set(A)),A5: set(set(A)),Y6: set(set(A))] :
( pp(member(set(set(A)),X6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(member(set(set(A)),Y6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( ! [Z6: set(set(A))] :
( pp(member(set(set(A)),Z6,pred_s596693808085603175closed(set(A),A5,ord_less(set(A)))))
=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Z6),Y6))
=> ( ( Z6 = Y6 )
| pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A5,ord_less(set(A)),Z6)),Y6)) ) ) )
=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),X6),Y6))
| pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pred_suc(set(A),A5,ord_less(set(A)),Y6)),X6)) ) ) ) ) ).
% subset.suc_Union_closed_total'
tff(fact_7170_length__remdups__leq,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)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).
% length_remdups_leq
tff(fact_7171_subset_Osuc__Union__closed__def,axiom,
! [A: $tType,A5: set(set(A))] : pred_s596693808085603175closed(set(A),A5,ord_less(set(A))) = collect(set(set(A)),pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A)))) ).
% subset.suc_Union_closed_def
tff(fact_7172_sorted__remdups,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A)] :
( sorted_wrt(A,ord_less_eq(A),Xs)
=> sorted_wrt(A,ord_less_eq(A),remdups(A,Xs)) ) ) ).
% sorted_remdups
tff(fact_7173_subset_Osuc__Union__closedp__suc__Union__closed__eq,axiom,
! [A: $tType,A5: set(set(A)),X3: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X3))
<=> pp(member(set(set(A)),X3,pred_s596693808085603175closed(set(A),A5,ord_less(set(A))))) ) ).
% subset.suc_Union_closedp_suc_Union_closed_eq
tff(fact_7174_subset_Osuc__Union__closedp_Osuc,axiom,
! [A: $tType,A5: set(set(A)),X6: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X6))
=> pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),pred_suc(set(A),A5,ord_less(set(A)),X6))) ) ).
% subset.suc_Union_closedp.suc
tff(fact_7175_subset_Osuc__Union__closedp_Osimps,axiom,
! [A: $tType,A5: set(set(A)),A3: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),A3))
<=> ( ? [X10: set(set(A))] :
( ( A3 = pred_suc(set(A),A5,ord_less(set(A)),X10) )
& pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X10)) )
| ? [X10: set(set(set(A)))] :
( ( A3 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X10) )
& powp(set(set(A)),pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X10) ) ) ) ).
% subset.suc_Union_closedp.simps
tff(fact_7176_subset_Osuc__Union__closedp_Ointros_I2_J,axiom,
! [A: $tType,A5: set(set(A)),Xa5: set(set(set(A)))] :
( powp(set(set(A)),pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),Xa5)
=> pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),Xa5))) ) ).
% subset.suc_Union_closedp.intros(2)
tff(fact_7177_subset_Osuc__Union__closedp_OUnion,axiom,
! [A: $tType,A5: set(set(A)),X6: set(set(set(A)))] :
( powp(set(set(A)),pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X6)
=> pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X6))) ) ).
% subset.suc_Union_closedp.Union
tff(fact_7178_subset_Osuc__Union__closedp_Ocases,axiom,
! [A: $tType,A5: set(set(A)),A3: set(set(A))] :
( pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),A3))
=> ( ! [X9: set(set(A))] :
( ( A3 = pred_suc(set(A),A5,ord_less(set(A)),X9) )
=> ~ pp(aa(set(set(A)),bool,pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X9)) )
=> ~ ! [X9: set(set(set(A)))] :
( ( A3 = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),X9) )
=> ~ powp(set(set(A)),pred_s7749564232668923593losedp(set(A),A5,ord_less(set(A))),X9) ) ) ) ).
% subset.suc_Union_closedp.cases
tff(fact_7179_int_Odomain,axiom,
! [DR1: fun(nat,bool),DR2: fun(nat,bool)] :
( ( domainp(nat,nat,fequal(nat)) = DR1 )
=> ( ( domainp(nat,nat,fequal(nat)) = DR2 )
=> ( domainp(product_prod(nat,nat),int,pcr_int) = basic_pred_prod(nat,nat,DR1,DR2) ) ) ) ).
% int.domain
tff(fact_7180_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),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_case_Heap(A,B,F,heap_Time_Heap2(A,X)) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B,F,X) ).
% Heap.case
tff(fact_7181_Heap_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: 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,H2,heap_Time_case_Heap(A,B,F,Heap)) = 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_sz(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)),H2),F),Heap) ).
% Heap.case_distrib
tff(fact_7182_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),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_rec_Heap(A,C,F,heap_Time_Heap2(A,X)) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,F,X) ).
% Heap.rec
tff(fact_7183_enumerate__Suc__eq,axiom,
! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,aa(nat,nat,suc,N),Xs) = map(product_prod(nat,A),product_prod(nat,A),product_apfst(nat,nat,A,suc),enumerate(A,N,Xs)) ).
% enumerate_Suc_eq
tff(fact_7184_Heap__ordE,axiom,
! [A: $tType,X: heap_Time_Heap(A),Y: heap_Time_Heap(A),H2: 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),X),Y))
=> ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,X),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))),heap_Time_execute(A,X),H2) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Y),H2) ) ) ) ).
% Heap_ordE
tff(fact_7185_Heap__ordI,axiom,
! [A: $tType,X: heap_Time_Heap(A),Y: heap_Time_Heap(A)] :
( ! [H4: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,X),H4) = 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))),heap_Time_execute(A,X),H4) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Y),H4) ) )
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X),Y)) ) ).
% Heap_ordI
tff(fact_7186_heap_Oleq__refl,axiom,
! [A: $tType,X: 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),X),X)) ).
% heap.leq_refl
tff(fact_7187_heap_Oleq__trans,axiom,
! [A: $tType,X: 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),X),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),X),Z2)) ) ) ).
% heap.leq_trans
tff(fact_7188_heap_Oleq__antisym,axiom,
! [A: $tType,X: 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),X),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),X))
=> ( X = Y ) ) ) ).
% heap.leq_antisym
tff(fact_7189_heap_Olub__least,axiom,
! [A: $tType,A5: set(heap_Time_Heap(A)),Z2: heap_Time_Heap(A)] :
( comple1602240252501008431_chain(heap_Time_Heap(A),heap_Time_Heap_ord(A),A5)
=> ( ! [X2: heap_Time_Heap(A)] :
( pp(member(heap_Time_Heap(A),X2,A5))
=> 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)) )
=> 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),A5)),Z2)) ) ) ).
% heap.lub_least
tff(fact_7190_heap_Olub__upper,axiom,
! [A: $tType,A5: set(heap_Time_Heap(A)),X: heap_Time_Heap(A)] :
( comple1602240252501008431_chain(heap_Time_Heap(A),heap_Time_Heap_ord(A),A5)
=> ( pp(member(heap_Time_Heap(A),X,A5))
=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),heap_Time_Heap_ord(A),X),aa(set(heap_Time_Heap(A)),heap_Time_Heap(A),heap_Time_Heap_lub(A),A5))) ) ) ).
% heap.lub_upper
tff(fact_7191_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_ta(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),C2)) ).
% heap.const_mono
tff(fact_7192_heap__interpretation,axiom,
! [A: $tType] : partia7178651479351089652itions(heap_Time_Heap(A),heap_Time_Heap_ord(A),heap_Time_Heap_lub(A)) ).
% heap_interpretation
tff(fact_7193_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)),C6: 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)),C6,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_tc(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),C6)) ) ) ).
% Heap_Time_Monad.bind_mono
tff(fact_7194_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_td(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_7195_heap_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U3: fun(C,fun(B,heap_Time_Heap(A))),F3: fun(C,C),C6: fun(fun(B,heap_Time_Heap(A)),C),F: C,P: fun(fun(B,heap_Time_Heap(A)),bool)] :
( ! [X2: 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_te(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),F3),C6),X2))
=> ( ( F = aa(fun(B,heap_Time_Heap(A)),C,C6,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_tf(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),F3),C6))) )
=> ( ! [F2: fun(B,heap_Time_Heap(A))] : aa(C,fun(B,heap_Time_Heap(A)),U3,aa(fun(B,heap_Time_Heap(A)),C,C6,F2)) = F2
=> ( 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_tg(B,heap_Time_Heap(A))))
=> ( ! [F2: C] :
( pp(aa(fun(B,heap_Time_Heap(A)),bool,P,aa(C,fun(B,heap_Time_Heap(A)),U3,F2)))
=> pp(aa(fun(B,heap_Time_Heap(A)),bool,P,aa(C,fun(B,heap_Time_Heap(A)),U3,aa(C,C,F3,F2)))) )
=> 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_7196_heap_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F3: fun(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)))] :
( ! [X2: 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_th(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))),F3),X2))
=> ( 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)),F3) = aa(fun(B,heap_Time_Heap(A)),fun(B,heap_Time_Heap(A)),F3,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)),F3)) ) ) ).
% heap.mono_body_fixp
tff(fact_7197_heap_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U3: fun(C,fun(B,heap_Time_Heap(A))),F3: fun(C,C),C6: fun(fun(B,heap_Time_Heap(A)),C),F: C] :
( ! [X2: 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_te(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),F3),C6),X2))
=> ( ( F = aa(fun(B,heap_Time_Heap(A)),C,C6,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_tf(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),F3),C6))) )
=> ( ! [F2: C] : aa(fun(B,heap_Time_Heap(A)),C,C6,aa(C,fun(B,heap_Time_Heap(A)),U3,F2)) = F2
=> ( F = aa(C,C,F3,F) ) ) ) ) ).
% heap.fixp_rule_uc
tff(fact_7198_ATP_Olambda__1,axiom,
! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_iu(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),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(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_7199_ATP_Olambda__2,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_sc(nat,set(old_node(A,product_unit))),Uu) = sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_rz(nat,set(old_node(A,product_unit))),aTP_Lamp_sb(nat,set(old_node(A,product_unit))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).
% ATP.lambda_2
tff(fact_7200_ATP_Olambda__3,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_ry(nat,set(old_node(A,product_unit))),Uu) = sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_rw(nat,set(old_node(A,product_unit))),aTP_Lamp_rx(nat,set(old_node(A,product_unit))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).
% ATP.lambda_3
tff(fact_7201_ATP_Olambda__4,axiom,
! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_kp(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),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).
% ATP.lambda_4
tff(fact_7202_ATP_Olambda__5,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Uu: A] :
( pp(aa(A,bool,aTP_Lamp_kn(A,bool),Uu))
<=> ( pp(member(A,Uu,ring_1_Ints(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Uu)) ) ) ) ).
% ATP.lambda_5
tff(fact_7203_ATP_Olambda__6,axiom,
! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_iv(product_prod(int,int),product_prod(int,int)),Uu) = aa(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_7204_ATP_Olambda__7,axiom,
! [Uu: nat] : aa(nat,int,aTP_Lamp_rm(nat,int),Uu) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Uu))),one_one(int)) ).
% ATP.lambda_7
tff(fact_7205_ATP_Olambda__8,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: nat] :
( pp(aa(nat,bool,aTP_Lamp_cd(nat,bool),Uu))
<=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).
% ATP.lambda_8
tff(fact_7206_ATP_Olambda__9,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_cq(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).
% ATP.lambda_9
tff(fact_7207_ATP_Olambda__10,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_mr(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).
% ATP.lambda_10
tff(fact_7208_ATP_Olambda__11,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_it(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).
% ATP.lambda_11
tff(fact_7209_ATP_Olambda__12,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] :
( pp(aa(A,bool,aTP_Lamp_qr(A,bool),Uu))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uu),one_one(A))) ) ) ).
% ATP.lambda_12
tff(fact_7210_ATP_Olambda__13,axiom,
! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_mu(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uu),zero_zero(nat)) ).
% ATP.lambda_13
tff(fact_7211_ATP_Olambda__14,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_sb(nat,set(old_node(A,product_unit))),Uu) = aa(product_prod(nat,nat),set(old_node(A,product_unit)),product_case_prod(nat,nat,set(old_node(A,product_unit)),aTP_Lamp_sa(nat,fun(nat,set(old_node(A,product_unit))))),aa(nat,product_prod(nat,nat),nat_prod_decode,Uu)) ) ).
% ATP.lambda_14
tff(fact_7212_ATP_Olambda__15,axiom,
! [Uu: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aTP_Lamp_mk(product_prod(int,int),bool),Uu))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(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),Uu)))) ) ).
% ATP.lambda_15
tff(fact_7213_ATP_Olambda__16,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_rk(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)) ).
% ATP.lambda_16
tff(fact_7214_ATP_Olambda__17,axiom,
! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_jo(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).
% ATP.lambda_17
tff(fact_7215_ATP_Olambda__18,axiom,
! [B: $tType,Uu: list(B)] :
( pp(aa(list(B),bool,aTP_Lamp_jp(list(B),bool),Uu))
<=> ( Uu != nil(B) ) ) ).
% ATP.lambda_18
tff(fact_7216_ATP_Olambda__19,axiom,
! [A: $tType,Uu: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_jm(list(A),bool),Uu))
<=> ( Uu != nil(A) ) ) ).
% ATP.lambda_19
tff(fact_7217_ATP_Olambda__20,axiom,
! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_mo(nat,set(nat)),Uu) = collect(nat,aTP_Lamp_bx(nat,fun(nat,bool),Uu)) ).
% ATP.lambda_20
tff(fact_7218_ATP_Olambda__21,axiom,
! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_jn(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ).
% ATP.lambda_21
tff(fact_7219_ATP_Olambda__22,axiom,
! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_jl(list(A),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu)) ).
% ATP.lambda_22
tff(fact_7220_ATP_Olambda__23,axiom,
! [Uu: num] : aa(num,option(num),aTP_Lamp_ni(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).
% ATP.lambda_23
tff(fact_7221_ATP_Olambda__24,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hn(int,fun(int,product_prod(int,int))),Uu) = product_Pair(int,int,aa(int,int,uminus_uminus(int),Uu)) ).
% ATP.lambda_24
tff(fact_7222_ATP_Olambda__25,axiom,
! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_im(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).
% ATP.lambda_25
tff(fact_7223_ATP_Olambda__26,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_rz(nat,set(old_node(A,product_unit))),Uu) = old_Leaf(A,product_unit,aa(nat,A,from_nat(A),Uu)) ) ).
% ATP.lambda_26
tff(fact_7224_ATP_Olambda__27,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_rx(nat,set(old_node(A,product_unit))),Uu) = old_In1(A,product_unit,nth_item(A,Uu)) ) ).
% ATP.lambda_27
tff(fact_7225_ATP_Olambda__28,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_rw(nat,set(old_node(A,product_unit))),Uu) = old_In0(A,product_unit,nth_item(A,Uu)) ) ).
% ATP.lambda_28
tff(fact_7226_ATP_Olambda__29,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Uu: A] :
( pp(aa(A,bool,aTP_Lamp_lr(A,bool),Uu))
<=> ? [N5: int] :
( ( Uu = aa(int,A,ring_1_of_int(A),N5) )
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N5)) ) ) ) ).
% ATP.lambda_29
tff(fact_7227_ATP_Olambda__30,axiom,
! [Uu: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,aTP_Lamp_sf(set(product_prod(nat,nat)),bool),Uu))
<=> ? [X5: product_prod(nat,nat)] :
( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X5),X5))
& ( Uu = collect(product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X5)) ) ) ) ).
% ATP.lambda_30
tff(fact_7228_ATP_Olambda__31,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_rn(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),Uu))
<=> ? [F4: fun(nat,sum_sum(B,nat)),X5: sum_sum(A,nat),K4: nat] :
( ( Uu = aa(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),F4),X5) )
& ( aa(nat,sum_sum(B,nat),F4,K4) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ) ).
% ATP.lambda_31
tff(fact_7229_ATP_Olambda__32,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_tg(B,heap_Time_Heap(A)),Uu) = heap_Time_Heap2(A,aTP_Lamp_oa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).
% ATP.lambda_32
tff(fact_7230_ATP_Olambda__33,axiom,
! [B: $tType,Uu: nat] : aa(nat,sum_sum(B,nat),aTP_Lamp_rr(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ).
% ATP.lambda_33
tff(fact_7231_ATP_Olambda__34,axiom,
! [A: $tType,Uu: nat] : aa(nat,sum_sum(A,nat),aTP_Lamp_ro(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).
% ATP.lambda_34
tff(fact_7232_ATP_Olambda__35,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ds(nat,fun(nat,A),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).
% ATP.lambda_35
tff(fact_7233_ATP_Olambda__36,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_is(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),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),bit0(one2))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).
% ATP.lambda_36
tff(fact_7234_ATP_Olambda__37,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_ie(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_37
tff(fact_7235_ATP_Olambda__38,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_gv(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_38
tff(fact_7236_ATP_Olambda__39,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ib(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),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(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_39
tff(fact_7237_ATP_Olambda__40,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_mt(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),product_Pair(int,int,zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),product_Pair(int,int,Uu),Uua)) ).
% ATP.lambda_40
tff(fact_7238_ATP_Olambda__41,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gq(nat,fun(nat,A),Uu),Uua) = if(A,fNot(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).
% ATP.lambda_41
tff(fact_7239_ATP_Olambda__42,axiom,
! [A: $tType] :
( ( lattice(A)
& order_top(A) )
=> ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_mq(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).
% ATP.lambda_42
tff(fact_7240_ATP_Olambda__43,axiom,
! [A: $tType] :
( ( lattice(A)
& order_bot(A) )
=> ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_kz(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),bot_bot(A)) ) ).
% ATP.lambda_43
tff(fact_7241_ATP_Olambda__44,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_pa(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_44
tff(fact_7242_ATP_Olambda__45,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_pr(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).
% ATP.lambda_45
tff(fact_7243_ATP_Olambda__46,axiom,
! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_iz(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iy(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ).
% ATP.lambda_46
tff(fact_7244_ATP_Olambda__47,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hd(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_hc(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).
% ATP.lambda_47
tff(fact_7245_ATP_Olambda__48,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gy(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_gx(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).
% ATP.lambda_48
tff(fact_7246_ATP_Olambda__49,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_iw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(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_49
tff(fact_7247_ATP_Olambda__50,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gh(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_50
tff(fact_7248_ATP_Olambda__51,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gj(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),bit0(one2))),Uua)) ) ).
% ATP.lambda_51
tff(fact_7249_ATP_Olambda__52,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_ix(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),Uu),Uua) = aa(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_52
tff(fact_7250_ATP_Olambda__53,axiom,
! [Uu: rat,Uua: int] :
( pp(aa(int,bool,aTP_Lamp_op(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_53
tff(fact_7251_ATP_Olambda__54,axiom,
! [A: $tType] :
( comm_ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gn(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_54
tff(fact_7252_ATP_Olambda__55,axiom,
! [Uu: rat,Uua: product_prod(int,int)] :
( pp(aa(product_prod(int,int),bool,aTP_Lamp_or(rat,fun(product_prod(int,int),bool),Uu),Uua))
<=> ( ( Uu = aa(int,rat,aa(int,fun(int,rat),fract,aa(product_prod(int,int),int,product_fst(int,int),Uua)),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
& 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),Uua)))
& algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).
% ATP.lambda_55
tff(fact_7253_ATP_Olambda__56,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fh(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_56
tff(fact_7254_ATP_Olambda__57,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gp(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),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).
% ATP.lambda_57
tff(fact_7255_ATP_Olambda__58,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fw(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_58
tff(fact_7256_ATP_Olambda__59,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_ir(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)),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),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_59
tff(fact_7257_ATP_Olambda__60,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lu(nat,fun(nat,bool)),Uu),Uua))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
& ( Uu != Uua ) ) ) ).
% ATP.lambda_60
tff(fact_7258_ATP_Olambda__61,axiom,
! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
( pp(aa(set(set(A)),bool,aTP_Lamp_kq(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_61
tff(fact_7259_ATP_Olambda__62,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ga(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).
% ATP.lambda_62
tff(fact_7260_ATP_Olambda__63,axiom,
! [Uu: set(int),Uua: int] :
( pp(aa(int,bool,aTP_Lamp_pd(set(int),fun(int,bool),Uu),Uua))
<=> ( pp(member(int,Uua,Uu))
& ! [X5: int] :
( pp(member(int,X5,Uu))
=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X5),Uua)) ) ) ) ).
% ATP.lambda_63
tff(fact_7261_ATP_Olambda__64,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_nj(set(A),fun(set(A),bool)),Uu),Uua))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uu),Uua))
& pp(aa(set(A),bool,finite_finite2(A),Uua)) ) ) ).
% ATP.lambda_64
tff(fact_7262_ATP_Olambda__65,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_bp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uua)),Uua) ).
% ATP.lambda_65
tff(fact_7263_ATP_Olambda__66,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_fb(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_66
tff(fact_7264_ATP_Olambda__67,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ez(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_67
tff(fact_7265_ATP_Olambda__68,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bv(A,fun(A,bool),Uu),Uua))
<=> ( pp(member(A,Uua,ring_1_Ints(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).
% ATP.lambda_68
tff(fact_7266_ATP_Olambda__69,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: set(A),Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_px(set(A),fun(list(A),bool),Uu),Uua))
<=> ( sorted_wrt(A,ord_less(A),Uua)
& ( set2(A,Uua) = Uu ) ) ) ) ).
% ATP.lambda_69
tff(fact_7267_ATP_Olambda__70,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_mz(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).
% ATP.lambda_70
tff(fact_7268_ATP_Olambda__71,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_nb(nat,fun(nat,bool),Uu),Uua))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).
% ATP.lambda_71
tff(fact_7269_ATP_Olambda__72,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_de(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),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),bit0(one2))),Uua)))) ) ).
% ATP.lambda_72
tff(fact_7270_ATP_Olambda__73,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fu(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),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),bit0(one2))),Uua)))) ) ).
% ATP.lambda_73
tff(fact_7271_ATP_Olambda__74,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_es(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_74
tff(fact_7272_ATP_Olambda__75,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ci(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_75
tff(fact_7273_ATP_Olambda__76,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ev(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_76
tff(fact_7274_ATP_Olambda__77,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ed(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).
% ATP.lambda_77
tff(fact_7275_ATP_Olambda__78,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fr(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_78
tff(fact_7276_ATP_Olambda__79,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fo(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_79
tff(fact_7277_ATP_Olambda__80,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_et(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_80
tff(fact_7278_ATP_Olambda__81,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_rh(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ( pp(aa(A,bool,Uu,Uua))
& ! [Y5: A] :
( pp(aa(A,bool,Uu,Y5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5)) ) ) ) ) ).
% ATP.lambda_81
tff(fact_7279_ATP_Olambda__82,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ri(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ( pp(aa(A,bool,Uu,Uua))
& ! [Y5: A] :
( pp(aa(A,bool,Uu,Y5))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),Uua)) ) ) ) ) ).
% ATP.lambda_82
tff(fact_7280_ATP_Olambda__83,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_cn(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_ord_lessThan(nat,Uua)) ) ).
% ATP.lambda_83
tff(fact_7281_ATP_Olambda__84,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fa(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_ord_lessThan(nat,Uua)) ) ).
% ATP.lambda_84
tff(fact_7282_ATP_Olambda__85,axiom,
! [B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_rp(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu),Uua))
<=> ( aa(nat,sum_sum(B,nat),Uu,Uua) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ).
% ATP.lambda_85
tff(fact_7283_ATP_Olambda__86,axiom,
! [A: $tType,Uu: fun(nat,sum_sum(A,nat)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_rl(fun(nat,sum_sum(A,nat)),fun(nat,bool),Uu),Uua))
<=> ( aa(nat,sum_sum(A,nat),Uu,Uua) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ).
% ATP.lambda_86
tff(fact_7284_ATP_Olambda__87,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(A,A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ps(fun(A,A),fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua)) ) ) ).
% ATP.lambda_87
tff(fact_7285_ATP_Olambda__88,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_en(fun(B,A),fun(B,bool),Uu),Uua))
<=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).
% ATP.lambda_88
tff(fact_7286_ATP_Olambda__89,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,A),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_br(fun(B,A),fun(B,bool),Uu),Uua))
<=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).
% ATP.lambda_89
tff(fact_7287_ATP_Olambda__90,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_ij(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)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ii(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_90
tff(fact_7288_ATP_Olambda__91,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_pw(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_oi(list(A),fun(A,list(A))),Uua),Uu) ).
% ATP.lambda_91
tff(fact_7289_ATP_Olambda__92,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hi(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_92
tff(fact_7290_ATP_Olambda__93,axiom,
! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_md(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),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).
% ATP.lambda_93
tff(fact_7291_ATP_Olambda__94,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_su(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)),collect(product_prod(A,B),product_case_prod(A,B,bool,Uu)))) ) ).
% ATP.lambda_94
tff(fact_7292_ATP_Olambda__95,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lt(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_95
tff(fact_7293_ATP_Olambda__96,axiom,
! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_ms(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_96
tff(fact_7294_ATP_Olambda__97,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_sa(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_97
tff(fact_7295_ATP_Olambda__98,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_gd(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).
% ATP.lambda_98
tff(fact_7296_ATP_Olambda__99,axiom,
! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_jt(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_99
tff(fact_7297_ATP_Olambda__100,axiom,
! [Uu: set(nat),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_kf(set(nat),fun(nat,bool),Uu),Uua))
<=> pp(member(nat,aa(nat,nat,suc,Uua),Uu)) ) ).
% ATP.lambda_100
tff(fact_7298_ATP_Olambda__101,axiom,
! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aTP_Lamp_rb(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_101
tff(fact_7299_ATP_Olambda__102,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_eg(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(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),bit0(one2))),Uua)),one_one(A))) ) ).
% ATP.lambda_102
tff(fact_7300_ATP_Olambda__103,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_ef(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(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),bit0(one2))),Uua)) ) ).
% ATP.lambda_103
tff(fact_7301_ATP_Olambda__104,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aa(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),bit0(one2)))) ) ).
% ATP.lambda_104
tff(fact_7302_ATP_Olambda__105,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_gl(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_105
tff(fact_7303_ATP_Olambda__106,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_il(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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,Uua),zero_zero(nat))) ).
% ATP.lambda_106
tff(fact_7304_ATP_Olambda__107,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_ik(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)),product_Pair(A,product_prod(heap_ext(product_unit),nat),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,Uua),one_one(nat))) ).
% ATP.lambda_107
tff(fact_7305_ATP_Olambda__108,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(A,A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_qc(fun(A,A),fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(A,A,Uu,Uua))) ) ) ).
% ATP.lambda_108
tff(fact_7306_ATP_Olambda__109,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fs(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(A,fun(nat,bool),bit_se5641148757651400278ts_bit(A),Uu),Uua))) ) ).
% ATP.lambda_109
tff(fact_7307_ATP_Olambda__110,axiom,
! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_hp(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),fNot(aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).
% ATP.lambda_110
tff(fact_7308_ATP_Olambda__111,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_na(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).
% ATP.lambda_111
tff(fact_7309_ATP_Olambda__112,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] :
( pp(aa(list(A),bool,aTP_Lamp_jf(nat,fun(list(A),bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).
% ATP.lambda_112
tff(fact_7310_ATP_Olambda__113,axiom,
! [A: $tType] :
( ( semiring_char_0(A)
& semidom_divide(A) )
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_di(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_113
tff(fact_7311_ATP_Olambda__114,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dh(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_114
tff(fact_7312_ATP_Olambda__115,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ct(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_115
tff(fact_7313_ATP_Olambda__116,axiom,
! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_pb(nat,fun(nat,list(nat))),Uu),Uua) = cons(nat,Uu,aa(nat,list(nat),nat_list_decode,Uua)) ).
% ATP.lambda_116
tff(fact_7314_ATP_Olambda__117,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_nl(nat,fun(nat,bool)),Uu),Uua))
<=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).
% ATP.lambda_117
tff(fact_7315_ATP_Olambda__118,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ae(nat,fun(nat,bool)),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).
% ATP.lambda_118
tff(fact_7316_ATP_Olambda__119,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lm(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_119
tff(fact_7317_ATP_Olambda__120,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pu(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_120
tff(fact_7318_ATP_Olambda__121,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_os(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_121
tff(fact_7319_ATP_Olambda__122,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pj(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_122
tff(fact_7320_ATP_Olambda__123,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ml(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_123
tff(fact_7321_ATP_Olambda__124,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ru(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_124
tff(fact_7322_ATP_Olambda__125,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bi(A,fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).
% ATP.lambda_125
tff(fact_7323_ATP_Olambda__126,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_je(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).
% ATP.lambda_126
tff(fact_7324_ATP_Olambda__127,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_an(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).
% ATP.lambda_127
tff(fact_7325_ATP_Olambda__128,axiom,
! [A: $tType] :
( field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oc(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).
% ATP.lambda_128
tff(fact_7326_ATP_Olambda__129,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).
% ATP.lambda_129
tff(fact_7327_ATP_Olambda__130,axiom,
! [A: $tType] :
( bounde4967611905675639751up_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ln(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_130
tff(fact_7328_ATP_Olambda__131,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pv(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_131
tff(fact_7329_ATP_Olambda__132,axiom,
! [A: $tType] :
( order_bot(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ot(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_132
tff(fact_7330_ATP_Olambda__133,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pi(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_133
tff(fact_7331_ATP_Olambda__134,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_pt(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_134
tff(fact_7332_ATP_Olambda__135,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_rv(A,fun(A,bool)),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_135
tff(fact_7333_ATP_Olambda__136,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bh(A,fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).
% ATP.lambda_136
tff(fact_7334_ATP_Olambda__137,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_cv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_137
tff(fact_7335_ATP_Olambda__138,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).
% ATP.lambda_138
tff(fact_7336_ATP_Olambda__139,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_az(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).
% ATP.lambda_139
tff(fact_7337_ATP_Olambda__140,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ku(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).
% ATP.lambda_140
tff(fact_7338_ATP_Olambda__141,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).
% ATP.lambda_141
tff(fact_7339_ATP_Olambda__142,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ce(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).
% ATP.lambda_142
tff(fact_7340_ATP_Olambda__143,axiom,
! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_cb(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).
% ATP.lambda_143
tff(fact_7341_ATP_Olambda__144,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_bx(nat,fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).
% ATP.lambda_144
tff(fact_7342_ATP_Olambda__145,axiom,
! [Uu: int,Uua: int] :
( pp(aa(int,bool,aTP_Lamp_bo(int,fun(int,bool),Uu),Uua))
<=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).
% ATP.lambda_145
tff(fact_7343_ATP_Olambda__146,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bg(A,fun(A,bool),Uu),Uua))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).
% ATP.lambda_146
tff(fact_7344_ATP_Olambda__147,axiom,
! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ls(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uua),Uu) ).
% ATP.lambda_147
tff(fact_7345_ATP_Olambda__148,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_eq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).
% ATP.lambda_148
tff(fact_7346_ATP_Olambda__149,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),nat,aTP_Lamp_bb(array(A),fun(heap_ext(product_unit),nat),Uu),Uua) = aa(array(A),nat,array_length(A,Uua),Uu) ) ).
% ATP.lambda_149
tff(fact_7347_ATP_Olambda__150,axiom,
! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_oi(list(A),fun(A,list(A))),Uu),Uua) = cons(A,Uua,Uu) ).
% ATP.lambda_150
tff(fact_7348_ATP_Olambda__151,axiom,
! [B: $tType,Uu: set(B),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_ol(set(B),fun(B,bool),Uu),Uua))
<=> pp(member(B,Uua,Uu)) ) ).
% ATP.lambda_151
tff(fact_7349_ATP_Olambda__152,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_qf(set(A),fun(A,bool),Uu),Uua))
<=> pp(member(A,Uua,Uu)) ) ) ).
% ATP.lambda_152
tff(fact_7350_ATP_Olambda__153,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_153
tff(fact_7351_ATP_Olambda__154,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_my(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Uua),Uu) ).
% ATP.lambda_154
tff(fact_7352_ATP_Olambda__155,axiom,
! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_jg(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).
% ATP.lambda_155
tff(fact_7353_ATP_Olambda__156,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ow(A,fun(A,bool),Uu),Uua))
<=> ( Uua = Uu ) ) ).
% ATP.lambda_156
tff(fact_7354_ATP_Olambda__157,axiom,
! [A: $tType,Uu: code_natural,Uua: A] :
( pp(aa(A,bool,aa(code_natural,fun(A,bool),aTP_Lamp_qt(code_natural,fun(A,bool)),Uu),Uua))
<=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),Uu)) ) ).
% ATP.lambda_157
tff(fact_7355_ATP_Olambda__158,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_qj(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).
% ATP.lambda_158
tff(fact_7356_ATP_Olambda__159,axiom,
! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_jq(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
<=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).
% ATP.lambda_159
tff(fact_7357_ATP_Olambda__160,axiom,
! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
( pp(aa(product_prod(A,nat),bool,aTP_Lamp_jr(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
<=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).
% ATP.lambda_160
tff(fact_7358_ATP_Olambda__161,axiom,
! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: product_prod(B,A)] : aa(product_prod(B,A),nat,aTP_Lamp_lh(fun(A,nat),fun(product_prod(B,A),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(B,A),A,product_snd(B,A),Uua)) ).
% ATP.lambda_161
tff(fact_7359_ATP_Olambda__162,axiom,
! [B: $tType,A: $tType,Uu: fun(A,nat),Uua: product_prod(A,B)] : aa(product_prod(A,B),nat,aTP_Lamp_li(fun(A,nat),fun(product_prod(A,B),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).
% ATP.lambda_162
tff(fact_7360_ATP_Olambda__163,axiom,
! [Uu: fun(nat,bool),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_qg(fun(nat,bool),fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).
% ATP.lambda_163
tff(fact_7361_ATP_Olambda__164,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_164
tff(fact_7362_ATP_Olambda__165,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_el(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).
% ATP.lambda_165
tff(fact_7363_ATP_Olambda__166,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).
% ATP.lambda_166
tff(fact_7364_ATP_Olambda__167,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_nm(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_min(A),Uu),Uua)) ) ).
% ATP.lambda_167
tff(fact_7365_ATP_Olambda__168,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ns(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).
% ATP.lambda_168
tff(fact_7366_ATP_Olambda__169,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_nt(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).
% ATP.lambda_169
tff(fact_7367_ATP_Olambda__170,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_nu(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).
% ATP.lambda_170
tff(fact_7368_ATP_Olambda__171,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_nq(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) = 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_np(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_171
tff(fact_7369_ATP_Olambda__172,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_no(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) = 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_nn(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_172
tff(fact_7370_ATP_Olambda__173,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_mx(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_mw(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_173
tff(fact_7371_ATP_Olambda__174,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_mc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_mb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_174
tff(fact_7372_ATP_Olambda__175,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_ma(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lz(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_175
tff(fact_7373_ATP_Olambda__176,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_ly(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_lx(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_176
tff(fact_7374_ATP_Olambda__177,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_lw(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_lv(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_177
tff(fact_7375_ATP_Olambda__178,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_lp(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = product_case_prod(nat,nat,product_prod(nat,nat),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_lo(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_178
tff(fact_7376_ATP_Olambda__179,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_ld(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lc(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ).
% ATP.lambda_179
tff(fact_7377_ATP_Olambda__180,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_le(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_la(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ).
% ATP.lambda_180
tff(fact_7378_ATP_Olambda__181,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_lf(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lc(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),top_top(set(B)))) ) ).
% ATP.lambda_181
tff(fact_7379_ATP_Olambda__182,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_lb(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,aa(B,fun(C,A),aTP_Lamp_la(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),top_top(set(C)))) ) ).
% ATP.lambda_182
tff(fact_7380_ATP_Olambda__183,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_nd(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)),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),product_Pair(heap_ext(product_unit),nat,Uua),one_one(nat)))) ).
% ATP.lambda_183
tff(fact_7381_ATP_Olambda__184,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_dc(nat,fun(nat,bool),Uu),Uua))
<=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).
% ATP.lambda_184
tff(fact_7382_ATP_Olambda__185,axiom,
! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_kl(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),image(nat,set(A),Uu,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).
% ATP.lambda_185
tff(fact_7383_ATP_Olambda__186,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dm(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_186
tff(fact_7384_ATP_Olambda__187,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_dl(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_187
tff(fact_7385_ATP_Olambda__188,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_po(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),image(B,A,Uu,Uua)) ) ).
% ATP.lambda_188
tff(fact_7386_ATP_Olambda__189,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_pn(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),image(B,A,Uu,Uua)) ) ).
% ATP.lambda_189
tff(fact_7387_ATP_Olambda__190,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kc(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uu),Uua)) ).
% ATP.lambda_190
tff(fact_7388_ATP_Olambda__191,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kd(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Uua),Uu)) ).
% ATP.lambda_191
tff(fact_7389_ATP_Olambda__192,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ka(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uu),Uua)) ).
% ATP.lambda_192
tff(fact_7390_ATP_Olambda__193,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_kb(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Uua),Uu)) ).
% ATP.lambda_193
tff(fact_7391_ATP_Olambda__194,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_be(set(A),fun(A,bool),Uu),Uua))
<=> ~ pp(member(A,Uua,Uu)) ) ).
% ATP.lambda_194
tff(fact_7392_ATP_Olambda__195,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_qy(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)),product_Pair(product_unit,product_prod(heap_ext(product_unit),nat),product_Unity),aa(nat,product_prod(heap_ext(product_unit),nat),product_Pair(heap_ext(product_unit),nat,Uua),Uu))) ).
% ATP.lambda_195
tff(fact_7393_ATP_Olambda__196,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_pq(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).
% ATP.lambda_196
tff(fact_7394_ATP_Olambda__197,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_qb(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).
% ATP.lambda_197
tff(fact_7395_ATP_Olambda__198,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_ec(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_198
tff(fact_7396_ATP_Olambda__199,axiom,
! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_ax(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).
% ATP.lambda_199
tff(fact_7397_ATP_Olambda__200,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_1(A)
=> ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_ak(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).
% ATP.lambda_200
tff(fact_7398_ATP_Olambda__201,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_du(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).
% ATP.lambda_201
tff(fact_7399_ATP_Olambda__202,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_ks(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).
% ATP.lambda_202
tff(fact_7400_ATP_Olambda__203,axiom,
! [A: $tType,B: $tType] :
( comple489889107523837845lgebra(A)
=> ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ky(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).
% ATP.lambda_203
tff(fact_7401_ATP_Olambda__204,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_eh(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).
% ATP.lambda_204
tff(fact_7402_ATP_Olambda__205,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_co(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_205
tff(fact_7403_ATP_Olambda__206,axiom,
! [A: $tType,B: $tType] :
( comm_ring_1(A)
=> ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ac(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).
% ATP.lambda_206
tff(fact_7404_ATP_Olambda__207,axiom,
! [A: $tType,B: $tType] :
( ring_1(A)
=> ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_dv(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).
% ATP.lambda_207
tff(fact_7405_ATP_Olambda__208,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_nf(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_208
tff(fact_7406_ATP_Olambda__209,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fc(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_209
tff(fact_7407_ATP_Olambda__210,axiom,
! [B: $tType,A: $tType] :
( ordere166539214618696060dd_abs(B)
=> ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_dy(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).
% ATP.lambda_210
tff(fact_7408_ATP_Olambda__211,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_km(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_211
tff(fact_7409_ATP_Olambda__212,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ca(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).
% ATP.lambda_212
tff(fact_7410_ATP_Olambda__213,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_bf(fun(A,bool),fun(A,bool),Uu),Uua))
<=> ~ pp(aa(A,bool,Uu,Uua)) ) ).
% ATP.lambda_213
tff(fact_7411_ATP_Olambda__214,axiom,
! [A: $tType,B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: sum_sum(A,nat)] : aa(sum_sum(A,nat),nat,aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat),aTP_Lamp_rq(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),Uu),Uua) = ord_Least(nat,aTP_Lamp_rp(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu)) ).
% ATP.lambda_214
tff(fact_7412_ATP_Olambda__215,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_qh(nat,fun(nat,set(nat)),Uu),Uua) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ad(nat,fun(nat,bool)),Uu)) ).
% ATP.lambda_215
tff(fact_7413_ATP_Olambda__216,axiom,
! [A: $tType,Uu: list(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_lk(list(A),fun(A,bool),Uu),Uua))
<=> ? [I5: nat] :
( ( Uua = aa(nat,A,nth(A,Uu),I5) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).
% ATP.lambda_216
tff(fact_7414_ATP_Olambda__217,axiom,
! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
( pp(aa(set(B),bool,aTP_Lamp_pp(set(set(B)),fun(set(B),bool),Uu),Uua))
<=> ? [F4: fun(set(B),B)] :
( ( Uua = image(set(B),B,F4,Uu) )
& ! [X5: set(B)] :
( pp(member(set(B),X5,Uu))
=> pp(member(B,aa(set(B),B,F4,X5),X5)) ) ) ) ).
% ATP.lambda_217
tff(fact_7415_ATP_Olambda__218,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: set(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_pk(set(set(A)),fun(set(A),bool),Uu),Uua))
<=> ? [F4: fun(set(A),A)] :
( ( Uua = image(set(A),A,F4,Uu) )
& ! [X5: set(A)] :
( pp(member(set(A),X5,Uu))
=> pp(member(A,aa(set(A),A,F4,X5),X5)) ) ) ) ) ).
% ATP.lambda_218
tff(fact_7416_ATP_Olambda__219,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: set(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_pl(set(set(A)),fun(set(A),bool),Uu),Uua))
<=> ? [F4: fun(set(A),A)] :
( ( Uua = image(set(A),A,F4,Uu) )
& ! [X5: set(A)] :
( pp(member(set(A),X5,Uu))
=> pp(member(A,aa(set(A),A,F4,X5),X5)) ) ) ) ) ).
% ATP.lambda_219
tff(fact_7417_ATP_Olambda__220,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [Uu: set(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_pm(set(set(A)),fun(set(A),bool),Uu),Uua))
<=> ? [F4: fun(set(A),A)] :
( ( Uua = image(set(A),A,F4,Uu) )
& ! [X5: set(A)] :
( pp(member(set(A),X5,Uu))
=> pp(member(A,aa(set(A),A,F4,X5),X5)) ) ) ) ) ).
% ATP.lambda_220
tff(fact_7418_ATP_Olambda__221,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_qa(set(A),fun(set(A),bool),Uu),Uua))
<=> ? [B9: set(A)] :
( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B9) )
& pp(member(set(A),Uu,pow2(A,B9))) ) ) ).
% ATP.lambda_221
tff(fact_7419_ATP_Olambda__222,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_pg(set(A),fun(A,bool),Uu),Uua))
<=> ! [X5: A] :
( pp(member(A,X5,Uu))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X5)) ) ) ) ).
% ATP.lambda_222
tff(fact_7420_ATP_Olambda__223,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_pf(set(A),fun(A,bool),Uu),Uua))
<=> ! [X5: A] :
( pp(member(A,X5,Uu))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X5)) ) ) ) ).
% ATP.lambda_223
tff(fact_7421_ATP_Olambda__224,axiom,
! [A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ph(set(A),fun(A,bool),Uu),Uua))
<=> ! [X5: A] :
( pp(member(A,X5,Uu))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Uua)) ) ) ) ).
% ATP.lambda_224
tff(fact_7422_ATP_Olambda__225,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_pe(set(A),fun(A,bool),Uu),Uua))
<=> ! [X5: A] :
( pp(member(A,X5,Uu))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X5),Uua)) ) ) ) ).
% ATP.lambda_225
tff(fact_7423_ATP_Olambda__226,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_td(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))
<=> ! [X5: A,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: B,N5: nat] :
( heap_Time_effect(B,aa(A,heap_Time_Heap(B),Uua,X5),H6,H7,R5,N5)
=> 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,X5),H6),H7),R5),N5)) ) ) ).
% ATP.lambda_226
tff(fact_7424_ATP_Olambda__227,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_om(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).
% ATP.lambda_227
tff(fact_7425_ATP_Olambda__228,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_rg(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).
% ATP.lambda_228
tff(fact_7426_ATP_Olambda__229,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_id(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),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),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),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),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_229
tff(fact_7427_ATP_Olambda__230,axiom,
! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ia(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),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),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),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_230
tff(fact_7428_ATP_Olambda__231,axiom,
! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ic(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),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),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),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).
% ATP.lambda_231
tff(fact_7429_ATP_Olambda__232,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_go(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),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),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),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).
% ATP.lambda_232
tff(fact_7430_ATP_Olambda__233,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_em(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,member(B,Uub,Uua),aa(B,A,Uu,Uub),zero_zero(A)) ) ).
% ATP.lambda_233
tff(fact_7431_ATP_Olambda__234,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_bq(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_234
tff(fact_7432_ATP_Olambda__235,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dx(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),zero_zero(A)) ) ).
% ATP.lambda_235
tff(fact_7433_ATP_Olambda__236,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_ai(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_236
tff(fact_7434_ATP_Olambda__237,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dw(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),zero_zero(A)) ) ).
% ATP.lambda_237
tff(fact_7435_ATP_Olambda__238,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_ah(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_238
tff(fact_7436_ATP_Olambda__239,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_iq(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),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),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_239
tff(fact_7437_ATP_Olambda__240,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_jw(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),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),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_240
tff(fact_7438_ATP_Olambda__241,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_ip(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),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),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_241
tff(fact_7439_ATP_Olambda__242,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_ek(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),zero_zero(A)) ) ).
% ATP.lambda_242
tff(fact_7440_ATP_Olambda__243,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_jj(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),zero_zero(A)) ) ).
% ATP.lambda_243
tff(fact_7441_ATP_Olambda__244,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_au(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_244
tff(fact_7442_ATP_Olambda__245,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_nk(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_245
tff(fact_7443_ATP_Olambda__246,axiom,
! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_od(fun(A,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uub),Uu),Uua) ).
% ATP.lambda_246
tff(fact_7444_ATP_Olambda__247,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_nr(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) = 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))),product_case_prod(A,product_prod(heap_ext(product_unit),nat),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nq(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))),heap_Time_execute(A,Uu),Uub)) ).
% ATP.lambda_247
tff(fact_7445_ATP_Olambda__248,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_sl(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),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),bit0(one2))),Uub)))) ).
% ATP.lambda_248
tff(fact_7446_ATP_Olambda__249,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lc(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).
% ATP.lambda_249
tff(fact_7447_ATP_Olambda__250,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_hc(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_250
tff(fact_7448_ATP_Olambda__251,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_gx(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_251
tff(fact_7449_ATP_Olambda__252,axiom,
! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ss(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_252
tff(fact_7450_ATP_Olambda__253,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_cj(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_253
tff(fact_7451_ATP_Olambda__254,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_ew(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_254
tff(fact_7452_ATP_Olambda__255,axiom,
! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sj(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_255
tff(fact_7453_ATP_Olambda__256,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_th(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_256
tff(fact_7454_ATP_Olambda__257,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_tb(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_257
tff(fact_7455_ATP_Olambda__258,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_la(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_258
tff(fact_7456_ATP_Olambda__259,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_rt(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_259
tff(fact_7457_ATP_Olambda__260,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_dp(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(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_260
tff(fact_7458_ATP_Olambda__261,axiom,
! [A: $tType,Uu: set(set(A)),Uua: set(set(A)),Uub: set(set(A))] :
( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_ra(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),Uu,ord_less(set(A))),Uub))
& pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),Uua),Uub)) ) ) ).
% ATP.lambda_261
tff(fact_7459_ATP_Olambda__262,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_ck(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_cj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).
% ATP.lambda_262
tff(fact_7460_ATP_Olambda__263,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_ex(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_ew(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).
% ATP.lambda_263
tff(fact_7461_ATP_Olambda__264,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hy(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(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_hx(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_264
tff(fact_7462_ATP_Olambda__265,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hw(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(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_hv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_265
tff(fact_7463_ATP_Olambda__266,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hu(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(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_ht(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_266
tff(fact_7464_ATP_Olambda__267,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hs(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(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_hr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_267
tff(fact_7465_ATP_Olambda__268,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_hm(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aa(int,fun(int,fun(int,bool)),aTP_Lamp_hl(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_268
tff(fact_7466_ATP_Olambda__269,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_hk(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> pp(aa(product_prod(int,int),bool,product_case_prod(int,int,bool,aa(int,fun(int,fun(int,bool)),aTP_Lamp_hj(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_269
tff(fact_7467_ATP_Olambda__270,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_gt(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)),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),product_Pair(heap_ext(product_unit),nat,Uub),one_one(nat))) ) ).
% ATP.lambda_270
tff(fact_7468_ATP_Olambda__271,axiom,
! [A: $tType] :
( field(A)
=> ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ee(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = 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,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).
% ATP.lambda_271
tff(fact_7469_ATP_Olambda__272,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_bt(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_272
tff(fact_7470_ATP_Olambda__273,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_ho(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)),product_Pair(list(A),list(A),Uua),Uub),lex(A,Uu))) ) ) ) ).
% ATP.lambda_273
tff(fact_7471_ATP_Olambda__274,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_bu(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_274
tff(fact_7472_ATP_Olambda__275,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_dn(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)),set2(A,Uub)),Uua)) ) ) ).
% ATP.lambda_275
tff(fact_7473_ATP_Olambda__276,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_do(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)),set2(A,Uub)),Uu)) ) ) ).
% ATP.lambda_276
tff(fact_7474_ATP_Olambda__277,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))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set2(A,Uub)),Uu))
& ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).
% ATP.lambda_277
tff(fact_7475_ATP_Olambda__278,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_cu(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set2(A,Uub)),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).
% ATP.lambda_278
tff(fact_7476_ATP_Olambda__279,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_oj(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set2(A,Uub)),Uu))
& ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).
% ATP.lambda_279
tff(fact_7477_ATP_Olambda__280,axiom,
! [Uu: nat,Uua: nat,Uub: list(nat)] :
( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_bs(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_280
tff(fact_7478_ATP_Olambda__281,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_bz(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_281
tff(fact_7479_ATP_Olambda__282,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_ju(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_282
tff(fact_7480_ATP_Olambda__283,axiom,
! [Uu: nat,Uua: nat,Uub: set(nat)] :
( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_pz(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
<=> ( pp(member(set(nat),Uub,pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
& ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).
% ATP.lambda_283
tff(fact_7481_ATP_Olambda__284,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_ke(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
& pp(member(nat,Uub,Uua)) ) ) ).
% ATP.lambda_284
tff(fact_7482_ATP_Olambda__285,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_jk(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
& pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).
% ATP.lambda_285
tff(fact_7483_ATP_Olambda__286,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_nc(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,set2(A,Uu)))
& pp(aa(A,bool,Uua,Uub)) ) ) ) ).
% ATP.lambda_286
tff(fact_7484_ATP_Olambda__287,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_df(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).
% ATP.lambda_287
tff(fact_7485_ATP_Olambda__288,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_qi(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).
% ATP.lambda_288
tff(fact_7486_ATP_Olambda__289,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ov(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,Uu))
& ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),image(A,B,Uua,Uu)) ) ) ) ) ).
% ATP.lambda_289
tff(fact_7487_ATP_Olambda__290,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_rf(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( order_ofilter(A,Uu,Uua)
& ( Uua != field2(A,Uu) )
& order_ofilter(A,Uu,Uub)
& ( Uub != field2(A,Uu) )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub)) ) ) ).
% ATP.lambda_290
tff(fact_7488_ATP_Olambda__291,axiom,
! [Uu: set(nat),Uua: set(nat),Uub: nat] :
( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_se(set(nat),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(member(nat,Uub,Uu))
& pp(member(nat,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_sd(set(nat),fun(nat,fun(nat,bool)),Uu),Uub))),Uua)) ) ) ).
% ATP.lambda_291
tff(fact_7489_ATP_Olambda__292,axiom,
! [A: $tType,B: $tType,Uu: nat,Uua: set(old_node(A,B)),Uub: old_node(A,B)] :
( pp(aa(old_node(A,B),bool,aa(set(old_node(A,B)),fun(old_node(A,B),bool),aTP_Lamp_rs(nat,fun(set(old_node(A,B)),fun(old_node(A,B),bool)),Uu),Uua),Uub))
<=> ( pp(member(old_node(A,B),Uub,Uua))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),old_ndepth(A,B,Uub)),Uu)) ) ) ).
% ATP.lambda_292
tff(fact_7490_ATP_Olambda__293,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_by(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_293
tff(fact_7491_ATP_Olambda__294,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_mh(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))),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_294
tff(fact_7492_ATP_Olambda__295,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fi(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub)) ).
% ATP.lambda_295
tff(fact_7493_ATP_Olambda__296,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_in(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).
% ATP.lambda_296
tff(fact_7494_ATP_Olambda__297,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_af(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).
% ATP.lambda_297
tff(fact_7495_ATP_Olambda__298,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_io(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).
% ATP.lambda_298
tff(fact_7496_ATP_Olambda__299,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_al(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).
% ATP.lambda_299
tff(fact_7497_ATP_Olambda__300,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_am(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).
% ATP.lambda_300
tff(fact_7498_ATP_Olambda__301,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ag(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).
% ATP.lambda_301
tff(fact_7499_ATP_Olambda__302,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_qo(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)),Uua),Uub))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).
% ATP.lambda_302
tff(fact_7500_ATP_Olambda__303,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_mn(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uua))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uub),Uu)) ) ) ).
% ATP.lambda_303
tff(fact_7501_ATP_Olambda__304,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_mp(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uua))
& pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uub),Uu)) ) ) ).
% ATP.lambda_304
tff(fact_7502_ATP_Olambda__305,axiom,
! [Uu: set(nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_sd(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),Uua)) ) ) ).
% ATP.lambda_305
tff(fact_7503_ATP_Olambda__306,axiom,
! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_hz(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uu,Uub)),aa(A,nat,Uua,Uub)) ).
% ATP.lambda_306
tff(fact_7504_ATP_Olambda__307,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_hq(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uua,Uub)),aa(A,nat,Uu,Uub)) ).
% ATP.lambda_307
tff(fact_7505_ATP_Olambda__308,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_at(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_308
tff(fact_7506_ATP_Olambda__309,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_oy(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,Uu))
& pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_309
tff(fact_7507_ATP_Olambda__310,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_ar(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_310
tff(fact_7508_ATP_Olambda__311,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_jb(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,Uu))
& ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).
% ATP.lambda_311
tff(fact_7509_ATP_Olambda__312,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_ao(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_312
tff(fact_7510_ATP_Olambda__313,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_jc(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(member(B,Uub,Uua))
& ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).
% ATP.lambda_313
tff(fact_7511_ATP_Olambda__314,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_mf(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_314
tff(fact_7512_ATP_Olambda__315,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_mm(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),times_times(nat),Uub),Uu)),Uua)) ) ).
% ATP.lambda_315
tff(fact_7513_ATP_Olambda__316,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_gw(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_316
tff(fact_7514_ATP_Olambda__317,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_gz(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_317
tff(fact_7515_ATP_Olambda__318,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cz(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_318
tff(fact_7516_ATP_Olambda__319,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cx(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_319
tff(fact_7517_ATP_Olambda__320,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cy(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_320
tff(fact_7518_ATP_Olambda__321,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_cw(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_321
tff(fact_7519_ATP_Olambda__322,axiom,
! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iy(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ).
% ATP.lambda_322
tff(fact_7520_ATP_Olambda__323,axiom,
! [A: $tType] :
( archim2362893244070406136eiling(A)
=> ! [Uu: A,Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_bm(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(member(A,Uub,ring_1_Ints(A)))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).
% ATP.lambda_323
tff(fact_7521_ATP_Olambda__324,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_tc(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_tb(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_324
tff(fact_7522_ATP_Olambda__325,axiom,
! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_bn(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_325
tff(fact_7523_ATP_Olambda__326,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: B,Uub: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_ja(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_326
tff(fact_7524_ATP_Olambda__327,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_av(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_327
tff(fact_7525_ATP_Olambda__328,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_fk(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_328
tff(fact_7526_ATP_Olambda__329,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_hh(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_329
tff(fact_7527_ATP_Olambda__330,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ha(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_330
tff(fact_7528_ATP_Olambda__331,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_jd(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_331
tff(fact_7529_ATP_Olambda__332,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ba(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uu,Uub))
=> pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_332
tff(fact_7530_ATP_Olambda__333,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qq(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uu,Uub))
| pp(aa(A,bool,Uua,Uub)) ) ) ) ).
% ATP.lambda_333
tff(fact_7531_ATP_Olambda__334,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_sw(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_334
tff(fact_7532_ATP_Olambda__335,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: B,Uua: fun(B,A),Uub: B] :
( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_mg(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).
% ATP.lambda_335
tff(fact_7533_ATP_Olambda__336,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_eb(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uua,Uub))) ) ).
% ATP.lambda_336
tff(fact_7534_ATP_Olambda__337,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_rj(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uua,Uub))
& ! [Y5: A] :
( pp(aa(A,bool,Uua,Y5))
=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y5)) ) ) ) ).
% ATP.lambda_337
tff(fact_7535_ATP_Olambda__338,axiom,
! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_bl(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_338
tff(fact_7536_ATP_Olambda__339,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_aw(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_339
tff(fact_7537_ATP_Olambda__340,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_fd(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_340
tff(fact_7538_ATP_Olambda__341,axiom,
! [B: $tType,A: $tType] :
( semiring_0(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_bk(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_341
tff(fact_7539_ATP_Olambda__342,axiom,
! [B: $tType,A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kt(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_342
tff(fact_7540_ATP_Olambda__343,axiom,
! [B: $tType,A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kg(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_343
tff(fact_7541_ATP_Olambda__344,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_mj(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_344
tff(fact_7542_ATP_Olambda__345,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_ne(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_345
tff(fact_7543_ATP_Olambda__346,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_qm(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(B,A,Uu,Uub) = Uua ) ) ).
% ATP.lambda_346
tff(fact_7544_ATP_Olambda__347,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_ig(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)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_if(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,map(nat,A,Uua,upt(zero_zero(nat),Uu)),Uub)) ) ).
% ATP.lambda_347
tff(fact_7545_ATP_Olambda__348,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_ih(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)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_if(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_348
tff(fact_7546_ATP_Olambda__349,axiom,
! [A: $tType] :
( field_char_0(A)
=> ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dg(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),Uub))) ) ).
% ATP.lambda_349
tff(fact_7547_ATP_Olambda__350,axiom,
! [A: $tType,B: $tType] :
( semiring_1(A)
=> ! [Uu: fun(B,bool),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ea(fun(B,bool),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uub))),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_350
tff(fact_7548_ATP_Olambda__351,axiom,
! [A: $tType] :
( comple9053668089753744459l_ccpo(A)
=> ! [Uu: fun(A,A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qn(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( ? [X5: A] :
( ( Uub = aa(A,A,Uu,X5) )
& pp(aa(A,bool,Uua,X5)) )
| ? [M9: set(A)] :
( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M9) )
& comple1602240252501008431_chain(A,ord_less_eq(A),M9)
& ! [X5: A] :
( pp(member(A,X5,M9))
=> pp(aa(A,bool,Uua,X5)) ) ) ) ) ) ).
% ATP.lambda_351
tff(fact_7549_ATP_Olambda__352,axiom,
! [A: $tType] :
( ord(A)
=> ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_lj(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
<=> ( ? [Y5: A,Ys3: list(A)] :
( ( Uua = nil(A) )
& ( Uub = cons(A,Y5,Ys3) ) )
| ? [X5: A,Y5: A,Xs3: list(A),Ys3: list(A)] :
( ( Uua = cons(A,X5,Xs3) )
& ( Uub = cons(A,Y5,Ys3) )
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5)) )
| ? [X5: A,Y5: A,Xs3: list(A),Ys3: list(A)] :
( ( Uua = cons(A,X5,Xs3) )
& ( Uub = cons(A,Y5,Ys3) )
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),Y5))
& ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X5))
& pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys3)) ) ) ) ) ).
% ATP.lambda_352
tff(fact_7550_ATP_Olambda__353,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_cl(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_353
tff(fact_7551_ATP_Olambda__354,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_ey(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_354
tff(fact_7552_ATP_Olambda__355,axiom,
! [A: $tType] :
( euclid5411537665997757685th_nat(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gm(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_355
tff(fact_7553_ATP_Olambda__356,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_gg(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_356
tff(fact_7554_ATP_Olambda__357,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_if(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)),product_Pair(array(A),product_prod(heap_ext(product_unit),nat),Uua),aa(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_357
tff(fact_7555_ATP_Olambda__358,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_ii(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)),product_Pair(array(A),product_prod(heap_ext(product_unit),nat),Uua),aa(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_358
tff(fact_7556_ATP_Olambda__359,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_fv(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_359
tff(fact_7557_ATP_Olambda__360,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_dq(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_360
tff(fact_7558_ATP_Olambda__361,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_gs(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_361
tff(fact_7559_ATP_Olambda__362,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_eo(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_362
tff(fact_7560_ATP_Olambda__363,axiom,
! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_gk(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_363
tff(fact_7561_ATP_Olambda__364,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_qe(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> pp(member(B,Uub,image(A,B,Uu,Uua))) ) ) ).
% ATP.lambda_364
tff(fact_7562_ATP_Olambda__365,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_jv(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_365
tff(fact_7563_ATP_Olambda__366,axiom,
! [A: $tType,B: $tType] :
( semiring_0(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_bj(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_366
tff(fact_7564_ATP_Olambda__367,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kv(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_367
tff(fact_7565_ATP_Olambda__368,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ki(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_368
tff(fact_7566_ATP_Olambda__369,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_fe(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_369
tff(fact_7567_ATP_Olambda__370,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_jh(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).
% ATP.lambda_370
tff(fact_7568_ATP_Olambda__371,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_db(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_371
tff(fact_7569_ATP_Olambda__372,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: B] :
( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aTP_Lamp_qw(fun(A,B),fun(fun(A,bool),fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub))) ) ).
% ATP.lambda_372
tff(fact_7570_ATP_Olambda__373,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_bw(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_373
tff(fact_7571_ATP_Olambda__374,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_ep(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_374
tff(fact_7572_ATP_Olambda__375,axiom,
! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sp(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_375
tff(fact_7573_ATP_Olambda__376,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_oz(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),product_Pair(A,B,Uua),Uub))) ) ).
% ATP.lambda_376
tff(fact_7574_ATP_Olambda__377,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_sx(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_377
tff(fact_7575_ATP_Olambda__378,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_sz(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_378
tff(fact_7576_ATP_Olambda__379,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pc(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_379
tff(fact_7577_ATP_Olambda__380,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ko(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_380
tff(fact_7578_ATP_Olambda__381,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_mv(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).
% ATP.lambda_381
tff(fact_7579_ATP_Olambda__382,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_jx(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).
% ATP.lambda_382
tff(fact_7580_ATP_Olambda__383,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_qv(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).
% ATP.lambda_383
tff(fact_7581_ATP_Olambda__384,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_qx(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).
% ATP.lambda_384
tff(fact_7582_ATP_Olambda__385,axiom,
! [C: $tType,B: $tType,A: $tType] :
( semiring_1(C)
=> ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_fl(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_385
tff(fact_7583_ATP_Olambda__386,axiom,
! [T: $tType,Uu: fun(code_natural,T),Uua: code_natural,Uub: T] : aa(T,T,aa(code_natural,fun(T,T),aTP_Lamp_qu(fun(code_natural,T),fun(code_natural,fun(T,T)),Uu),Uua),Uub) = aa(code_natural,T,Uu,Uua) ).
% ATP.lambda_386
tff(fact_7584_ATP_Olambda__387,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_nh(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_387
tff(fact_7585_ATP_Olambda__388,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_qd(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),case_option(A,A,Uua,aa(A,fun(A,A),Uu,Uua),Uub)) ).
% ATP.lambda_388
tff(fact_7586_ATP_Olambda__389,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_ff(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),aa(B,A,Uu,Uub))) ) ) ).
% ATP.lambda_389
tff(fact_7587_ATP_Olambda__390,axiom,
! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_hg(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),collect(B,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_hf(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).
% ATP.lambda_390
tff(fact_7588_ATP_Olambda__391,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_lg(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [I5: nat] :
( ( Uub = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Uu),I5)),aa(nat,B,nth(B,Uua),I5)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(nat,nat,aa(nat,fun(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_391
tff(fact_7589_ATP_Olambda__392,axiom,
! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_ll(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
<=> ? [I5: nat] :
( ( Uub = aa(nat,A,nth(A,Uu),I5) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I5),aa(list(A),nat,size_size(list(A)),Uu)))
& pp(member(nat,I5,Uua)) ) ) ).
% ATP.lambda_392
tff(fact_7590_ATP_Olambda__393,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_nz(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ? [A8: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A8) )
& pp(member(A,A8,Uu)) ) ) ) ).
% ATP.lambda_393
tff(fact_7591_ATP_Olambda__394,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_nw(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ? [A8: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A8) )
& pp(member(A,A8,Uu)) ) ) ) ).
% ATP.lambda_394
tff(fact_7592_ATP_Olambda__395,axiom,
! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ob(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ? [N5: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N5),Uu),Uua) ) ).
% ATP.lambda_395
tff(fact_7593_ATP_Olambda__396,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ny(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [A8: A,B8: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A8),B8) )
& pp(member(A,A8,Uu))
& pp(member(A,B8,Uua)) ) ) ) ).
% ATP.lambda_396
tff(fact_7594_ATP_Olambda__397,axiom,
! [A: $tType] :
( distrib_lattice(A)
=> ! [Uu: set(A),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_nx(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [A8: A,B8: A] :
( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A8),B8) )
& pp(member(A,A8,Uu))
& pp(member(A,B8,Uua)) ) ) ) ).
% ATP.lambda_397
tff(fact_7595_ATP_Olambda__398,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_st(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_ss(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),set_ord_atMost(nat,Uuc)) ).
% ATP.lambda_398
tff(fact_7596_ATP_Olambda__399,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_si(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_399
tff(fact_7597_ATP_Olambda__400,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_sm(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_ord_lessThan(nat,Uuc)) ).
% ATP.lambda_400
tff(fact_7598_ATP_Olambda__401,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_gb(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_401
tff(fact_7599_ATP_Olambda__402,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_dj(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_402
tff(fact_7600_ATP_Olambda__403,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_dk(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_403
tff(fact_7601_ATP_Olambda__404,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_gc(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_404
tff(fact_7602_ATP_Olambda__405,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_sh(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_405
tff(fact_7603_ATP_Olambda__406,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(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_cr(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_406
tff(fact_7604_ATP_Olambda__407,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_da(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_407
tff(fact_7605_ATP_Olambda__408,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: B,Uub: set(A),Uuc: A] : aa(A,B,aa(set(A),fun(A,B),aa(B,fun(set(A),fun(A,B)),aTP_Lamp_oq(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,member(A,Uuc,Uub),Uu,Uua) ).
% ATP.lambda_408
tff(fact_7606_ATP_Olambda__409,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(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_cf(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_409
tff(fact_7607_ATP_Olambda__410,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_er(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_410
tff(fact_7608_ATP_Olambda__411,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_sr(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_411
tff(fact_7609_ATP_Olambda__412,axiom,
! [A: $tType,B: $tType] :
( comm_semiring_0(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_kr(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uuc),aa(A,fun(A,A),times_times(A),Uua)),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))) ) ).
% ATP.lambda_412
tff(fact_7610_ATP_Olambda__413,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType] :
( ( order(C)
& order(A) )
=> ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_lq(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).
% ATP.lambda_413
tff(fact_7611_ATP_Olambda__414,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_sy(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_414
tff(fact_7612_ATP_Olambda__415,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,B,aa(set(C),fun(A,B),aa(fun(C,B),fun(set(C),fun(A,B)),aTP_Lamp_ej(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ei(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).
% ATP.lambda_415
tff(fact_7613_ATP_Olambda__416,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,bool)),Uub: set(A),Uuc: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_qz(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(A),bool,pred_chain(A,Uu,Uua),Uuc))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uuc)) ) ) ).
% ATP.lambda_416
tff(fact_7614_ATP_Olambda__417,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_lo(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(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_417
tff(fact_7615_ATP_Olambda__418,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_dr(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)),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),product_Pair(heap_ext(product_unit),nat,array_update(A,Uub,Uu,Uua,Uuc)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).
% ATP.lambda_418
tff(fact_7616_ATP_Olambda__419,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_ou(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),product_Pair(A,A,Uub),Uuc),Uua)) ) ) ) ).
% ATP.lambda_419
tff(fact_7617_ATP_Olambda__420,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_fy(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_420
tff(fact_7618_ATP_Olambda__421,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_ft(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_421
tff(fact_7619_ATP_Olambda__422,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_fx(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_422
tff(fact_7620_ATP_Olambda__423,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_fq(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_423
tff(fact_7621_ATP_Olambda__424,axiom,
! [A: $tType,B: $tType,Uu: set(B),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_hf(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(B,Uuc,Uu))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_424
tff(fact_7622_ATP_Olambda__425,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,bool)),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),aTP_Lamp_he(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(member(A,Uuc,Uu))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uuc),Uub)) ) ) ).
% ATP.lambda_425
tff(fact_7623_ATP_Olambda__426,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_fm(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_426
tff(fact_7624_ATP_Olambda__427,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_fp(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_427
tff(fact_7625_ATP_Olambda__428,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_fj(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_428
tff(fact_7626_ATP_Olambda__429,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_hl(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(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_429
tff(fact_7627_ATP_Olambda__430,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_lx(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_430
tff(fact_7628_ATP_Olambda__431,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_hj(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(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_431
tff(fact_7629_ATP_Olambda__432,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_lv(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_432
tff(fact_7630_ATP_Olambda__433,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_lz(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(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_433
tff(fact_7631_ATP_Olambda__434,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_mb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(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_434
tff(fact_7632_ATP_Olambda__435,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_ox(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( Uu = Uub )
& ( Uua = Uuc ) ) ) ).
% ATP.lambda_435
tff(fact_7633_ATP_Olambda__436,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_mw(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_436
tff(fact_7634_ATP_Olambda__437,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_ap(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_437
tff(fact_7635_ATP_Olambda__438,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_as(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_438
tff(fact_7636_ATP_Olambda__439,axiom,
! [B: $tType,A: $tType] :
( comm_semiring_1(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_fz(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).
% ATP.lambda_439
tff(fact_7637_ATP_Olambda__440,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_qp(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(A,bool,Uu,Uuc))
& ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).
% ATP.lambda_440
tff(fact_7638_ATP_Olambda__441,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_js(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_441
tff(fact_7639_ATP_Olambda__442,axiom,
! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_oo(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).
% ATP.lambda_442
tff(fact_7640_ATP_Olambda__443,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_ei(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).
% ATP.lambda_443
tff(fact_7641_ATP_Olambda__444,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_kw(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_444
tff(fact_7642_ATP_Olambda__445,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_kj(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_445
tff(fact_7643_ATP_Olambda__446,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_ji(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).
% ATP.lambda_446
tff(fact_7644_ATP_Olambda__447,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_1(C)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_fn(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),collect(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_fm(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).
% ATP.lambda_447
tff(fact_7645_ATP_Olambda__448,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_gr(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)),product_Pair(array(A),product_prod(heap_ext(product_unit),nat),Uub),aa(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),bit0(one2)))) ) ).
% ATP.lambda_448
tff(fact_7646_ATP_Olambda__449,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_gu(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)),product_Pair(array(A),product_prod(heap_ext(product_unit),nat),Uub),aa(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_449
tff(fact_7647_ATP_Olambda__450,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_nn(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))),heap_Time_execute(A,aa(B,heap_Time_Heap(A),Uu,Uua)),Uub)) ).
% ATP.lambda_450
tff(fact_7648_ATP_Olambda__451,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_np(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))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),Uu,Uua)),Uub)) ).
% ATP.lambda_451
tff(fact_7649_ATP_Olambda__452,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_ch(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_452
tff(fact_7650_ATP_Olambda__453,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_eu(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_453
tff(fact_7651_ATP_Olambda__454,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_so(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_454
tff(fact_7652_ATP_Olambda__455,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_cs(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_455
tff(fact_7653_ATP_Olambda__456,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_fg(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_456
tff(fact_7654_ATP_Olambda__457,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_sn(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_457
tff(fact_7655_ATP_Olambda__458,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_tf(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_458
tff(fact_7656_ATP_Olambda__459,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_kk(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_kj(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).
% ATP.lambda_459
tff(fact_7657_ATP_Olambda__460,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_kx(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),image(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_kw(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc),Uub)) ) ).
% ATP.lambda_460
tff(fact_7658_ATP_Olambda__461,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_hx(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(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_461
tff(fact_7659_ATP_Olambda__462,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_hv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(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_462
tff(fact_7660_ATP_Olambda__463,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_ht(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(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_463
tff(fact_7661_ATP_Olambda__464,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_hr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(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_464
tff(fact_7662_ATP_Olambda__465,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_sk(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_sj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uuc)) ).
% ATP.lambda_465
tff(fact_7663_ATP_Olambda__466,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_sq(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_466
tff(fact_7664_ATP_Olambda__467,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_sg(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_467
tff(fact_7665_ATP_Olambda__468,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_te(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_468
tff(fact_7666_ATP_Olambda__469,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_gi(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_469
tff(fact_7667_ATP_Olambda__470,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_ge(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_470
tff(fact_7668_ATP_Olambda__471,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_gf(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_471
tff(fact_7669_ATP_Olambda__472,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_of(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ? [A8: A,B8: C] :
( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A8)),aa(C,B,Uua,B8)) )
& pp(member(A,A8,Uub))
& pp(member(C,B8,Uuc)) ) ) ) ).
% ATP.lambda_472
tff(fact_7670_ATP_Olambda__473,axiom,
! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_ng(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).
% ATP.lambda_473
tff(fact_7671_ATP_Olambda__474,axiom,
! [B: $tType,A: $tType,Uu: heap_Time_Heap(A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_ta(heap_Time_Heap(A),fun(B,heap_Time_Heap(A)),Uu),Uua) = Uu ).
% ATP.lambda_474
tff(fact_7672_ATP_Olambda__475,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_oe(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_475
tff(fact_7673_ATP_Olambda__476,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_og(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_476
tff(fact_7674_ATP_Olambda__477,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_ok(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).
% ATP.lambda_477
tff(fact_7675_ATP_Olambda__478,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_re(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_478
tff(fact_7676_ATP_Olambda__479,axiom,
! [C: $tType,B: $tType] :
( semiring_1(B)
=> ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_cg(B,fun(C,B),Uu),Uua) = Uu ) ).
% ATP.lambda_479
tff(fact_7677_ATP_Olambda__480,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_mi(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_480
tff(fact_7678_ATP_Olambda__481,axiom,
! [B: $tType,A: $tType] :
( semiring_1(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_dz(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_481
tff(fact_7679_ATP_Olambda__482,axiom,
! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cc(A,fun(nat,A),Uu),Uua) = Uu ).
% ATP.lambda_482
tff(fact_7680_ATP_Olambda__483,axiom,
! [Uu: typerep,Uua: nat] : aa(nat,nat,aa(typerep,fun(nat,nat),aTP_Lamp_rd(typerep,fun(nat,nat)),Uu),Uua) = zero_zero(nat) ).
% ATP.lambda_483
tff(fact_7681_ATP_Olambda__484,axiom,
! [Uu: typerep,Uua: nat] : aa(nat,list(nat),aa(typerep,fun(nat,list(nat)),aTP_Lamp_rc(typerep,fun(nat,list(nat))),Uu),Uua) = nil(nat) ).
% ATP.lambda_484
tff(fact_7682_ATP_Olambda__485,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_cp(nat,nat),Uu) = Uu ).
% ATP.lambda_485
tff(fact_7683_ATP_Olambda__486,axiom,
! [Uu: int] : aa(int,int,aTP_Lamp_ay(int,int),Uu) = Uu ).
% ATP.lambda_486
tff(fact_7684_ATP_Olambda__487,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_me(A,A),Uu) = Uu ) ).
% ATP.lambda_487
tff(fact_7685_ATP_Olambda__488,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_bc(A,A),Uu) = Uu ) ).
% ATP.lambda_488
tff(fact_7686_ATP_Olambda__489,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_qs(A,A),Uu) = Uu ) ).
% ATP.lambda_489
tff(fact_7687_ATP_Olambda__490,axiom,
! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ql(A,A),Uu) = Uu ).
% ATP.lambda_490
tff(fact_7688_ATP_Olambda__491,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_on(A,set(B)),Uu) = top_top(set(B)) ).
% ATP.lambda_491
tff(fact_7689_ATP_Olambda__492,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_dt(B,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_492
tff(fact_7690_ATP_Olambda__493,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_aj(B,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_493
tff(fact_7691_ATP_Olambda__494,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_bd(A,A),Uu) = zero_zero(A) ) ).
% ATP.lambda_494
tff(fact_7692_ATP_Olambda__495,axiom,
! [A: $tType,B: $tType] :
( zero(B)
=> ! [Uu: A] : aa(A,B,aTP_Lamp_qk(A,B),Uu) = zero_zero(B) ) ).
% ATP.lambda_495
tff(fact_7693_ATP_Olambda__496,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_ab(B,A),Uu) = one_one(A) ) ).
% ATP.lambda_496
tff(fact_7694_ATP_Olambda__497,axiom,
! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_hb(A,nat),Uu) = one_one(nat) ).
% ATP.lambda_497
tff(fact_7695_ATP_Olambda__498,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_oa(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_498
tff(fact_7696_ATP_Olambda__499,axiom,
! [Uu: nat] :
( pp(aa(nat,bool,aTP_Lamp_jz(nat,bool),Uu))
<=> $false ) ).
% ATP.lambda_499
tff(fact_7697_ATP_Olambda__500,axiom,
! [Uu: nat] :
( pp(aa(nat,bool,aTP_Lamp_jy(nat,bool),Uu))
<=> $true ) ).
% ATP.lambda_500
tff(fact_7698_ATP_Olambda__501,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_sv(A,bool),Uu))
<=> $true ) ).
% ATP.lambda_501
tff(fact_7699_ATP_Olambda__502,axiom,
! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_py(B,fun(nat,nat)),Uu) = suc ).
% ATP.lambda_502
tff(fact_7700_ATP_Olambda__503,axiom,
! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_nv(A,fun(nat,nat)),Uu) = suc ).
% ATP.lambda_503
% Subclasses (3)
tff(subcl_Heap_Oheap___HOL_Otype,axiom,
! [A: $tType] :
( heap(A)
=> type(A) ) ).
tff(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
! [A: $tType] :
( heap(A)
=> typerep2(A) ) ).
tff(subcl_Heap_Oheap___Countable_Ocountable,axiom,
! [A: $tType] :
( heap(A)
=> countable(A) ) ).
% Type constructors (657)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
bounded_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
finite_finite(product_unit) ).
tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
! [A15: $tType,A16: $tType] :
( ( finite_finite(A15)
& finite_finite(A16) )
=> finite_finite(product_prod(A15,A16)) ) ).
tff(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
! [A15: $tType] :
( finite_finite(A15)
=> finite_finite(option(A15)) ) ).
tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A15: $tType,A16: $tType] :
( ( finite_finite(A15)
& finite_finite(A16) )
=> finite_finite(sum_sum(A15,A16)) ) ).
tff(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
finite_finite(char) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice_5,axiom,
bounded_lattice(bool) ).
tff(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
finite_finite(bool) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice_7,axiom,
! [A15: $tType] : bounded_lattice(set(A15)) ).
tff(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A15: $tType] :
( finite_finite(A15)
=> finite_finite(set(A15)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice_9,axiom,
! [A15: $tType,A16: $tType] :
( bounded_lattice(A16)
=> bounded_lattice(fun(A15,A16)) ) ).
tff(tcon_fun___Finite__Set_Ofinite_10,axiom,
! [A15: $tType,A16: $tType] :
( ( finite_finite(A15)
& finite_finite(A16) )
=> finite_finite(fun(A15,A16)) ) ).
tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A15: $tType,A16: $tType] :
( comple6319245703460814977attice(A16)
=> condit1219197933456340205attice(fun(A15,A16)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A15: $tType,A16: $tType] :
( comple592849572758109894attice(A16)
=> comple592849572758109894attice(fun(A15,A16)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A15: $tType,A16: $tType] :
( comple489889107523837845lgebra(A16)
=> comple489889107523837845lgebra(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A15: $tType,A16: $tType] :
( bounded_lattice(A16)
=> bounde4967611905675639751up_bot(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A15: $tType,A16: $tType] :
( bounded_lattice(A16)
=> bounde4346867609351753570nf_top(fun(A15,A16)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A15: $tType,A16: $tType] :
( comple6319245703460814977attice(A16)
=> comple6319245703460814977attice(fun(A15,A16)) ) ).
tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A15: $tType,A16: $tType] :
( boolea8198339166811842893lgebra(A16)
=> boolea8198339166811842893lgebra(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A15: $tType,A16: $tType] :
( bounded_lattice(A16)
=> bounded_lattice_top(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A15: $tType,A16: $tType] :
( bounded_lattice(A16)
=> bounded_lattice_bot(fun(A15,A16)) ) ).
tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A15: $tType,A16: $tType] :
( comple6319245703460814977attice(A16)
=> comple9053668089753744459l_ccpo(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A15: $tType,A16: $tType] :
( semilattice_sup(A16)
=> semilattice_sup(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A15: $tType,A16: $tType] :
( semilattice_inf(A16)
=> semilattice_inf(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A15: $tType,A16: $tType] :
( distrib_lattice(A16)
=> distrib_lattice(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Oorder__top,axiom,
! [A15: $tType,A16: $tType] :
( order_top(A16)
=> order_top(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Oorder__bot,axiom,
! [A15: $tType,A16: $tType] :
( order_bot(A16)
=> order_bot(fun(A15,A16)) ) ).
tff(tcon_fun___Countable_Ocountable,axiom,
! [A15: $tType,A16: $tType] :
( ( finite_finite(A15)
& countable(A16) )
=> countable(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Opreorder,axiom,
! [A15: $tType,A16: $tType] :
( preorder(A16)
=> preorder(fun(A15,A16)) ) ).
tff(tcon_fun___Lattices_Olattice,axiom,
! [A15: $tType,A16: $tType] :
( lattice(A16)
=> lattice(fun(A15,A16)) ) ).
tff(tcon_fun___Typerep_Otyperep,axiom,
! [A15: $tType,A16: $tType] :
( ( typerep2(A15)
& typerep2(A16) )
=> typerep2(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Oorder,axiom,
! [A15: $tType,A16: $tType] :
( order(A16)
=> order(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Otop,axiom,
! [A15: $tType,A16: $tType] :
( top(A16)
=> top(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Oord,axiom,
! [A15: $tType,A16: $tType] :
( ord(A16)
=> ord(fun(A15,A16)) ) ).
tff(tcon_fun___Orderings_Obot,axiom,
! [A15: $tType,A16: $tType] :
( bot(A16)
=> bot(fun(A15,A16)) ) ).
tff(tcon_fun___Groups_Ouminus,axiom,
! [A15: $tType,A16: $tType] :
( uminus(A16)
=> uminus(fun(A15,A16)) ) ).
tff(tcon_fun___Groups_Ominus,axiom,
! [A15: $tType,A16: $tType] :
( minus(A16)
=> minus(fun(A15,A16)) ) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder(int) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
condit1219197933456340205attice(int) ).
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___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative(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___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel(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___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict(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_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring(int) ).
tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
euclid5891614535332579305n_ring(int) ).
tff(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict(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___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize(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_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring(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__semiring,axiom,
linordered_semiring(int) ).
tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__sup_12,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_13,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Lattices_Odistrib__lattice_14,axiom,
distrib_lattice(int) ).
tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel(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___Rings_Oordered__semiring,axiom,
ordered_semiring(int) ).
tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(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___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_15,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_16,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_Oidom__modulo,axiom,
idom_modulo(int) ).
tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide(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___Lattices_Olattice_17,axiom,
lattice(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___Typerep_Otyperep_18,axiom,
typerep2(int) ).
tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring(int) ).
tff(tcon_Int_Oint___Orderings_Oorder_19,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___Rings_Osemidom,axiom,
semidom(int) ).
tff(tcon_Int_Oint___Orderings_Oord_20,axiom,
ord(int) ).
tff(tcon_Int_Oint___Groups_Ouminus_21,axiom,
uminus(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if(int) ).
tff(tcon_Int_Oint___Groups_Ominus_22,axiom,
minus(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___Rings_Oidom,axiom,
idom(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_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_23,axiom,
condit6923001295902523014norder(nat) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
condit1219197933456340205attice(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_25,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_26,axiom,
semiri1453513574482234551roduct(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_27,axiom,
euclid5411537665997757685th_nat(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_28,axiom,
ordere1937475149494474687imp_le(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_29,axiom,
euclid3128863361964157862miring(nat) ).
tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_30,axiom,
euclid4440199948858584721cancel(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_31,axiom,
normal6328177297339901930cative(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_32,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_33,axiom,
semiri6575147826004484403cancel(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_34,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_35,axiom,
ordere580206878836729694up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_36,axiom,
ordere2412721322843649153imp_le(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_37,axiom,
bit_se359711467146920520ations(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_38,axiom,
linord2810124833399127020strict(nat) ).
tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_39,axiom,
strict7427464778891057005id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_40,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_41,axiom,
euclid3725896446679973847miring(nat) ).
tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_42,axiom,
linord4140545234300271783up_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_43,axiom,
semiri2026040879449505780visors(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_44,axiom,
linord181362715937106298miring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_45,axiom,
semido2269285787275462019factor(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_46,axiom,
linord8928482502909563296strict(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_47,axiom,
semiri3467727345109120633visors(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_48,axiom,
ordere6658533253407199908up_add(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_49,axiom,
semiri6843258321239162965malize(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_50,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_51,axiom,
cancel2418104881723323429up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
cancel1802427076303600483id_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
comm_s4317794764714335236cancel(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
bit_semiring_bits(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_55,axiom,
ordere2520102378445227354miring(nat) ).
tff(tcon_Nat_Onat___Rings_Onormalization__semidom_56,axiom,
normal8620421768224518004emidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_57,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semiring_58,axiom,
linordered_semiring(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_59,axiom,
ordered_semiring_0(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_60,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_61,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_62,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_63,axiom,
distrib_lattice(nat) ).
tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64,axiom,
ab_semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65,axiom,
semiring_1_cancel(nat) ).
tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66,axiom,
algebraic_semidom(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67,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_68,axiom,
ab_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Oordered__semiring_69,axiom,
ordered_semiring(nat) ).
tff(tcon_Nat_Onat___Parity_Osemiring__parity_70,axiom,
semiring_parity(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__modulo_72,axiom,
semiring_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73,axiom,
comm_semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74,axiom,
comm_semiring_0(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__mult_75,axiom,
semigroup_mult(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__modulo_76,axiom,
semidom_modulo(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom__divide_77,axiom,
semidom_divide(nat) ).
tff(tcon_Nat_Onat___Num_Osemiring__numeral_78,axiom,
semiring_numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Osemigroup__add_79,axiom,
semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__less__one_80,axiom,
zero_less_one(nat) ).
tff(tcon_Nat_Onat___Rings_Ocomm__semiring_81,axiom,
comm_semiring(nat) ).
tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder__bot_82,axiom,
order_bot(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_83,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Countable_Ocountable_84,axiom,
countable(nat) ).
tff(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
zero_neq_one(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
monoid_add(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
no_top(nat) ).
tff(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
lattice(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Typerep_Otyperep_96,axiom,
typerep2(nat) ).
tff(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
mult_zero(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
semiring(nat) ).
tff(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
semidom(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_101,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Orderings_Obot_102,axiom,
bot(nat) ).
tff(tcon_Nat_Onat___Groups_Ominus_103,axiom,
minus(nat) ).
tff(tcon_Nat_Onat___Power_Opower_104,axiom,
power(nat) ).
tff(tcon_Nat_Onat___Num_Onumeral_105,axiom,
numeral(nat) ).
tff(tcon_Nat_Onat___Groups_Ozero_106,axiom,
zero(nat) ).
tff(tcon_Nat_Onat___Groups_Oplus_107,axiom,
plus(nat) ).
tff(tcon_Nat_Onat___Groups_Oone_108,axiom,
one(nat) ).
tff(tcon_Nat_Onat___Rings_Odvd_109,axiom,
dvd(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_110,axiom,
heap(nat) ).
tff(tcon_Nat_Onat___Nat_Osize,axiom,
size(nat) ).
tff(tcon_Num_Onum___Orderings_Opreorder_111,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_112,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Typerep_Otyperep_113,axiom,
typerep2(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_114,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_115,axiom,
ord(num) ).
tff(tcon_Num_Onum___Groups_Oplus_116,axiom,
plus(num) ).
tff(tcon_Num_Onum___Nat_Osize_117,axiom,
size(num) ).
tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_118,axiom,
semiri1453513574482234551roduct(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_119,axiom,
ordere1937475149494474687imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_120,axiom,
semiri6575147826004484403cancel(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_121,axiom,
strict9044650504122735259up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_122,axiom,
ordere580206878836729694up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_123,axiom,
ordere2412721322843649153imp_le(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_124,axiom,
linord2810124833399127020strict(rat) ).
tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_125,axiom,
strict7427464778891057005id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_126,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_127,axiom,
linord715952674999750819strict(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_128,axiom,
linord4140545234300271783up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_129,axiom,
semiri2026040879449505780visors(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_130,axiom,
linord181362715937106298miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_131,axiom,
linord8928482502909563296strict(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_132,axiom,
semiri3467727345109120633visors(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_133,axiom,
ordere6658533253407199908up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_134,axiom,
ordere166539214618696060dd_abs(rat) ).
tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_135,axiom,
ordere6911136660526730532id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_136,axiom,
linord5086331880401160121up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_137,axiom,
cancel2418104881723323429up_add(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_138,axiom,
ring_15535105094025558882visors(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_139,axiom,
cancel1802427076303600483id_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_140,axiom,
linord4710134922213307826strict(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_141,axiom,
comm_s4317794764714335236cancel(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_142,axiom,
ordere2520102378445227354miring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_143,axiom,
linord6961819062388156250ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_144,axiom,
ordered_ab_group_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_145,axiom,
cancel_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semiring_146,axiom,
linordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_147,axiom,
ordered_semiring_0(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semidom_148,axiom,
linordered_semidom(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_149,axiom,
semilattice_sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_150,axiom,
semilattice_inf(rat) ).
tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_151,axiom,
distrib_lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_152,axiom,
ab_semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_153,axiom,
semiring_1_cancel(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_154,axiom,
comm_monoid_mult(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_155,axiom,
ab_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__semiring_156,axiom,
ordered_semiring(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_157,axiom,
ordered_ring_abs(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_158,axiom,
comm_monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring_159,axiom,
linordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__idom_160,axiom,
linordered_idom(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_161,axiom,
comm_semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_162,axiom,
comm_semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__mult_163,axiom,
semigroup_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Osemidom__divide_164,axiom,
semidom_divide(rat) ).
tff(tcon_Rat_Orat___Num_Osemiring__numeral_165,axiom,
semiring_numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Osemigroup__add_166,axiom,
semigroup_add(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
field_abs_sgn(rat) ).
tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__less__one_167,axiom,
zero_less_one(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__semiring_168,axiom,
comm_semiring(rat) ).
tff(tcon_Rat_Orat___Nat_Osemiring__char__0_169,axiom,
semiring_char_0(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__group__add_170,axiom,
ab_group_add(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0(rat) ).
tff(tcon_Rat_Orat___Countable_Ocountable_171,axiom,
countable(rat) ).
tff(tcon_Rat_Orat___Rings_Ozero__neq__one_172,axiom,
zero_neq_one(rat) ).
tff(tcon_Rat_Orat___Rings_Oordered__ring_173,axiom,
ordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_174,axiom,
idom_abs_sgn(rat) ).
tff(tcon_Rat_Orat___Orderings_Opreorder_175,axiom,
preorder(rat) ).
tff(tcon_Rat_Orat___Orderings_Olinorder_176,axiom,
linorder(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__mult_177,axiom,
monoid_mult(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom__divide_178,axiom,
idom_divide(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_179,axiom,
comm_ring_1(rat) ).
tff(tcon_Rat_Orat___Groups_Omonoid__add_180,axiom,
monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1_181,axiom,
semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__0_182,axiom,
semiring_0(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__top_183,axiom,
no_top(rat) ).
tff(tcon_Rat_Orat___Orderings_Ono__bot_184,axiom,
no_bot(rat) ).
tff(tcon_Rat_Orat___Lattices_Olattice_185,axiom,
lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_186,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Typerep_Otyperep_187,axiom,
typerep2(rat) ).
tff(tcon_Rat_Orat___Rings_Omult__zero_188,axiom,
mult_zero(rat) ).
tff(tcon_Rat_Orat___Rings_Ocomm__ring_189,axiom,
comm_ring(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_190,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Num_Oneg__numeral_191,axiom,
neg_numeral(rat) ).
tff(tcon_Rat_Orat___Nat_Oring__char__0_192,axiom,
ring_char_0(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring_193,axiom,
semiring(rat) ).
tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse(rat) ).
tff(tcon_Rat_Orat___Rings_Osemidom_194,axiom,
semidom(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_195,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Groups_Ouminus_196,axiom,
uminus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_197,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Oabs__if_198,axiom,
abs_if(rat) ).
tff(tcon_Rat_Orat___Groups_Ominus_199,axiom,
minus(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
tff(tcon_Rat_Orat___Power_Opower_200,axiom,
power(rat) ).
tff(tcon_Rat_Orat___Num_Onumeral_201,axiom,
numeral(rat) ).
tff(tcon_Rat_Orat___Groups_Ozero_202,axiom,
zero(rat) ).
tff(tcon_Rat_Orat___Groups_Oplus_203,axiom,
plus(rat) ).
tff(tcon_Rat_Orat___Rings_Oring_204,axiom,
ring(rat) ).
tff(tcon_Rat_Orat___Rings_Oidom_205,axiom,
idom(rat) ).
tff(tcon_Rat_Orat___Groups_Oone_206,axiom,
one(rat) ).
tff(tcon_Rat_Orat___Rings_Odvd_207,axiom,
dvd(rat) ).
tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_208,axiom,
! [A15: $tType] : condit1219197933456340205attice(set(A15)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_209,axiom,
! [A15: $tType] : comple592849572758109894attice(set(A15)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_210,axiom,
! [A15: $tType] : comple489889107523837845lgebra(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_211,axiom,
! [A15: $tType] : bounde4967611905675639751up_bot(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_212,axiom,
! [A15: $tType] : bounde4346867609351753570nf_top(set(A15)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_213,axiom,
! [A15: $tType] : comple6319245703460814977attice(set(A15)) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_214,axiom,
! [A15: $tType] : boolea8198339166811842893lgebra(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_215,axiom,
! [A15: $tType] : bounded_lattice_top(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_216,axiom,
! [A15: $tType] : bounded_lattice_bot(set(A15)) ).
tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_217,axiom,
! [A15: $tType] : comple9053668089753744459l_ccpo(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_218,axiom,
! [A15: $tType] : semilattice_sup(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_219,axiom,
! [A15: $tType] : semilattice_inf(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Odistrib__lattice_220,axiom,
! [A15: $tType] : distrib_lattice(set(A15)) ).
tff(tcon_Set_Oset___Orderings_Oorder__top_221,axiom,
! [A15: $tType] : order_top(set(A15)) ).
tff(tcon_Set_Oset___Orderings_Oorder__bot_222,axiom,
! [A15: $tType] : order_bot(set(A15)) ).
tff(tcon_Set_Oset___Countable_Ocountable_223,axiom,
! [A15: $tType] :
( finite_finite(A15)
=> countable(set(A15)) ) ).
tff(tcon_Set_Oset___Orderings_Opreorder_224,axiom,
! [A15: $tType] : preorder(set(A15)) ).
tff(tcon_Set_Oset___Lattices_Olattice_225,axiom,
! [A15: $tType] : lattice(set(A15)) ).
tff(tcon_Set_Oset___Typerep_Otyperep_226,axiom,
! [A15: $tType] :
( typerep2(A15)
=> typerep2(set(A15)) ) ).
tff(tcon_Set_Oset___Orderings_Oorder_227,axiom,
! [A15: $tType] : order(set(A15)) ).
tff(tcon_Set_Oset___Orderings_Otop_228,axiom,
! [A15: $tType] : top(set(A15)) ).
tff(tcon_Set_Oset___Orderings_Oord_229,axiom,
! [A15: $tType] : ord(set(A15)) ).
tff(tcon_Set_Oset___Orderings_Obot_230,axiom,
! [A15: $tType] : bot(set(A15)) ).
tff(tcon_Set_Oset___Groups_Ouminus_231,axiom,
! [A15: $tType] : uminus(set(A15)) ).
tff(tcon_Set_Oset___Groups_Ominus_232,axiom,
! [A15: $tType] : minus(set(A15)) ).
tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_233,axiom,
condit1219197933456340205attice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_234,axiom,
comple592849572758109894attice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_235,axiom,
comple489889107523837845lgebra(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_236,axiom,
bounde4967611905675639751up_bot(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_237,axiom,
bounde4346867609351753570nf_top(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_238,axiom,
comple6319245703460814977attice(bool) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_239,axiom,
boolea8198339166811842893lgebra(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_240,axiom,
bounded_lattice_top(bool) ).
tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_241,axiom,
bounded_lattice_bot(bool) ).
tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_242,axiom,
comple9053668089753744459l_ccpo(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_243,axiom,
semilattice_sup(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_244,axiom,
semilattice_inf(bool) ).
tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_245,axiom,
distrib_lattice(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder__top_246,axiom,
order_top(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder__bot_247,axiom,
order_bot(bool) ).
tff(tcon_HOL_Obool___Countable_Ocountable_248,axiom,
countable(bool) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_249,axiom,
preorder(bool) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_250,axiom,
linorder(bool) ).
tff(tcon_HOL_Obool___Lattices_Olattice_251,axiom,
lattice(bool) ).
tff(tcon_HOL_Obool___Typerep_Otyperep_252,axiom,
typerep2(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder_253,axiom,
order(bool) ).
tff(tcon_HOL_Obool___Orderings_Otop_254,axiom,
top(bool) ).
tff(tcon_HOL_Obool___Orderings_Oord_255,axiom,
ord(bool) ).
tff(tcon_HOL_Obool___Orderings_Obot_256,axiom,
bot(bool) ).
tff(tcon_HOL_Obool___Groups_Ouminus_257,axiom,
uminus(bool) ).
tff(tcon_HOL_Obool___Groups_Ominus_258,axiom,
minus(bool) ).
tff(tcon_HOL_Obool___Heap_Oheap_259,axiom,
heap(bool) ).
tff(tcon_List_Olist___Countable_Ocountable_260,axiom,
! [A15: $tType] :
( countable(A15)
=> countable(list(A15)) ) ).
tff(tcon_List_Olist___Typerep_Otyperep_261,axiom,
! [A15: $tType] :
( typerep2(A15)
=> typerep2(list(A15)) ) ).
tff(tcon_List_Olist___Heap_Oheap_262,axiom,
! [A15: $tType] :
( heap(A15)
=> heap(list(A15)) ) ).
tff(tcon_List_Olist___Nat_Osize_263,axiom,
! [A15: $tType] : size(list(A15)) ).
tff(tcon_Heap_Oarray___Countable_Ocountable_264,axiom,
! [A15: $tType] : countable(array(A15)) ).
tff(tcon_Heap_Oarray___Typerep_Otyperep_265,axiom,
! [A15: $tType] : typerep2(array(A15)) ).
tff(tcon_Heap_Oarray___Heap_Oheap_266,axiom,
! [A15: $tType] : heap(array(A15)) ).
tff(tcon_Heap_Oarray___Nat_Osize_267,axiom,
! [A15: $tType] : size(array(A15)) ).
tff(tcon_String_Ochar___Countable_Ocountable_268,axiom,
countable(char) ).
tff(tcon_String_Ochar___Typerep_Otyperep_269,axiom,
typerep2(char) ).
tff(tcon_String_Ochar___Heap_Oheap_270,axiom,
heap(char) ).
tff(tcon_String_Ochar___Nat_Osize_271,axiom,
size(char) ).
tff(tcon_Nitpick_Oword___Typerep_Otyperep_272,axiom,
! [A15: $tType] :
( typerep2(A15)
=> typerep2(word(A15)) ) ).
tff(tcon_Sum__Type_Osum___Countable_Ocountable_273,axiom,
! [A15: $tType,A16: $tType] :
( ( countable(A15)
& countable(A16) )
=> countable(sum_sum(A15,A16)) ) ).
tff(tcon_Sum__Type_Osum___Typerep_Otyperep_274,axiom,
! [A15: $tType,A16: $tType] :
( ( typerep2(A15)
& typerep2(A16) )
=> typerep2(sum_sum(A15,A16)) ) ).
tff(tcon_Sum__Type_Osum___Heap_Oheap_275,axiom,
! [A15: $tType,A16: $tType] :
( ( heap(A15)
& heap(A16) )
=> heap(sum_sum(A15,A16)) ) ).
tff(tcon_Sum__Type_Osum___Nat_Osize_276,axiom,
! [A15: $tType,A16: $tType] : size(sum_sum(A15,A16)) ).
tff(tcon_Option_Ooption___Countable_Ocountable_277,axiom,
! [A15: $tType] :
( countable(A15)
=> countable(option(A15)) ) ).
tff(tcon_Option_Ooption___Typerep_Otyperep_278,axiom,
! [A15: $tType] :
( typerep2(A15)
=> typerep2(option(A15)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_279,axiom,
! [A15: $tType] :
( heap(A15)
=> heap(option(A15)) ) ).
tff(tcon_Option_Ooption___Nat_Osize_280,axiom,
! [A15: $tType] : size(option(A15)) ).
tff(tcon_String_Oliteral___Groups_Osemigroup__add_281,axiom,
semigroup_add(literal) ).
tff(tcon_String_Oliteral___Countable_Ocountable_282,axiom,
countable(literal) ).
tff(tcon_String_Oliteral___Orderings_Opreorder_283,axiom,
preorder(literal) ).
tff(tcon_String_Oliteral___Orderings_Olinorder_284,axiom,
linorder(literal) ).
tff(tcon_String_Oliteral___Groups_Omonoid__add_285,axiom,
monoid_add(literal) ).
tff(tcon_String_Oliteral___Typerep_Otyperep_286,axiom,
typerep2(literal) ).
tff(tcon_String_Oliteral___Orderings_Oorder_287,axiom,
order(literal) ).
tff(tcon_String_Oliteral___Orderings_Oord_288,axiom,
ord(literal) ).
tff(tcon_String_Oliteral___Groups_Ozero_289,axiom,
zero(literal) ).
tff(tcon_String_Oliteral___Groups_Oplus_290,axiom,
plus(literal) ).
tff(tcon_String_Oliteral___Heap_Oheap_291,axiom,
heap(literal) ).
tff(tcon_String_Oliteral___Nat_Osize_292,axiom,
size(literal) ).
tff(tcon_Nitpick_Ofun__box___Typerep_Otyperep_293,axiom,
! [A15: $tType,A16: $tType] :
( ( typerep2(A15)
& typerep2(A16) )
=> typerep2(fun_box(A15,A16)) ) ).
tff(tcon_Typerep_Otyperep___Countable_Ocountable_294,axiom,
countable(typerep) ).
tff(tcon_Typerep_Otyperep___Typerep_Otyperep_295,axiom,
typerep2(typerep) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_296,axiom,
heap(typerep) ).
tff(tcon_Typerep_Otyperep___Nat_Osize_297,axiom,
size(typerep) ).
tff(tcon_Old__Datatype_Onode___Typerep_Otyperep_298,axiom,
! [A15: $tType,A16: $tType] :
( ( typerep2(A15)
& typerep2(A16) )
=> typerep2(old_node(A15,A16)) ) ).
tff(tcon_Product__Type_Oprod___Countable_Ocountable_299,axiom,
! [A15: $tType,A16: $tType] :
( ( countable(A15)
& countable(A16) )
=> countable(product_prod(A15,A16)) ) ).
tff(tcon_Product__Type_Oprod___Typerep_Otyperep_300,axiom,
! [A15: $tType,A16: $tType] :
( ( typerep2(A15)
& typerep2(A16) )
=> typerep2(product_prod(A15,A16)) ) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_301,axiom,
! [A15: $tType,A16: $tType] :
( ( heap(A15)
& heap(A16) )
=> heap(product_prod(A15,A16)) ) ).
tff(tcon_Product__Type_Oprod___Nat_Osize_302,axiom,
! [A15: $tType,A16: $tType] : size(product_prod(A15,A16)) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_303,axiom,
condit6923001295902523014norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_304,axiom,
condit1219197933456340205attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_305,axiom,
comple592849572758109894attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_306,axiom,
comple489889107523837845lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_307,axiom,
bounde4967611905675639751up_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_308,axiom,
bounde4346867609351753570nf_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_309,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_310,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_311,axiom,
bounded_lattice_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_312,axiom,
bounded_lattice_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_313,axiom,
comple9053668089753744459l_ccpo(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_314,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_315,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_316,axiom,
distrib_lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Owellorder_317,axiom,
wellorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_318,axiom,
order_top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_319,axiom,
order_bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Countable_Ocountable_320,axiom,
countable(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_321,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_322,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Olattice_323,axiom,
lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Typerep_Otyperep_324,axiom,
typerep2(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_325,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Otop_326,axiom,
top(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_327,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Obot_328,axiom,
bot(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ouminus_329,axiom,
uminus(product_unit) ).
tff(tcon_Product__Type_Ounit___Groups_Ominus_330,axiom,
minus(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_331,axiom,
heap(product_unit) ).
tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_332,axiom,
! [A15: $tType] :
( typerep2(A15)
=> typerep2(heap_ext(A15)) ) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_333,axiom,
bit_un5681908812861735899ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_334,axiom,
semiri1453513574482234551roduct(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_335,axiom,
euclid5411537665997757685th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_336,axiom,
euclid8789492081693882211th_nat(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_337,axiom,
ordere1937475149494474687imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_338,axiom,
euclid3128863361964157862miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_339,axiom,
euclid4440199948858584721cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_340,axiom,
unique1627219031080169319umeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_341,axiom,
euclid8851590272496341667cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_342,axiom,
semiri6575147826004484403cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_343,axiom,
strict9044650504122735259up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_344,axiom,
ordere580206878836729694up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_345,axiom,
ordere2412721322843649153imp_le(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_346,axiom,
bit_se359711467146920520ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_347,axiom,
linord2810124833399127020strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_348,axiom,
strict7427464778891057005id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_349,axiom,
ordere8940638589300402666id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_350,axiom,
euclid3725896446679973847miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_351,axiom,
linord715952674999750819strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_352,axiom,
linord4140545234300271783up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_353,axiom,
bit_ri3973907225187159222ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_354,axiom,
semiri2026040879449505780visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_355,axiom,
linord181362715937106298miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_356,axiom,
euclid5891614535332579305n_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_357,axiom,
linord8928482502909563296strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_358,axiom,
semiri3467727345109120633visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_359,axiom,
ordere6658533253407199908up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_360,axiom,
ordere166539214618696060dd_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_361,axiom,
ordere6911136660526730532id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_362,axiom,
linord5086331880401160121up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_363,axiom,
cancel2418104881723323429up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_364,axiom,
ring_15535105094025558882visors(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_365,axiom,
cancel1802427076303600483id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_366,axiom,
linord4710134922213307826strict(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_367,axiom,
comm_s4317794764714335236cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_368,axiom,
bit_semiring_bits(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_369,axiom,
ordere2520102378445227354miring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_370,axiom,
linord6961819062388156250ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_371,axiom,
ordered_ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_372,axiom,
cancel_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_373,axiom,
linordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_374,axiom,
ordered_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_375,axiom,
linordered_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_376,axiom,
ab_semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_377,axiom,
semiring_1_cancel(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_378,axiom,
algebraic_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_379,axiom,
comm_monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_380,axiom,
ab_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_381,axiom,
ordered_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_382,axiom,
ordered_ring_abs(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_383,axiom,
semiring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_384,axiom,
comm_monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_385,axiom,
semiring_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_386,axiom,
linordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_387,axiom,
linordered_idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_388,axiom,
comm_semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_389,axiom,
comm_semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_390,axiom,
semigroup_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_391,axiom,
semidom_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_392,axiom,
semidom_divide(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_393,axiom,
semiring_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_394,axiom,
semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_395,axiom,
zero_less_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_396,axiom,
comm_semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_397,axiom,
semiring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_398,axiom,
ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_399,axiom,
zero_neq_one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_400,axiom,
ordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_401,axiom,
idom_abs_sgn(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_402,axiom,
ring_parity(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_403,axiom,
preorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_404,axiom,
linorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_405,axiom,
monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_406,axiom,
idom_modulo(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_407,axiom,
idom_divide(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_408,axiom,
comm_ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_409,axiom,
monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_410,axiom,
semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_411,axiom,
semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_412,axiom,
group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_413,axiom,
typerep2(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414,axiom,
mult_zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415,axiom,
comm_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416,axiom,
order(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417,axiom,
neg_numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418,axiom,
ring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419,axiom,
semiring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_420,axiom,
semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_421,axiom,
ord(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_422,axiom,
uminus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_423,axiom,
ring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_424,axiom,
abs_if(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_425,axiom,
minus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Power_Opower_426,axiom,
power(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_427,axiom,
numeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_428,axiom,
zero(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_429,axiom,
plus(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring_430,axiom,
ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_431,axiom,
idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oone_432,axiom,
one(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_433,axiom,
dvd(code_integer) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_434,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_435,axiom,
euclid5411537665997757685th_nat(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_436,axiom,
ordere1937475149494474687imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_437,axiom,
euclid3128863361964157862miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_438,axiom,
euclid4440199948858584721cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_439,axiom,
semiri6575147826004484403cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_440,axiom,
strict9044650504122735259up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_441,axiom,
ordere580206878836729694up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_442,axiom,
ordere2412721322843649153imp_le(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_443,axiom,
bit_se359711467146920520ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_444,axiom,
linord2810124833399127020strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_445,axiom,
strict7427464778891057005id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_446,axiom,
ordere8940638589300402666id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_447,axiom,
euclid3725896446679973847miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_448,axiom,
linord4140545234300271783up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_449,axiom,
semiri2026040879449505780visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_450,axiom,
linord181362715937106298miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_451,axiom,
linord8928482502909563296strict(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_452,axiom,
semiri3467727345109120633visors(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_453,axiom,
ordere6658533253407199908up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_454,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_455,axiom,
cancel2418104881723323429up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_456,axiom,
cancel1802427076303600483id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_457,axiom,
comm_s4317794764714335236cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_458,axiom,
bit_semiring_bits(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_459,axiom,
ordere2520102378445227354miring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_460,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_461,axiom,
linordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_462,axiom,
ordered_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_463,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_464,axiom,
ab_semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_465,axiom,
semiring_1_cancel(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_466,axiom,
algebraic_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_467,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_468,axiom,
comm_monoid_diff(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_469,axiom,
ab_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_470,axiom,
ordered_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_471,axiom,
semiring_parity(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_472,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_473,axiom,
semiring_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_474,axiom,
comm_semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_475,axiom,
comm_semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_476,axiom,
semigroup_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_477,axiom,
semidom_modulo(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_478,axiom,
semidom_divide(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_479,axiom,
semiring_numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_480,axiom,
semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_481,axiom,
zero_less_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_482,axiom,
comm_semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_483,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_484,axiom,
zero_neq_one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_485,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_486,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_487,axiom,
monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_488,axiom,
monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_489,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_490,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Typerep_Otyperep_491,axiom,
typerep2(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_492,axiom,
mult_zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_493,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_494,axiom,
semiring(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemidom_495,axiom,
semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_496,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ominus_497,axiom,
minus(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Power_Opower_498,axiom,
power(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Num_Onumeral_499,axiom,
numeral(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ozero_500,axiom,
zero(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oplus_501,axiom,
plus(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oone_502,axiom,
one(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Odvd_503,axiom,
dvd(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osize_504,axiom,
size(code_natural) ).
tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_505,axiom,
! [A15: $tType] :
( typerep2(A15)
=> typerep2(heap_Time_Heap(A15)) ) ).
tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_506,axiom,
! [A15: $tType] : size(heap_Time_Heap(A15)) ).
tff(tcon_Record_Otuple__isomorphism___Typerep_Otyperep_507,axiom,
! [A15: $tType,A16: $tType,A17: $tType] :
( ( typerep2(A15)
& typerep2(A16)
& typerep2(A17) )
=> typerep2(tuple_isomorphism(A15,A16,A17)) ) ).
tff(tcon_Record_Otuple__isomorphism___Nat_Osize_508,axiom,
! [A15: $tType,A16: $tType,A17: $tType] : size(tuple_isomorphism(A15,A16,A17)) ).
% Helper facts (16)
tff(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(fNot(P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(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,X: A,Y: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = 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 (2)
tff(conj_0,hypothesis,
$true ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------